Glossary

abandonment – in plurality or two-round runoff voting systems, the decision by a voter whose most preferred candidate is known to have insufficient electoral support to win the election to cast their vote not for that most preferred candidate, but for one with a realistic chance of winning. See also tactical voting.

above-the-line voting – in preferential voting systems, and in particular in Australian STV systems, a supplementary ballot paper option offered to voters, inviting them to indicate support simply for a political party, or to accept a party’s proposed order of preferences between individual candidates, in place of the individual candidate preference rankings which the voter might fill out for themselves.

Even in the early developmental stages of the STV method there were suggestions for this kind of device (ie: Baily’s method) however there was no actual usage of what are now termed above-the-line ballot design devices until Australian developments from the 1980s onwards.

There are two main variants of this ballot design: the group voting ticket approach, and the group-optional approach.

The group voting ticket approach was created in 1984 (but later abandoned in 2016) for use in the system for electing the Australian Senate, during a reform debate in which the government sponsoring the legislation was reluctant to shift from compulsory preferencing to optional preferencing, but wished to reduce the high rate of ballot papers then being ruled informal. The method is now (2018) only used for elections of the Legislative Council of Victoria. The technique is based on pre-election registration of preference orderings (called group voting tickets) by each party (or group of parties), which are made publicly available. In addition to a standard matrix of boxes for each individual candidate, the ballot papers are divided by a horizontal line and thus present a space, conventionally known as the above-the-line (ATL) area, containing a single box for each party/group. Voters may only mark a single vote for one party/group, which has the effect of adopting in entirety the preference order chosen by that party for all candidates. Where voting follows the compulsory full preferencing rule, the registered group voting tickets are required to indicate preferences for every candidate. Voters retain the option of voting below-the-line (BTL) among the entire offering of all individual candidates, and if a voter uses both parts of the ballot, the BTL markings take priority.

The group-optional approach, adopted in 2003 to elect the Legislative Council of New South Wales – and later adopted for the Australian Senate in 2016 and for the Legislative Councils of South Australia and Western Australia in 2017 – is significantly different in that it allows voters to mark preferential rankings for any number of the parties/groups above the line. These rankings are then interpreted as if the voter had marked individual candidate rankings firstly for all the candidates of their first-ranked party (in the order specified by the party), then for all those of their second-ranked party, and so on. The group-optional variant constitutes a form of optional preferential voting.

accessit voting – a supplementary procedure used in approval voting methods where a minimum threshold for election is set (such as a 50%, or two-thirds, rate of approval). After the initial round of voting is held and counted, one or more sub-rounds of voting in accessit may be held, in which voters may offer additional approvals to candidates, perhaps limited only to candidates who received at least one vote in the main round of voting. The procedure allows voters high levels of information about the mood of the electors as a whole, and can simplify what might otherwise be a multi-round contest.

Adams’ method – (also smallest divisors method); an adjusted quota form of quotient rounding apportionment method. This method determines an initial divisor by dividing the national total population by the intended total number of seats, rounding all the resulting quotients up, and taking the resulting values as the seat allocation to each state. The method is named after its proponent, US statesman John Quincy Adams. The method is the most generous of the common methods to small states: Adam’s concern in proposing it was to secure the best possible representation in the US House of Representatives for the states in New England, which were undergoing relative population decline as the nation expanded.

additional member – any small number of members elected (or appointed) to an assembly for a particular purpose, perhaps relating to ensuring the representation of women or racial or regional minorities. The electoral systems used to elect the lower houses of Colombia, Egypt, Ethiopia, Mauritius, New Zealand, Pakistan, the Philippines, Singapore and Tanzania use various additional member mechanisms. Denmark, France and Portugal allocate a small number of seats to citizens in overseas territories, and Italy and Portugal allocate a small number of seats to citizens resident in foreign nations. (Not to be confused with the British Additional Member System.)

adjusted divisor methods – a family of quotient rounding apportionment methods in which the divisors used to calculate the apportionment are adjusted part-way through the process to achieve more precisely proportional results. There are variants of the method, but each commences with an initial division of each state’s population by a common divisor which is (usually) derived by dividing the total national population by the pre-determined total number of seats to be filled. The resulting state quotients are then rounded according to one of several possible rounding rules, each of which is distinct to each variant of the method. The targeted total number of seats may be reached in this first calculation, but if the initial calculation allocates too few (or too many) seats, the divisor is adjusted downward (or upward) to the highest (or lowest) integer value which, after the apportionment calculation is re-run, results in the correct predetermined total number of seats. These calculations may be modified by a minimum seat rule, which must be taken into account for both the initial and adjusted calculations. Thomas Jefferson proposed the first modified divisor method, now known as Jefferson’s method, in 1790 to address the issue of allocating seats in the United States House of Representatives among the nation’s states. Other adjusted divisor method variants were developed and debated during the 19^th century, usually with the goal of avoiding the adverse effects of largest remainder methods – see the Alabama paradox, population paradox, migration paradox, and new state paradox. (Needless to say, the political goal of maximising seat allotments to specific states, regardless of the elegance of the proposed solution, was also prominent in the minds of many locally partisan advocates.) Variants include Webster’s method, Jefferson’s method, Adams’ method, Dean’s method, and Huntington’s method. From the late 19^th century, the alternative escalating divisor methods emerged – predominantly in Europe – and these generated results equivalent to the adjusted divisor methods in many ways.

The adjusted divisor methods are one of the categories of ways of apportioning assembly seats between states, electoral divisions or political parties

Alabama paradox – one of a number of controversial results generated when using the largest remainder method of quotient rounding to carry out an apportionment of assembly seats. This paradox occurs where an increase in the total number of seats causes a state to lose a seat (or vice versa – a decrease in the total number of seats may lead to a seat gaining a seat), even though the population of the state itself was unchanged. The impact is always only a single seat. The effect occurs due to the use of the relativities between the remainders rather than those between the full quotients. (See also population paradox, migration paradox, and new state paradox.)

alternative vote – (AV); another term, commonly used in the UK, for preferential voting (using sequential elimination) to fill single-person offices (including single-member divisions in elections for an assembly). The equivalent term instant runoff voting is used in the US.

Alternative Vote Plus – (AV+); an electoral system proposed by the 1998 Jenkins Commission reviewing the electoral system for electing the UK House of Commons. The proposal was a form of additional member system, similar in many ways to the British AMS system adopted in the late 1990s for the Parliament of Scotland, the Assembly of Wales and the London Assembly. In an AV+ system most assembly seats would be elected in single-member divisions through a ‘first vote’ using preferential voting (known as the alternative vote in the UK), but a small number of additional seats, suggested as 15-20% of the final total, would be awarded to political parties as top-up seats in each larger electoral region based on a ‘second vote’ across that region in which voters would select just one party. Essentially AV+ is a form of MMP with, unusually, preferential voting instead of plurality (first-past-the-post) voting for the ‘local’ seats. The AV+ system would in theory achieve a degree of party-proportionality within each region, although the aggregate of all the regions would achieve only a rough proportionality nationally. The degree of overall party proportionality would be lower than has been achieved in the other British AMS systems because of the use of only 15-20% of total seats as additional (or ‘top-up’) seats. By contrast with the established British AMS systems around 40% of total seats are additional to the SMD directly elected seats, and in German MMP systems typically 50% of seats are additional. The Jenkins proposal was rejected by the UK government of the day, and has never been trailed in any parliamentary election.

Andrae’s method – a variant of the single transferable vote (STV) voting system in which the process of elimination of minor candidates is disregarded in determining winners. As in ordinary STV, the first preference votes are initially examined and any candidates with a quota of votes are elected. The surpluses of all such elected candidates are then distributed as in ordinary STV, possibly leading to additional candidates reaching quotas and having their surpluses distributed in turn. However when no further surpluses remain to be distributed, the counting ceases and, if the number of candidates elected remains fewer than the number of seats to be filled, the final seats are filled by the unelected candidates with the highest current tallies of votes. The system, whilst requiring fewer vote counting stages, eliminates the potential for votes that are cast for minor candidates to be counted in determining the last few seats won, and in doing so it confers less equality of influence on voters than is achieved by standard STV. The system was used in Denmark in the 1850s at the proposal of prime minister Carl Andrae.

anticipation impact – any effects on the decisions made by a voter in casting their vote which arise from their anticipation of how the various candidates will perform in the ballot: put another way, the impact on a voter’s behaviour of a voter’s anticipation of the voting decisions of the other voters in the electorate (see ‘psychological effect’).

apparentment – (French); in party list seat allocation systems, a formal association of two or more political parties allowing them to combine their vote totals into one overall total for the purposes of meeting threshold requirements, and also for the calculation of seat entitlements, so as to maximise the number of seats won by the parties as a group. In some cases the parties will submit a single joint party list for the election. In other cases some mechanism must be devised for dividing the seats won by the list between the participating parties – mist usually the simple method of requiring the parties to settle the sequential order of individuals on the list in advance of the poll.

apportionment – any mathematical procedure for the distribution of seats among states (or among political parties, in seat allocation electoral systems). Apportionment is commonly used for two purposes: the allocation of assembly seats among a number of states (or other divisions) in proportion to their populations, and the allocation of seats in an electoral division to political parties on the basis of votes cast in elections. The simplest methods are those which simply divide populations (or vote totals) by a quota and round off the result, however that approach does not guarantee that the desired total number of seats will be achieved. The next simplest method is to use a quota and largest remainder approach. The more complex alternative classes of methods are the divisor methods. During the course of the 19th century an initial class of adjusted divisor methods were developed, largely in relation to the process for apportioning seats in the United States House of Representatives among the growing number of states. From the end of the 19th century the mathematically easier alternative of using escalating divisor methods (also known as highest average methods) largely replaced the use of the adjusted divisor methods.

There are several ways of apportioning assembly seats between states, electoral divisions or political parties

approval voting – a form of rate voting in which each voter either approves or disapproves of each available candidate (the method of ballot marking normally being to mark each candidate with an “x” (to indicate approval) or else leave them unmarked (to indicate disapproval). The method could also be described as ‘voting for’ as many candidates as the voter wishes, rather than being limited to voting for only a single candidate. The total number of approvals (or votes) given to each candidate are then tallied, and the candidate with the highest total is the winner. (An alternative counting method which yields identical results is to calculate the average approval rating, as a value between 0 and 1, for each candidate.) The method can also be used to award multiple seats to the N highest-rating candidates.

Arrow’s Theorem – one of a group of related social choice theorems, also including the Gibbard-Satterthwaite theorem and the Duggan-Schwartz theorem. The theory was published by economist Kenneth Arrow in 1950-51[i], and is widely used as a means of assessing the robustness of decision-making systems (including voting systems). Arrow’s theorem shows that every voting system (or ‘social welfare function’, or more specifically “procedure for deriving a social ordering or choice from a set of individual orderings”) which selects a single winner from among three or more candidates (or propositions) using the ranked preferences of voters will fail to meet at least one of a specified set of either four or five criteria. The four original criteria are ‘unrestricted domain’, ‘non-dictatorship’ (or ‘anonymity’), ‘Pareto efficiency’ (or the ‘consensus principle’), and ‘the independence of irrelevant alternatives’, and in some formulations of the theorem the fifth criterion of ‘collective rationality’ is included. These criteria are described for the theorem as being “minimally demanding normative requirements of fairness and logicality”. However the combination of criteria used in the theorem remains open to debate as a tool to measure the merit of electoral methods.

asset voting – a proposed form of voting where each voter essentially delegates their ballot to their preferred individual candidate, who is then empowered to trade the aggregate of votes they have been granted together with other candidates, either to directly make decisions or to engage in a further process of aggregating vote totals to select candidates to take seats in an assembly.

at-large (electoral district) – a term used in the United States to describe multi-member electoral divisions (or ‘districts’, in US parlance). The expression is most commonly used to refer to electing all the members of a state’s congressional delegation using the block voting method. The practice of at-large elections was used in several US states from the establishment of the Union until it was nominally barred by the federal Apportionment Act of 1842. It was used infrequently – despite this bar – over the following century, and was definitively barred by the current federal electoral law adopted in 1967.

The term ‘at-large’ has sometimes more generally (and loosely) been used to refer to any multi-member electoral district, even where the district does not consist of a whole state.

Ausgleich seats – see overhang and compensation seats.

Australian ballot – see secret ballot.

Baily’s method – a variant proposal for single transferable vote (STV) voting, using a preference ticket option involving lists of candidates. The method also derived the necessary vote totals for each candidate by using cumulative voting. (See Baily 1869, also Droop 1881, p 36).

Baldwin’s method – one of the ‘recursive elimination’ variants of the Borda count, others including the Nansen and Rouse methods. A Baldwin count proceeds as in a Borda count, except that a winner is not declared after the initial count, but after each such count the candidate with the lowest Borda Score is eliminated, and the Borda count re-conducted for the remaining pool of candidates (using a reduced array of scores aligned to each ballot’s order of preferences without the eliminated candidates). The process continues until only two candidates remain, at which point the one with the higher Borda score (now simply equivalent to having a majority of the two remaining preferences) is the winner.

Banzhof index – the original and simplest of the voting power indices, a mathematical method of estimating of the relative strength of political parties according to the number od members they have in a decision-making assembly. The index presents values between 0 and 1 for each party in an assembly, whereby 0 represents a total lack of political power and 1 indicates total control. The index is based on simple numerical count of the number of multi-party alliance scenarios in which a party may find itself in the majority group, based solely on the relative numbers of votes (ie: assembly members) of each party. The index does not have regard to factors such as the real political likelihood of specific combinations, nor does it deal with issues such as the likely frequency of such combinations, or the prospect of splits within parties or realignment of individual voters/members. The formula for the index was proposed separately by Penrose (1946)[ii], Banzhof (1965)[iii] and also Coleman (1971)[iv].

below quota – an element of the STV voting system, in which the last (or last few) candidates elected in an electoral division may fail to reach the required quota of votes. In this case the rule is the last available seats afe filled by the remaining un-eliminated candidate(s) with the highest vote tallies, even if those tallies are below the quota on which other candidates were awarded seats.

binomial system – an unusual form of seat allocation electoral system where each electoral division has exactly 2 seats available. Within each electoral division, each party nominates (up to) two candidates. Voters cast single non-transferable votes for one candidate among all those nominated. The party totals – the sum of the votes for the two candidates of each party – are used to allocate the two available seats, with the two parties having the largest totals each being allocated one seat, unless the largest party has twice the vote of the second-largest, in which case that party is allocated both seats. For a party allocated one seat, the successful individual candidate is the one who received the higher number of individual votes. This method tends to eliminate representation of parties other than the two dominant ones (except in regional areas where third parties have concentrated voter support), and also reduces the relative representation of the largest party and increases that of the second-largest, creating a strongly stable two-party system. The method was developed in Poland under a former constitutional regime, and is now used only in Chile.

Black’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. Black’s method is a dual-test approach, proposed by Scottish social choice theorist Duncan Black. The method first conducts pairwise comparisons of the preferences to determine the Condorcet winner, and if such a winner is found then that candidate wins the election. If there is no such winner (ie: there are multiple candidates in a Condorcet Cycle), then the preferences are re-examined as in a Borda Count to determine the winner.

block voting – (also the block vote, blockvote, or scrutin de liste (French)): a form or multiple non-transferable rate voting in which each voter expresses support (typically by marking the relevant number of ‘x’s on the ballot paper) for as many candidates as there are seats to be filled, with the seats then being awarded to the N candidates with the highest tallies of votes. Where parties nominate the same number of candidates as there are seats to be filled, and the supporters of political parties cast all their votes for the candidates of that party, the result is that the party which wins a plurality of votes wins all the seats. The use of the block vote in the UK and US in the 19^th century provided a historical driver to the development of electoral parties, by heavily rewarding the practice of nominating a list of candidates in the same number as the number of available places (and by punishing any alternative nomination pattern). The system is used in Mauritius and in Lebanon. See also party block vote.

Borda count – a rate vote counting system (though often mistakenly classed as a rank vote system) invented firstly by Nicholas of Cusa in 1430, and again by the French mathematician Jean-Charles de Borda in 1780. In a Borda count voters mark all the candidates with integers from N (the number of candidates) down to 1 in descending order of preference. The numbers are then tallied as points, with the candidate with the highest total (or ‘Borda score’) being the winner. Another way of defining this rule is that each candidate is given a score equal to the total number of times that candidate is ranked over any other minus the total number of times that candidate is ranked under any other (see Geller 2005, p268). While resembling preferential voting in ballot paper appearance (with the significance of the digits marked on the ballot reversed so that higher numbers indicate stronger support), formally the method is a specific variant of cumulative voting, and thus technically a form of rate voting rather than rank voting. Put another way, as a voter’s support does not in any sense transfer from one candidate to another in any circumstance, the method does not belong to the preferential or ‘ranking’ class of voting systems. However, Borda counting also has the property that it reveals the ‘highest average ranking’ of each candidate (on the premise that equal integers are the basis for the rankings), which can be seen as a useful criteria for assessing the relative merit of candidates[v]. If a ranked comparison of many candidates is desired, the Borda scores of each candidate may be referred to determine a ‘Borda ranking’ of the candidates. The Borda method is often favoured by voting theorists, but it does suffer from the serious defect that it is easily manipulated in real competitive elections by voters deliberately placing the strongest rivals to their favoured candidate in the lowest places on their ballots, even if this is insincere preferencing. Another defect is that Borda is not ‘clone independent’, in that parties have an incentive to nominate multiple similar candidates, thus generating additional Borda score points for all the closely related candidates. See also Nanson’s method (a specific method of elimination used in preferential voting which makes use of candidate’s Borda scores), the modified Borda count, and the Quota-Borda system.

Borda elimination – an alternative approach to eliminating candidates in preferential voting systems, in which each candidate’s Borda score is determined (usually at the initial count of first preferences), and thereafter where the elimination of a candidate is necessary the one with the lowest Borda score is the next eliminated. See also ‘Nansen’s method’, a specific method of bulk elimination which makes use of candidate’s Borda scores.

Borda fixed point – a variant of the Borda count voting method, proposed by Thomas Colignatus in 2008. Under this method multiple Borda counts are conducted using the same ballot papers. A number of preliminary Borda counts are conducted, one for each candidate, in which each candidate in turn is disregarded (as if they had not been nominated) and a winner of the count in their absence is identified. For each candidate A, the winner of the count in which they were disregarded is their complementary candidate B. Next, the same set of ballots are used to compare every candidate A with their complementary candidate B. Each candidate who loses one or more of these comparisons is eliminated. In the final round, the ballots are used once again to conduct a final Borda count between the remaining candidates, and this count determines the winner of the election.

Borda ranking – see Borda count.

Borda score – see Borda count.

British additional member system – (AMS); a modern variant of the MMP electoral system which directly integrates the counting procedure for awarding seats in local electoral divisions with the counting procedure for allocating seats to parties. With MMP (as used in Germany, New Zealand and others), the single-member division component of the Assembly is first determined, and then the allocation of members for the as ‘levelling’ component is determined so that the final total composition of the Assembly is party-proportional.

The AMS system was developed for – and first used in electing – the devolved assemblies of Scotland (1999) and Wales (2000).

With AMS as used in Scotland and Wales, the two stages are combined into one method of seat allocation, with the SMD-winning candidates being allocated seats as an integrated part of the allocation formula. The system can be applied to a whole nation as a single electorate, or the nation can be divided into a number electoral regions. In either case, every region must be divided into local divisions. The total number of seats for each electoral region will be set as substantially larger than the number of local divisions within the region. Voters cast two votes: one vote to elect a local division representative and one ‘list vote’ to support a party at the regional level. The seats within each region are allocated by a variant of the D’Hondt method in which the number of list votes cast for each party is divided successively during the count by a divisor equal to 1 plus the number of seats the party has already won. At each stage, another seat is awarded to the party with the highest current quotient, and the process is repeated until all seats are allocated. However this allocation process begins with initial places being awarded to the local division members who are directly elected through the local votes (typically by the plurality voting method). One part of the assembly is thus directly elected members, and the remainder are members awarded seats through party list seat allocation (typically using the closed list method). This system does not use, and does not require, vote thresholds for parties to be eligible to win seats. The counting process differs slightly in its seat allocation results from the standard D’Hondt method. Under this system a popular local independent candidate may be elected through a local division vote where a standard seat allocation process across the region would see them unelected.

Bucklin voting – (also Grand Junction voting): an unusual method of voting which has the appearance of being a preferential voting method, but is actually a rate voting method. Voters cast ballots by filling out ordinal preferences for each candidate. If a candidate has a majority (50%+) of formal votes on the initial count of first preferences, they win the seat. If not, the number of second preference votes marked for all candidates on all ballots are added to the tallies for each candidate (as if they were an additional set of single rate votes). The target for victory remains the number of votes equal to a majority of original number of formal votes cast. If 2 or more candidates exceed this target after the addition of second preferences, the winner is the candidate who has the highest new total. If no candidate reaches the target, the 3^rd preference votes marked on all ballots are added in the same manner, and the same victory rule applied. Addition of successive preferences continues in rounds until a round occurs in which the target is reached by at least one candidate. A variant of this method allows the target for victory to increase at each stage of counting to a majority of the new total of votes, but this option has the problem that it is possible that no candidate will in fact reach the target even after all preferences are added to the totals. The use of either compulsory (full) or optional preferencing is possible for Bucklin voting. The method was developed in America in the early 20th century, and used initially for local elections at the town of Grand Junction, Colorado. It was subsequently adopted in a small number of US States, but was later repealed or ruled unconstitutional by courts (in the judicial ruling in Minnesota, on the dubious basis that voting methods in which voters indicate votes for multiple candidates (a feature of all preferential or rank voting methods) was not permitted by the State constitution).

bulk elimination – any technique used in counting a (preferential or ranked) vote which eliminates multiple candidates in a single step, usually on the basis that they fall below a threshold. The elimination of all but the top two candidates, or of all candidates below a specified vote threshold (as in elections to the French Assemblée Nationale) are examples of bulk elimination.

bullet voting – see plumping.

Carey elimination – in preferential voting systems, the process of eliminating in stages all candidates with vote totals that are below the average of the vote totals of all the remaining candidates, until only two or one candidates remain, revealing the winner. (This method is only relevant to single-position contests.)

choice voting – a term used by some American voting reform advocates, including FairVote, for the single transferable vote voting method.

choice – the breadth of the range of options available to voters to select from between individual candidates. Choice is expanded where there is a vigorous market of candidates on offer to the voters. Choice may be artificially limited by the use of geographic electoral divisions (limiting the potential link between voters and candidates), by constraints on freedom of candidates to nominate, by formal or financial impediments placed on political parties or individual candidates in regard to entering electoral contests, by the actual mechanics of the electoral method used, and by the strategic impacts which a voting method has on the decisions which voters make.

class voting – electoral systems (largely historical only) in which voters are first categorised in different classes, which then elect their representatives separately. Logically such systems are consciously designed to give disproportionate levels of influence to the voters of the different classes. An example was the Prussian 3-class system of the mid-late 19^th century, in which voters were divided into three classes based on income/wealth levels, and each group elected one third of the representatives to the Prussian assembly.

closed list – a feature of some seat allocation methods in which an ordered list of a political party’s candidates, determined in advance of the election by the party (and normally published), is used to award to individual candidates the seats won by that party. In such systems there is no capacity for electors to vote for (or against) individual candidates. Closed list voting essentially removes the selection of representatives from the hands of the voters and confers it on political parties. This in turn significantly changes the decision-making loyalties, behavioural drivers and career traits of potential and serving assembly members.

coalition – any formal grouping of political parties as allies, usually resulting in a joint ticket for the purpose of electoral nominations. In the case of single-member districts, typically only one candidate from the parties in the coalition will be nominated (although in preferential voting systems rival candidates from coalition allies can be nominated with only minimal strategic disadvantage).

collective rationality – a term used frequently in economic theory and, specifically, a criterion of social choice theory. Collective rationality is a form of group process to arrive at a decision. Applied in relation to voting systems, collective rationality requires that…. Collective rationality is sometimes included as a fifth criterion used in Arrow’s theorem.

common divisor – an element of quotient rounding apportionment procedures, normally derived by dividing the total national population by the pre-determined total number of seats to be filled. The population of each state is then divided by this divisor to generate the quotient for each state.

common quota° – a quota, however determined, which has the same value across all the electoral divisions of an electoral system. (The single quota used for an electorate with no electoral divisions may also be regarded as a common quota.) A common quota is a form of fixed quota. The key virtue of such a quota is the uniformity of vote value it confers on each elector, subject only to any remaining variation in salience resulting from voters casting ballots which favour candidates who win very low levels of support from other voters.

compensation seats – see overhang and compensation seats.

composite assembly – (also mixed systems and less frequently combined systems) : an assembly consisting of two of more sections of seats, each constituted (that is, elected, allocated or appointed) by different electoral methods. There are a broad variety of combinations in use in world electoral systems. Systems in which two groups of members are elected by completely separate voting procedures are known generically as parallel voting systems. Conversely, some composite systems integrate two electoral methods in some way, usually so as to ensure that the total composition of the final assembly is proportional to the national vote totals of political parties. These latter systems are generally referred to as mixed member proportional or MMP systems. Constitutional systems which add to an assembly a small number of additional seats to represent special categories of voters (for example female MPs, representatives of ethnic minorities, or representatives of voters living overseas), or appoint a small number of officials ex officio, are not ‘composite’ systems in the primary sense of the term. Massicote and Blaise (1999) proposed a classification of ‘mixed’ electoral systems into 5 categories: (1) coexistence (where different geographical parts of the electorate use different voting systems). (2) superposition (where voters use two voting methods to elect two distinct bodies of representatives which are then combined as the assembly, as in ‘parallel systems’). (3) fusion (actually a form of reinforced majority system, where one part of the assembly is elected by some voting system, and another – perhaps half of all members – is constituted by awarding seats to the plurality party). (4) correction (equivalent to the German MMP approach, whereby seat allocation in one part of the assembly is used to cause the resulting final assembly to be party-proportional to the votes of electors). (5) conditional (where certain thresholds of party vote shares trigger specific awards of seats to the plurality party, under some reinforced-majority formula.) However of these categories only superposition and correction are in general use today as national electoral systems understood to be ‘mixed systems’.

compulsory enrolment – the imposition by law of a requirement that all persons with the right to vote become enrolled with the relevant electoral authority. Implemented in Australia, Belgium and other nations. Without full – or at least very high rates of – enrolment, key aspects of statistical analysis of election results, such as the legitimacy of assembly majorities and the ‘mandates’ claimed by elected representatives and governments, must be significantly qualified. Rates of enrolment in voluntary regimes vary in a range from as low as 50% (example) to around 95% (in Malta). The compulsory Australian enrolment rate is estimated to be around 94%, but has been declining slightly in recent years. In Australia the legal obligation is placed on each citizen to take the necessary steps to become enrolled and also to maintain the accuracy of their enrolment details as their place of residence changes. Electoral authorities can be empowered to take the initiative to maintain the accuracy of the electoral roll by accessing data such as school completion records, local government information, driver licence registration details, utility connection records at domestic homes and so on. Without such initiatives enrolment rates for certain categories of voters, particularly young adults but also people who have moved home, can be significantly lower than average rates.

compulsory preferencing – in relation to a preferential voting method, a rule than voters must mark a preference for every candidate on the ballot paper. The result is a pool of valid ballots all of which preference ‘fully’, that is to say that no ballots will become exhausted during vote counting. Implicit in compulsory preferencing is the rule that ballots which fail to fully preference will be held invalid. The invalidation rule has a significant impact in Australia’s compulsory preferential voting system: around 3% of voters who attempt to cast votes in House of Representatives elections (around 420,000 voters in 2010) have their votes invalidated due to inadvertent errors in completing ballot papers[vi]. Compulsory preferencing may also require some voters to record insincere or random preferences on their ballot. By contrast, optional preferencing permits voters to more accurately convey their true opinions in cases where they have no preference to indicate in regard to some candidates.

compulsory voting – a legal compulsion to vote. Implemented in Australia, Belgium, various Latin American nations and others (see Lundell 2012, Birch 2009). In the Australian case legislation requires a person to “vote” and gives specific instructions on the manner of marking the ballot paper to do so. There is a common misunderstanding that all that is required of voters is that they be marked off the roll, receive a ballot paper and place it in a ballot box – perhaps blank or else filled out informally – but this is on its face a misreading of the terms of the electoral law, which specifically requires an act of voting (in accordance with the instructions about how to validly fill out a ballot paper). In practice the secret nature of the ballot is a barrier to any prosecution so long as a person has apparently lodged a ballot. However there have been cases of individuals protesting against this legal requirement who have by their own confession provided evidence of their breach of the law.[vii]

concentration distortion – across a number of electoral divisions, a difference in the number of seats won by parties (compared to a result which is proportional to overall vote totals) resulting from uneven concentrations of party supporters across the divisions. For example where 50% of a population supports each of two parties and there are 10 single member divisions, the high concentration of one group of supporters in 4 or fewer seats, and a more even spread of the other party’s supporters across 6 seats, may result in the latter party winning a 6:4 majority of seats, instead of a more representative 5:5 return of members of each party. The phenomenon also represents an inequality in the effective value of each voter’s vote (or ‘salience’) – a lack of “one vote one value” – depending on the party they support and the vote share of that party in the voters electoral division. The effect occurs in all systems which use geographical electoral divisions, but is particularly prominent in systems made up of single member divisions. However since the difference does not appear to be a deliberate or direct consequence of system design (as is the case by contrast with malapportionment or gerrymandering) it is less often perceived as a problem in regard to voter equality. In fact the inequality of voter influence (or salience) caused by concentration distortion is often much more significant than that caused by malapportionment. Concentration distortion is also the basis for strategic ‘marginal seat’ campaigning, in which seats with high (ie: reliably seat-winning) concentrations of voters who support major parties can be given much less attention in election campaigns by all parties, because the results in those seats are largely forgone conclusions. This election strategy driver can have important implications for party policy-making, and for whether the elected assembly gives appropriate political attention to the needs of people living in all the divisions of the electorate.

Condorcet criterion – (also Condorcet efficiency or Condorcet principle): a criterion for assessing single-choice voting systems, indicating the extent to which the voting method will reliably elect the candidate who is the majority candidate (or Condorcet winner). For a voting system to meet the ‘Condorcet criterion’ strictly means that such a result is not merely likely but is in fact guaranteed.

Condorcet cycle – (also majority cycle or voter paradox): the occasional occurrence in a pairwise comparison between 3 or more candidates where the electorate prefers A over B, prefers B over C, but also prefers C over A. See majority candidate.

Condorcet winner – another term for the majority candidate.

confidence – the indication by a legislative chamber of which party (or individual leader) it supports to form an executive government. Confidence can be positively expressed by a formal vote, or assumed negatively by the failure of the government’s opponents to be able to persuade the assembly to pass a motion of no-confidence. Under the conventions of the Westminster system, when a prime minister determines that she no longer holds the confidence of the assembly, she is obliged to resign in favour of some alternative leader who appears likely to be able to command that confidence. The passage of a motion in the assembly may make the situation clear, but in any case it is one of the ongoing conventional responsibilities of the prime minister to remain aware of where the confidence of the assembly lies. If a prime minister is defeated on a no-confidence motion, their legal (or conventional) authority over such decisions as calling early elections may be reduced.

consistency – a criterion for assessing single-choice voting systems. A voting method is said to be consistent where, in regard to two subsets of the whole electorate, if voters would elect the same candidate in both subsets, that candidate should also be elected by the whole undivided pool of voters. Preferential methods that use sequential elimination may potentially fail this test.

contingent vote – (also instant runoff voting): a form of preferential voting for elections to single public offices (including members of an assembly elected in single member divisions). Ballot papers are marked preferentially in the usual manner. If a candidate secures 50% of the formal votes on first preferences, they are declared the winner. If no candidate achieves 50%, all candidates other than the two highest-placed candidates are simultaneously eliminated, and any preferences shown for either of the remaining two candidates are transferred to the tallies of those candidates, yielding a winner. With contingent voting only one round of counting of preferences is necessary, regardless of the number of candidates. If preferencing is optional, some votes will ‘exhaust’, meaning that the final tally of votes for the winner is not assured of being 50% or more of the original total formal vote, in which case perceptions that the winner has won a ‘majority’ may not be justified. Despite the different methods for eliminating candidates, contingent voting has very similar results to a full preference distribution using sequential elimination. Analysis of the history of preferential voting for the Australian House of Representatives from 1919 to 2010 (totalling around 4,200 individual seat contests) indicates that 99.36% of contests were eventually determined between the candidates who were placed 1^st and 2^nd on first preferences, even after sequential elimination. Thus, the contingent vote method would only have failed to replicate the final count between two candidates in these contests in 0.64% of cases, among which only 0.19% of all contests were actually won by a candidate placed 3^rd on first preferences (none were won by a candidate placed 4^th on first preferences, and indeed in all these elections no 4^th-placed candidate ever advanced to the final two places after preferences).

Note that during the 19th and early 20th century some writers, including Thomas Hare (1860), used the term ‘contingent voting’ simply as an equivalent term to preferential voting. The use of the term for the specific method of voting outlined above is more modern,

Coomb’s method – a form of preferential voting for single-member positions. In Coomb’s method, so long as no candidate has secured 50% of the vote, the candidate with the largest number of last preferences is eliminated: the votes currently tallied for that candidate are then transferred to the next most preferred remaining candidate, and vote totals are re-tallied. Eliminations continue until one candidate has a majority. To work as intended, Coomb’s method requires full preferencing, so that a tally of last preferences for every candidate can be identified at the beginning of the process. In two-party systems Coomb’s method can often have the surprising effect of eliminating the candidates for the two major candidates at the first two stages of counting, resulting in a surprisingly high and disproportionate number of minor party candidates being elected.[viii]

Copland number – in the examination of pairwise comparison victories to determine the Condorcet winner of a preferential vote, the total number of comparison ‘wins’ for each candidate is that candidate’s ‘Copland Number’. A Condorcet winner will be the sole candidate to have the highest possible Copland Number of N-1. Alternatively, where three (or more) candidates have equal highest Copland Numbers, a ‘Condorcet Cycle’ has occurred, and there is no Condorcet winner. See also Copland’s method (below).

Copland’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. The Copland method counts the number of pairwise victories for each candidate, counting a pairwise tie as half a victory: the winner is the candidate with the largest result. Copeland’s rule ignores the relative strengths of the pairwise comparisons. (The Dodgson and Simpson methods very rarely choose a Condorcet loser, but Copeland, Schulze’ and Tideman’s methods never do.)[ix]

countback – a practice for filling casual vacancies arising in an electoral division elected by the STV voting system. Where an elected member had resigned or died, all the ballot papers which contributed to the persons election in the original vote counting process (potentially including ballot papers with a transfer value of less than 1 if the Gregory method approach is used) are re-examined to choose a successor from among the candidates on those ballots. Normally, an initial process of contacting all former candidates and asking them to confirm that they wish to be considered, and those who do not so confirm are disregarded in the recount. The selected ballots are then re-examined as if an election for a single person by preferential voting were being conducted. The result is typically, though not necessarily, a member of the same political party as the retiring member.

CPO-STV – a variant of STV developed by British voting system theorist Nicholas Tideman to address the problems arising from sequential elimination in standard STV. In essence CPO-STV takes the concept of Condorcet pairwise comparisons and applies it to STV, by seeking to determine which out of all the possible sets of winners is the combination most preferred by voters, using the preferences marked on ballot papers. Whilst the sheer number of such possible sets may seem daunting, the range can be reduced significantly by examining the tallies of the candidates after the initial count of first preferences and the transfer of all quota excesses from candidate elected on those first preferences. The winning set must by definition include all candidates elected at that point, thus substantially reducing the number of possible sets of winners. Nonetheless, the examination of each possible set of additional winners one pair at a time requires computer processing to perform: physical counts of ballot papers in the numbers arising in real public elections is quite unrealistic. Tideman proposed a process for examining the preference rankings of any pair of winner sets and determining which of the pair prevailed. Once all the comparisons are performed, a Condorcet matrix emerges and the set with the most pairwise wins will become plain because one set will be the Condorcet winner among the sets considered (although there remains the minor possibility that no single winner set is dominant over all other sets – the Condorcet cycle possibility.) (See Tideman 1995, Tideman 2000)

cracking – a term used in relation to gerrymandering, in which those drawing manipulative electoral division boundaries divide a geographic population of voters with a predictable distinct voting trend (ie: a high concentration of Democratic party supporters) between two or more adjoining electoral divisions so that the population’s capacity to contribute to electing a representative of their preferred party is diminished or entirely thwarted.

cross-preferencing – (also exchanging preferences): in elections conducted using preferential voting systems, the deliberate ranking of the candidates of certain related, similar or otherwise allied political parties (or background) high in the rank order of the voters whose primary or first preference is for any of those candidates. Cross-preferencing agreements are a feature of election campaigning between such candidates and their parties. The implementation of a cross-preferencing arrangement requires some form of communication to voters, although it may be made to occur automatically by vote delegation mechanisms such as above-the-line voting systems used in some elections in Australia. Cross-preferencing may also arise naturally from the broad preferences of the voter communities in question.

cube law – an analysis tool used to predict and explain the disproportional results in plurality elections contested by two dominant parties. In brief, the ratio of seats won by two major parties is expected to be close to the cube of the ratio of votes between the two parties. The formula aims to quantify the large boost in seats won when one dominant party has a significant win over the other in terms of votes. The formula only operates reliably in strong two-party contests: the presence of a prominent third party, or a variety of smaller parties of various sizes, will weaken the predictive power of this formula. (Taagepera 1986)

cube root rule – the proposition that the ideal size of an assembly (in numbers of members) is the cube root of the population of the jurisdiction it represents. The rule was put forward by Finnish political scientist Rein Taagepera in 1972. Many national assemblies track close to the curve that this rule generates, Notable outliers are the UK House of Commons – which at 650 members is too large – and the US House of Representatives – which at 435 members is to small.

cumulation – in ballot designs which allow for the marking by a voter of multiple preference ‘votes’ among individual candidates, cumulation – if it is permitted – is the practice of giving two or more votes to any one candidate. Cumulation is used as part of the Swiss system for electing the Nationalrat (National Council).

cumulative voting – (also point-voting schemes and sometimes score voting): a form of rate voting in which voters have a certain number of points (or ‘votes’) to allocate as they wish among the available candidates. The points allocated to each candidate on the ballots are then tallied, and the candidate with the highest total is the winner. The method can also be used to award multiple seats to the N highest-rating candidates. Variants of cumulative voting may impose a condition that only a limited number of points (less than the total available to each voter) may be allocated to any one candidate. Such variants, which can be collectively termed ‘limited cumulative voting’, can be described by reference to the total votes a voter may use (T) and the limit on how many may be applied to any individual candidate (L). In other variants may define a strict scoring rule, under which the points/votes which each voter may use are prescribed. The Borda count and the descending fractions voting methods are examples of the latter, for these methods are in fact specific variants of cumulative voting.

Thomas Hare, writing in 1860, attributed the first proposal of the cumulative voting method to James Garth Marshall in 1839. The method was proposed for use in a South African colonial legislature in 1850. Thereafter during the mid-late 19^th century cumulative voting was sometimes advocated by Liberal politicians for use in House of Commons elections, but it was never adopted. Cumulative voting was used to elect members of the Illinois House of Representatives from 1870 to 1980, and is currently used in some US local government elections.

D’Hondt series – first proposed in 1880 by Belgian Victor D’Hondt, a very common divisor series used in the escalating divisor method for seat allocation to parties (or to allocate seats to states or other electoral divisions within a nation). A reference to the D’Hondt formula or the D’Hondt system is thus a reference to seat apportionment by the escalating divisor method using this series. The D’Hondt series requires that the number of votes for each party (or the population of each state or division, as appropriate) be divided by successive integers {1, 2, 3, 4, 5, etc}. In comparison to other divisor formula such as Sainte-Laguë, the D’Hondt formula tends to give political parties with larger vote shares a slightly larger portion of seats than their share of the vote merits. The D’Hondt formula escalating divisor method generates the same election results as the Jefferson modified divisor quotient rounding method.

Danish series – a type of escalating divisor series used for indirect election systems of seat allocation to parties (or, unusually, to allocate seats to states or other electoral divisions within a nation). The series is {1, 4, 7, 10, 13, etc}.

Dauer-Kelsay Index – the smallest percentage of the total values of a subset of a larger set of values required to produce more than 50% of the total of the larger set of values. Used as a measure of inequality, or malapportionent, between different populations, such as the enrolment numbers of several electoral divisions, this measure was popularised by American political scientists Manning Dauer and Robert Kelsay.

David-Eisenberg Index – the ratio of the largest of a set of values to the smallest. Used as a measure of inequality, or malapportionent, between different populations, such as the enrolment numbers of several electoral divisions, this measure was popularised by American political scientists Paul David and Ralph Eisenberg. The measure only indicates the relation of the extremities of the values, but does not measure the distribution within the set.

Dean’s method – an adjusted quota form of quotient rounding apportionment method. This method determines an initial divisor by dividing the national total population by the intended total number of seats, rounding all the resulting quotients up or down according to rounding at the harmonic mean, and taking the resulting values as the seat allocation to each state. The method is named after its proponent, US mathematician James Dean.

delegated ranked choice voting – a possible form of single transferable vote (STV) voting in which the transfer of votes from eliminated candidates is conducted not by using the expressed preference rankings of the individual voter, but instead at the direction of the eliminated candidate themselves. Ideally each candidate would publish their order of next preferred candidates in advance of the electorate voting, to allow voters to consider whether they approved of each order. The ballot design therefore does not require the marking of preferences, but simply a single vote for one candidate, ostensibly resembling plurality voting. The proposition was raised by Oregon commentator Kristen Eberhard of the Sightline Institute in early 2018. Cf Gove system, which deals with surplus transfers rather than eliminated candidates.

delegated voting – any situation where the electoral rules require, or permit, a voter to delegate her vote to a political party, including cases where a voter may (or is required to) adopt a party ticket as her vote (such as the Australian Senate ‘above-the-line’ voting option).

descending fractions method – (also the Dowdall system): a form of cumulating voting, somewhat resembling the Borda count. Voters mark ballot papers with preferential integer rankings from their most favoured candidate (marked “1”) downward. Votes are tallied by scoring 1 point for every 1^st preference, ½ of a point (0.5) for each 2^nd preference, 1/3^rd of a point (0.333) for each 3^rd preference, ¼ of a point for each 4^th preference, and so on. Once all the points are tallied, the winner is the candidate with the plurality of points. The method can also be used for multi-member divisions, with the N highest scoring candidates being awarded seats. The method is used in Nauru and has also been used in recent elections in Fiji. The Oklahoma Primary method was a variant of the descending fractions method.

direct election – (also direct voting): the use of electoral methods featuring voter selection of individual candidates to become their representatives. Distinguished from electoral methods in which voters cast a ballot to indicate support for a political party, which is then used to facilitate seat allocation to those parties, and also distinguished from systems where representatives are chosen by any intermediate body such as an electoral college or a state assembly.

directness – the electoral system feature that voters are able to directly choose between individual candidates. By contrast, in indirect systems voters may be invited to select between political parties, or other forms of candidates list proposed by parties or coalitions. Directness plays a central role in several aspects of an electoral system including the forces at work in relation to candidate nomination (including processes of pre-selection by parties), the choices available to voters, the nature of the perceived role of elected members (ie: whether it is representation of local, regional or national issues, of political opinions in the community, of the policy manifestos of political parties, or of the tactical interests of political parties), and the question of whether accountability of elected members is to the electorate, or is to political party leaders or party structures.

district magnitude – the number of members elected from, or allocated to, an electoral division. See also N, as used in formulae.

division – (also district, constituency, riding, ward, precinct: in some usages also electorate): a geographical subdivision of a nation or state which allows the electorate to be divided according to the place of primary residence of each voter.

divisor series – any of several possible series of divisors used in escalating divisor methods of apportionment. The two divisor series in most common use are the D’Hondt series {1, 2, 3, 4, 5 etc} and the Sainte-Laguë series {1, 3, 5, 7, 9 etc}, each of which sometimes has its first divisor modified. Other escalating divisor series proposed, but not in common use, include the Imperiali series {1, 1.5, 2, 2.5, 3 etc} (not to be confused with the Imperiali quota), the Danish series {1, 4, 7, 10, 13 etc} and the Huntington (or ‘equal proportions’) series in which the divisors are the square roots of the products of successive pairs of integers {√(1*2), √(2*3), √(3*4), √(4*5), √(5*6) etc}.

Dodgson matrix – a counting device named after Charles Dodgson, and consisting of a simple matrix mapping each candidate against each other on two axes, and indicating the candidate most preferred by voters (and by how many votes) in each comparison. The matrix is used in making pairwise comparisons between candidates with the aim of identifying the Condorcet winner in a preferential ballot. The total number of comparison ‘wins’ for each candidate is known as that candidate’s Copland Number. See Dodgson’s method (below).

Dodgson’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. The Dodgson method sums each candidate’s margins of defeat: the winner is the candidate with the smallest such sum. Dodgson can be said to pick the candidate “closest” to being a Condorcet winner where no such winner exists. (The Dodgson and Simpson methods very rarely choose a Condorcet loser, but Copeland, Schulze and Tideman never do.)[x]

donkey vote – (slang): in preferential elections, votes (or at least, some of them) which are cast for the first name appearing on the ballot paper. Some elections appear to show a vote tally for such first-placed candidates higher than seems likely to political analysts, given the expected public support for the candidate in question. The conclusion is that at least some voters simply fulfil their task of voting by voting for the first candidate they encounter. Any ballot paper numbered down the page 1, 2, 3, etc in sequence can be suspected of being a donkey vote. Donkey voting is therefore the act of filling out a preferential ballot paper by simply numbering the candidates from 1 to N down the page, by a voter indifferent to the outcome of the election. Of course, it may be that some voters are marking such ballots quite deliberately to reflect their true preferences, so donkey votes cannot be effectively separated out from other votes. Obviously, first place on the ballot creates the potential for such advantage, and for this reason ballot paper position is often determined randomly. A much more effective solution is Robson rotation of candidate names on ballot papers.

double complement rule – a requirement relating to plurality, single-candidate elections in which a leading candidate who achieves less than 50% of the vote does not win the election unless the margin (or difference in vote share) between that leading candidate’s vote share and 50% is less that the margin between the leading candidate’s vote share and that of the second-placed candidate. In other words, the lead candidate must be closer in vote share to 50% than they are to the vote share of the runner-up. If this condition is not met, resort is had to an alternative procedure such as a runoff election between the top two candidates. The rule was proposed by Shugart and Taagepera in a paper published in 1994. See also qualified plurality rule and margin requirement. (See Engstrom and Engstrom 2008, 415)

Droop quota – a quota (or divisor) for determining the allocation of seats in an assembly, defined as Q = (P/N+1)+1, where Q is the quota, P is the population (or equivalent such as formal vote total), and N is the number of seats available. The Droop quota has the feature that it is the lowest number of votes which cannot be obtained – even after any preferential transfers – by more candidates (or states) than there are seats available to be filled. The Droop quota is also used in single transferable vote (STV) voting systems to determine the quota of the formal votes cast that is required for each candidate’s election in each division. (See also simple quota, Hagenbach-Bischoff quota and Imperiali quota.)

dual member proportional: (DMP); a proposed hybrid voting system developed by Albertan political scientist and mathematician Sean Graham in 2013. The electorate is geographically structured into regions and in turn into electoral subdivisions. Two winning candidates will be identified for each subdivision. Candidates nominate within the subdivisions, and voter ballots provide for voters to mark a single vote within their subdivision for a party (which will list two candidates) or an individual independent. The aggregate vote for each party within each region is then used to determine a nominal party-proportional seat allocation of the total number of seats in the region (ie: twice the number of subdivisions), using the simple (Hare) quota and largest remainder formula.

Despite the party-proportional calculation, one in each electoral subdivision is initially awarded to the plurality winner in each subdivision (taking the first-listed candidate of each party to be the relevant winning candidate, or an independent candidate if one should happen be the plurality winner).

The remaining seats are then awarded to parties to make up the numbers to their nominal proportional allocation. The individual candidates selected for the second group of allocated seats are drawn in order from a ranking of candidates in descending order of their original party (or independent candidate) votes, with a party’s second-listed candidates deemed to have votes equal to half the vote for the party which has already elected its first candidate by plurality. Once a second-group seat has been awarded from an electoral subdivision, all other candidates in that subdivision are deleted from the list of those eligible for the remaining second-group candidates, so that only one second-group candidate from each subdivision is selected. By this means, every subdivision will see two candidates selected overall.

In essence DMP resembles MMP (or more closely, British AMS), applied at regional level and with the unique additional rule for selecting the successful (closed) party-list candidates in a manner that ensures that one nominee is selected from each local electoral district. With DMP the first group of successful candidates are essentially elected by plurality (first-past-the-post) voting, while the second group are elected by a uniquely weighted and constrained form of partially-open party list seat allocation.

The published description of DMP does not address anomalies which might occur due to ‘overhang seats’, arising where a party with geographically concentrated voter support wins a number of initial local plurality seats that exceeds its nominal region-wide allocation. In similar manner, DMP does not address the impact of independent candidates winning a plurality seat in the first group of winners. In most cases such anomalies could be avoided by disregarding more of the below-quota fractions during the party-proportional seat allocation calculations, but in some cases this might not be possible.

DMP was offered as an option in the Prince Edward Island referendum on electoral reform in 2016, at which it placed 3^rd out of 5 options. In May 2018 the government of British Columbia announced that DMP would be one of three options offered at a November 2018 plebiscite on reform of the electoral system for electing the provinces Legislative Assembly.

Duggan-Schwartz theorem – one of a group of related social choice theorems of which the Duggan-Schwartz theorem is the most general, the Gibbard-Satterthwaite theorem is a narrower application of Duggan-Schwartz, and Arrow’s theorem is in turn a narrower application of Gibbard-Satterthwaite (although these three theories were actually published in the reverse of that order). The Duggan-Schwartz theorem states that every voting system which selects one or more winners using the ranked preferences of voters suffers from at least one of four defects, namely (i) some voters are treated differently from others (the system is not ‘anonymous’), (ii) some candidates can never win (the system is ‘imposed’), (iii) the system is sufficiently generous in outcome that every voter’s first preference is included in the set of winners, or (iv) the rule is susceptible to strategic voting, in the sense that there are conditions under which a voter with knowledge of how the other voters intend to vote, and of the voting method being used, would have an incentive to vote in a manner that does not reflect her true preferences.

Duvergers’ Law – the axiom that a plurality election system tends to favour, generate and/or perpetuate a two-party party configuration. The idea was stated by Henry Droop as early as 1869[xi]. However, the axiom has come to be attributed to French sociologist Maurice Duverger, who developed and popularised the concept in papers published in the 1950s and ’60s.

The law may be stated as follows: the single-member division plurality voting system favours the emergence of a two-party system and thereby the alteration of power between two major parties[xii]. Duverger also proposed the complimentary proposition that preferential and proportional voting systems tend to favour (and perpetuate) multi-party party configurations. The outcome of these propositions is that preferential and especially proportional representation systems create electoral conditions that foster party development, while the plurality system inhibits the emergence of new political parties.

From the 1990s onward, Spanish political scientist Josep Colomer promulgated a converse proposition to Duverger’s, namely that the electoral system of a nation arises (and changes) as a result of the party configuration in effect at the time, as the parliamentary parties in control of enacting electoral legislation make risk-averse strategic choices to minimise threats to their continued political strength.

effective number of parties – (ENP): a comparative statistical measure of the number of political parties active in elections, or actually elected to assemblies. The original measure was proposed by Laakso and Taagepera in 1979[xiii]. Values range from 1.0 (which would indicate a strict one party-state, to as high as 9.0-10.0 (seen in Belgium and Israel). A value of 2.0 represents a two-party system with no other significant contestants. Most political systems will have values between around 2.1 and around 5.5. Duverger’s axioms about the differing effects of majoritarian and proportional electoral systems used this measure to draw out conclusions from elections held under different classes of electoral system. In regard to elections the measure is often termed ‘effective number of electoral parties’ (ENEP), and in regard to assemblies once elected it is correspondingly termed ‘effective number of parliamentary parties’ (ENPP). Different results can be expected for the two measures, as many parties will fail to secure assembly seats. The acronym ENP might refer to either version, depending on context. [See Norris 1996, p.306 and Ljiphardt 1994.]

effective vote – the converse of wasted vote: any vote which is influential in the final stage of counting of an election determining whether one candidate is elected and another not. In plurality elections, any vote which is cast for neither the winning candidate or the runner-up is generally regarded as a wasted vote. Any exhausted ballot cast in optional preferential voting can be regarded as a wasted vote. A voter who votes in the first round of a two-round runoff election for a candidate who is eliminated, and who then does not cast a vote in the final round, can be said to have cast a wasted vote. (See Knight and Baxter-Moore 1973)

effective votes – votes cast in an election which actually result in the election of the primary desired candidate to an assembly. The rate of effective voting – presumably a percentage ranging up to, but not normally reaching, 100% – is therefore a metric of actual representation.

efficiency gap: a measure of the ratio of respective rates of ‘vote wastage’ of two major political parties occurring in an election. Specifically, the term was proposed by Nicholas Stephanopolous and Eric McGhee in 2015 to describe the degree of partisan bias arising from gerrymandering of electoral district boundaries used in United States legislative elections, in which legislators are elected in single-member districts by plurality voting.

In such electoral systems vote wastage, in respect of one individual electoral district, can be calculated as the sum of all votes cast for unsuccessful candidates (which therefore fail to elect a representative) and votes cast for the sole successful candidate in excess of the number needed to win the seat (that is, votes in excess of the number of votes cast for the largest runner-up candidate). Once vote wastage for each electoral district is calculated, the aggregate vote wastage for each political party across a number of electoral districts (ie: all those across one state) can be determined.

The efficiency gap is the ratio of one of the two major party’s wasted vote aggregate to that of the other party. A significant efficiency gap reveals that one party is benefiting from concentration distortion arising from the pattern of district boundaries that have been used. A particularly large efficiency gap, and/or one which has increased as a result of a recent redrawing of those boundaries, provides objective evidence of deliberate partisan gerrymandering. As a standard, Stephanopolous and McGhee propose that an efficiency ratio between the two parties of 1.07 of greater (ie, a ‘gap’ of 7% or greater) is unacceptably high, and requires the drawing of ‘fairer’ boundaries with a lesser ratio.

In most states in the United States electoral district boundary drawing for both state legislature and Congressional elections is effected directly by legislation, and is thus controlled by sitting partisan state legislators. Boundaries are traditionally redrawn once the demographic results of the decennial US Census, held in years ending in “0”, are available. There is abundant evidence that many of the boundary maps legislated during the 2011-12 decennial round of redrawing of district boundaries has deliberately exaggerated the degree of concentration distortion across districts in many states; that is, the maps were gerrymandered. The result has been partisan bias in many election outcomes from 2012 onward. The great majority of these effects have been seen in states where the Republican Party controlled the redistricting outcome, and clearly favour that party. In reaction to this situation litigants have initiated legal challenges to the current boundary maps in several states. The work of Stephanopolous and McGhee was undertaken to develop robust and objective statistical measures to help illuminate where, and to what extent, such partisan bias could be said to occur, and thus to help resolve the issues being litigated. In late 2016 the federal US District Court adopted the use of the measure in a judicial decision relating to district boundaries used in the state of Wisconsin (Whitford v Gill), but the decision has been appealed to the US Supreme Court [June 2017].

A detailed discussion of the measure is provided in Extreme Maps, Laura Royden and Michael Li, Brennan Centre for Justice, 2017.

See also partisan symmetry, the mean-median district vote share difference and the seats-to-votes curve, which are other methods to attempt to measure partisan bias in district boundary-making.

electoral college – an institution (and also the voting system used to elect it) used in a form of indirect voting system in which voters first elect a panel of representatives, who in turn vote to elect the person actually chosen to hold the public office in question. The only prominent use of this method in modern democracies is for the election of the President of the United States, in which voters in each of 50 States elect college members (in numbers semi-proportional to State populations) as a group by the party block voting method. The college members then submit written votes for President (and also Vice-President). These are then counted by the plurality method. However, if no candidate wins a majority of the total votes, the procedure terminates and the President is instead chosen by the US House of Representatives by a different method.

electorate – (sometimes also constituency): refers collectively to the people making up the body of voters within a nation, or within a specific electoral division.

enrolment rate – the proportion of a nations enfranchised population that is enrolled.

enrolment – (also registration): the administrative practice of recording the details of all persons entitled to vote on an official roll or register: also, the fact of an individual voter being on such a roll.

equal proportions method – see Huntington’s method.

equality – the measure of the extent to which vote salience is equal for all voters across an electorate at a general election. Uniformity can be reduced by many electoral system design features, including the electoral method chosen (rate voting systems being prone to wide variations in uniformity), and certain specific features of particular electoral division boundaries (including malapportionment, concentration distortion, and gerrymandering). Variations in vote salience can also be caused by the specific vote choices of voters in the context of the overall pattern of choices by all voters.

escalating divisor methods – (sometimes simply called the divisor methods, and also frequently termed highest averages methods [xiv]): a family of apportionment methods which use one of various possible divisor series to resolve, in a single procedure, the allocation of a predetermined number of seats to political parties at elections (based on votes cast at an election), or among electoral divisions such as states or provinces before elections (based on population figures).

The first such method was that proposed by the Belgian mathematician Victor D’Hondt in 1880.

The escalating divisor methods can be applied in either of two ways. In the priority ranking method, the number of votes for each party – or the population of each state or electoral division – is divided in turn by the series of divisors particular to each formula (for example, in the case of the D’Hondt series, the divisors are defined by the formula ‘NS+1’, or the number of seats awarded to the party/province in preceding steps plus 1 {1, 2, 3, 4, 5 etc} while with the Sainte-Laguë series the divisor is ‘(NSx2)+1’) {1, 3, 5, 7, 9 etc}. At each step the party or province with the largest current quotient value is awarded the next available seat. The quotient value for that party is then recalculated using the next divisor in the series, and the results for each party re-examined for the following step. This procedure is repeated step by step until the required number of seats is fully allocated.

In the tabular method, a table is created in which the first column (reading down) sets out the values of the chosen series of divisors. To the right, one column is allotted to each party (most conveniently in decreasing order of their vote total). The resulting grid is then filled with a matrix of the vote totals won by each party divided by the divisors in the left-hand column. For as many seats as are available, the highest values in the matrix each win one seat for the party in whose column they appear.

The results generated by the priority ranking method and the tabular method are identical.

Different divisor series may generate different results for individual states or parties, especially where large numbers of seats are available. For example, the D’Hondt series tends to give larger parties a slightly larger portion of seats than their share of the vote, whereas the Sainte-Laguë series does not inherently favour larger parties over smaller ones, and the latter is thus more proportional overall than the former.

These series are sometimes modified by adjusting the value of the first divisor used in the calculations, which will either increase or decrease the effective threshold, thus affecting the ability of small parties to win their first (and only) seat in the relevant electoral division. For example, in Lithuania a value of 0.9 is used in place of 1.0 in a D’Hondt series, specifically to assist small parties. By contrast, a variant known as modified Sainte-Laguë uses a first divisor value of 1.4 in place of 1.0, specifically to disadvantage small parties.

Election rules using these methods often require parties to win a specified threshold of votes to be eligible to win seats. Such thresholds are set at proportions of the national vote total in Denmark and Israel (2%), Spain (3%), Poland (5%, or 8% for coalitions), Russia (7%), and Turkey (10%), or as proportions of the vote in regional electoral divisions, as in Belgium (5%). In any case, the number of seats available in the electoral division alone will generate an implicit threshold.

exchanging preferences – See cross-preferencing.

exhausted ballot – in optional preferential voting, a voter’s ballot may not express preferences for all candidates. Under the optional preferencing rule these ballots are formal (or ‘valid’) votes, in contrast to counts conducted under the compulsory preferencing rule, which invalidates such ballots, discarding them prior to commencing the vote counting process. If during the conduct of the count all the candidates for whom a preference is marked on a ballot have been eliminated (or, in the case of multi-member electoral divisions, have already been elected with quotas), the ballot is set aside from the count and tallied as an exhausted ballot. Various forms of validity rules relating to duplicated preference numbers, skipped numbers, and other permutations of marks on ballot papers may result in different treatments of ballots as exhausted votes.

exhaustive ballot – a form of rate voting in which separate votes are taken repeatedly until a vote occurs in which one candidate secures the necessary majority to be awarded the position in question. While that target might be a simple majority (50%), higher majorities such as 2/3rds majorities are common. For example, a new Catholic Pope is identified by the College of Cardinals through an exhaustive ballot requiring a 2/3rds majority.

exhaustive elimination – a method of elimination described by Duncan Black (who used the expression exhaustive voting) in which there are successive rounds of votes, at each of which voters indicate one candidate which they wish to see eliminated, and the single candidate with the plurality of such votes is accordingly eliminated at the end of each round. (In Black’s method, each voter casts approval votes for one less than the number of remaining candidates, but the result is mathematically the same as identifying a single candidate to eliminate.) No winner is declared until the final round where two candidates remain, at which point the final vote is a simple ballot between two candidates.

This method is in practice the same as the rule for the game of ‘musical chairs’, and it is also used to eliminate contestants in many reality (sic) TV shows. This method does not require fully preferenced ballot papers, and in practice individual voters are able to ‘vote optionally’ by sitting out selected rounds of the voting.

This elimination method will always ensure that a ‘majority candidate’ (that is, a Condorcet winner), if one exists, will prevailing in the election. If no such candidate exists, this approach will have the general effect of excluding candidates with higher rates of disapproval, in particular those who are seen as politically ‘extremist’, and generally directing results towards ‘moderate’ or ‘centrist’ candidates.

extent [of representation] – a term of measurement used to indicate the proportion of the electorate who are directly represented in an assembly following an election. The measure can also be used in regard to single-office positions, such as the election of presidents. The measure is subject to degrees of intensity, and in addition may be measured against different definitions of the ‘electorate’ (i.e.: all citizens, all eligible voters, all enrolled voters, or all actual voters). In the most intense sense, a voter may be said to be directly represented if a candidate whom they supported (in preferential voting systems, this being their first preference choice) is elected to the assembly. Alternatively, they may be said to be ‘party-represented’ if in their relevant electoral division at least one candidate from the political party of the candidate who received their first vote is elected to the assembly. An even wider sense of party representation may not even require a member elected in the relevant electoral division, but may count any party member elected nationally.

In preferential voting systems a voter may be said to be preferentially represented if, after the distribution of preferences is complete to the point where all seats in the assembly are determined, the preferences on their ballot have been used in the election of at least one member to the assembly. In regard to the electorate, the measure may be taken against the entire ‘entitled’ population of enfranchised people (which can only be an estimate), or the known population of enrolled people, or the known subset of enrolled people who actually voted.

fair majority voting: (FMV); a hybrid system of seat allocation proposed by US mathematician Michael Balinski in 2008. Candidates would nominate in local electoral divisions, but such divisions would also be grouped together in larger electoral regions. Voters would cast single votes for individual candidates in a local division, as in plurality (first past the post) voting. The number of successful candidates for each political party in each electoral region would, however, be initially determined by applying the D’Hondt divisor seat allocation formula (Balinski referred to the “Jefferson” formula, which is mathematically identical to D’Hondt in outcome).

The elected individual candidates would then be discovered by scaling up all the vote tallies of parties other than the most successful across all the local electoral divisions, until the rescaled tallies produced the numbers of seat wins for each party matching the initial region-wide proportional allocation. The result would be region-wide party-proportional seat allocation, but with exactly one successful candidate identified in each local division. This scaling approach would operate simply in a two-party political milieu such as the United States, but is not clear how the scaling would work in a milieu involving additional minor parties and independents.

false majority – a term for an election result in which a majority of seats in an assembly is won by a party (or coalition) when that party has not won a majority of the votes in the electorate.

federal malapportionment – any rule which affects the apportionment of seats in an assembly between various states or territories of a federal nation away from a mathematical proportionality. While technically constituting mallaportionment, such arrangements are normally regarded as an acceptable feature of federalism. The rule that every jurisdiction must have at least one seat is almost universal, and does not usually create significant disproportionality of voter influence between the jurisdictions. At the other extreme are rules that every jurisdiction is allotted the same number of seats – such as with the US Senate and Australian Senate – which can create very significant disproportionality of voter influence. In between may lie various options for minimum-seat rules, for example the Australian House of Representatives rule that no state is allotted fewer than 5 seats, or the bespoke Canadian House of Commons arrangements which provide that no province will be allotted fewer members than (i) the number of national Senators for that province, and (ii) the number that province had in the year 1985.

first preferences – in any preferential voting system, the indications of a voter’s most preferred candidate, numbered “1” on ballots. A candidate who achieves election by having more than 50% of all first preferences is thus said to ‘win on first preferences’.

flexible district PR – a proposed Canadian variant of single transferable vote (STV) voting, also initially termed rural-urban PR (or RU-PR), put forward by FairVote Canada as part of the parliamentary inquiry held during 2016 for reform of the Canadian electoral system. The system involves a geographic mix of multi-member electoral divisions where members would be elected by STV voting together with some single-member divisions, intended especially for the low-population regions of Canada, where members would be elected by either plurality or preferential (AV) voting. Finally the party-proportionality of the regional electoral areas (or alternatively, of the whole jurisdiction) would be supplemented through a second tier of allocated seats (probably fairly few in number), which would be awarded to parties in a manner similar to MMP systems.

A further variant of the flexible district PR proposal termed riding-centric-RU-PR was put forward by academic Byron Weber Becker and British Columbia electoral reform advocate Anthony Hodgson as part of the electoral reform process in British Columbia in 2018.

In May 2018 the government of British Columbia announced that flexible district PR would be one of three options offered at a November 2018 plebiscite on reform of the electoral system for electing the provinces Legislative Assembly.

first-past-the-post – (FPTP; the phrase may be hyphenated, but more usually is not): the common name given to the plurality voting method when used in single member divisions, and therefore also known as single-member division (or district) plurality (SMD, or SMD plurality) or single-member plurality (SMP).

The term is widely used in Britain, Canada and the other nations still using the system. The expression has been generally criticised as an illogical description of SMD plurality elections because there is no conceptual ‘post’ that candidates must pass during the counting of votes.

The origins of the expression lie in Britain in the 1880s, where it was adopted from usage in the horse racing industry. Horse races in that era were often subject to protests which might alter the result for the purposes of prizes and gambling payoffs. However one accepted form of betting, particularly in the less regulated ‘suburban’ races, was the ‘first-past-the-post’ bet, which paid out simply on the horse that passed the finish line first, regardless of any disqualifications or penalties. By the 1880s alternatives to plurality voting (both in its multi-member and single-member district forms), including cumulative voting and preferential voting (including STV) were being widely debated by politicians and other electoral experts. It appears that the expression first-past-the-post was taken up in electoral science by analogy with the horse-racing usage, connoting that SMD plurality was the voting method that lacked any means of correction to ensure fairness in the outcome of the election ‘race’. It may be relevant to this linguistic history that by this era, at least in the United States, elections had come to be termed ‘races’.

Plurality voting in single-member divisions is widely assumed to be the ‘original’ British voting system, but in fact while plurality vote counting has been used in England (and later the United Kingdom) for House of Commons elections for several centuries, the generalised use of single member divisions only dates from 1884. Voting in the UK had hitherto mainly used plurality voting in two-member divisions, with voters having two votes, the practice thus amounting to the block voting method. Britain retained the use of a few multi-member constituencies until they were finally discontinued the late 1940s.

Earlier, single-member division plurality voting was mainly the norm in electing the legislative assemblies of the provinces of Upper and Lower Canada from the late 18th century, a policy which continued after 1841 in the United Province of Canada. In the legislative assemblies of the Australian colonies that were established from the 1850s onwards there were a mixture of single- and multi- (mainly two-) member electoral divisions. The use of multi-member divisions was largely extinct – and thus replaced by SMD plurality – in the Canadian and Australian jurisdictions by the early 20th century. From 1919 onward Australian federal and state parliamentary elections then adopted the practice of preferential voting.

Elections for the United States House of Representatives also initially used a mix of single-member and multi-member districts (the latter generally known as ‘at-large‘ elections) until an initial attempt by the federal Congress to universally require single-member districts was made in 1842. The United States did not finally legislate for universal single-member divisions until 1967.

fixed quota – a pre-determined number of votes (or proportion of the vote total) used in a proportional election such as one using the STV method, or in a process of seat allocation. The most common forms of quota used as fixed quotas are the Hare quota and the Droop quota. However any pre-determined number – indeed any arbitrary number – can be used as a fixed quota.

floating total – the total number of members elected in a multi-member electoral division in an electoral system which does not set the number of elected members in advance. Multiple-division electoral systems using common quotas will result in floating totals. Systems using a floating total will more successfully avoid the variations in vote strength uniformity that are caused by concentration distortion, and will reduce the variations in salience which inevitably affect voting systems which fix in advance the number of members to be elected.

formal participation (rate) – (also formal turnout, valid turnout, or valid vote): the proportion of the enrolled electorate which casts a formal (or ‘valid’) vote in an election.

fractional method – (also Gregory method known in Eire as Senatorial rules): in an STV election, a formula used in transferring the surplus of votes from a candidate who has won a seat at the immediately preceding stage of counting. The method is attributed to John Gregory of Melbourne, who proposed it in 1880.

The Gregory method dictates that instead of transferring a number of whole ballot papers equal to the surplus to be transferred, all relevant ballot papers are used, but at a fractional value such that the total weight of votes transferred is equal to the value of the surplus. This approach eliminates the need to select a subset of the ballot papers (whether by random selection or some other method) for the transfer. The fractional value applied to such ballot papers must be preserved at all future stages of counting. It is also possible that a ballot paper may undergo two or more transfers, and if so such a ballot’s value in terms of ‘votes’ decreases by the product of all transfer fractions it has undergone. This method has the virtue that the variety of next preferences represented by the voters who have contributed to election of the candidate with the quota is captured more equitably.

The original Gregory method was usually applied to the ‘last parcel’ of votes which had caused the candidate to go over the seat quota. However since this potentially quite small pool of votes does not necessarily represent the full variety of next preferences of all the voters who have contributed to the election of the candidate, a more appropriate variant of the formula is to use all of the ballots making up the quota of the elected candidate (including the surplus) in the Gregory transfer, obviously with a somewhat lower fractional value for each ballot. This variant is known as inclusive Gregory method.

franchise – the legal entitlement to vote, normally before any statutory exclusions relating to individual voters (such as mental incapacity or criminal convictions).

free list – an early name for an open party list seat allocation electoral system, specifically requiring the option of multiple (rather than single) candidate preferences to be marked by voters, perhaps allowing specifically the same number of preferences as there are seats to be filled. The preference vote tallies for each candidate within each party list would be the aggregate of all the preferences cast. Cumulation (ie: the marking by a voter two or more preference votes for any individual candidate) would not normally be permitted. The free list system was used in the earliest form of party list seat allocation in the Canton of Ticino in the mid-19^th century.

free riding – [pending]

full preference distribution – in relation to a preferential election, the practice of completely counting and transferring all preference votes until only two candidates remain, even if that extent of counting is not required to determine the winner of the election (that is, the winner may be apparent on the count of first preferences, or after only a few rounds of elimination of minor candidates. Only in Australia is this practice conducted, and only since 1984. In modern elections the Australian Electoral Commission conducts full distributions of preferences for House of Representatives contests down to two final candidates in all divisions, and also (for information purposes only) conducts indicative distributions between the candidates of the two major parties, even where such distributions are not required to determine the seat winner. This information permits the determination of comparative ‘two-party-preferred’ statistics for each seat and for state and national totals.

fused voting – an electoral technique where the same ballots papers are used to determine the winners of two of more offices. For example, ballots might be first used to determine individual members of parliament, and the same ballots then used again to elect an executive president. Such ballots will normally indicate support for political parties, not individual candidates. In some systems for electing composite assemblies, a single ballot indicating a voter’s support for a party is used both to elect a local member, and also to support a party list seat allocation process, or alternatively to support seat allocation processes for both a regional and a national tier.

fusion voting – the practice in US elections of allowing candidates to be nominated (or endorsed) on a ballot paper by two or more political parties. Candidate’ names can therefore appear two or more times on the ballot as the nominee of distinct political parties. For example a specific candidate may primarily be the nominee of the Republican Party, but might also be nominated by the Conservative Party. In determining the winner of the election, each candidate’s vote tally is the sum of the votes received through all their nominations. The system allows minor parties some influence on major party candidate selection and policies, by means of the process of negotiating whether or not to grant ‘endorsements’, as the secondary nominations are usually termed. Fusion voting was largely disused after the adoption of secret balloting in the late 19th century, but is still used in many elections in New York state.

Gallagher index – (a name used in electoral science for the least squares index); a statistical measure of proportionality used in relation to seat allocation systems, either for elections to allocate seats among parties or (less commonly) for pre-election procedures for allocating seats among electoral divisions, states, etc. The index attributes to each party a value equal to the square of the difference between the percentage of the total votes cast that supported that party and the percentage of seats which it won, and then measures the disproportionality of the election outcome as a whole as being the square root of half the sum of the values attributed each party.

The resulting values allow comparison of degrees of disproportionality between multiple elections, and in turn allows comparisons of average degrees of disproportionality between different seat allocation formulae.

The least squares index is a long-established statistical tool. The method acquired the name Gallagher index after the use of the method was recommended by Irish political scientist Michael Gallagher in 1991 as an alternative to earlier measures of disproportionality such as the Loosemore-Hanby Index. (See Gallagher: Proportionality, Disproportionality and Electoral Systems, 1991.) [xv]

general ticket – a term used in the US to refer to the election of all of a state’s congressional delegates by a single statewide ballot, typically by the block voting method. General ticket elections were used in several US states from 1789 until 1840, and thereafter infrequently. See at-large voting.

geometric mean – one of the three classical ‘Pythagorean means’ (the others being the standard ‘arithmetic mean’ and the ‘harmonic mean’): the geometric mean of two successive integers is the square root of the product of the two integers, or √(N*(N+1)).

geometric rounding – the practice of rounding remainders that are below the geometric mean of the two integers on either side of the quotient down to the integer below, and rounding remainders at or above that value up to the integer above. The technique is used in Huntington’s modified divisor method.

Gergonne’s method – a multi-member quota-proportional direct election method proposed by the French mathematician Joseph Gergonne in the early 19th century. The system is premised on tasking the voters to organise themselves into groups equal in number to a pre-determined quota. Each group would then, either by unanimity or by some other voting method, select one representative. A version of the method was adopted for municipal elections in the colony of South Australia in 1839, but was later abandoned.

gerrymander – the practice of deliberately drawing the boundaries of electoral divisions so that concentration distortion (and also malapportionment if it is present), favours certain political parties to win more seats across a number of neighbouring electoral divisions. The term derived from the use of the practice in Massachusetts in 1812, signed into law by Governor Elbridge Gerry (although he was not the author of the proposal), which included the creation of a division in the western part of the state the map outline of which resembled the shape of salamander.

The practice of gerrymandering is still commonplace in the United States, where in many states the process of drawing boundaries of federal electoral divisions is in the hands of partisan officials. In Australia, Canada, the UK and many other nations, and in some US states, the task of boundary determination is placed in the hands of independent authorities and governed by legislative constraints which act to prevent gerrymandering.

Gibbard-Satterthwaite theorem – one of a group of related social choice theorems, and a narrower application of the Duggan-Schwartz theorem. The Gibbard-Satterthwaite theorem states that every voting system which selects a single winner from among three or more candidates (or propositions) using the ranked preferences of voters suffers from at least one of three defects, namely (i) there is a single voter who is effectively able to choose the winner (the ‘dictatorial’ flaw), (ii) there is some candidate who can never win, or (iii) the rule is susceptible to strategic voting, in the sense that there are conditions under which a voter with knowledge of how the other voters intend to vote, and of the voting method being used, would have an incentive to vote in a manner that does not reflect her true preferences. Arrow’s theorem is a more narrowly defined application of the Gibbard-Satterthwaite theorem.

Gini Index – a well-known statistical measure, which can be used as a measure of inequality, or malapportionment, between different populations, such as the enrolment numbers of several electoral divisions. (See Hughes, 1977, pp127-8).

Gove system – a late 19^th-century proposal for a form of single transferable vote (STV) voting in which the transfer of surplus votes from candidates who are declared elected with a quota is conducted not by using the expressed preference rankings of the individual voter, but instead at the direction of the eliminated candidate themselves. Ideally each candidate would publish their order of next preferred candidates in advance of the electorate voting, to allow voters to consider whether they approved of each order. Cf delegated ranked choice voting, which deals with eliminated candidates rather than surplus transfers.

greatest divisors method – see Jefferson’s method.

group voting ticket – a device used in the STV elections for elections to the Australian Senate – see above-the-line voting.

Hagenbach-Bischoff method – a hybrid approach to apportionment for use in seat allocation systems, under which an initial allocation of seats is made by applying the Hagenbach-Bischoff quota to a set of vote totals, and taking the integers of each party’s resulting quotient (ie: rounding down) as the number of seats won by each party. If there are remaining seats unallocated by that calculation, the party vote totals are re-examined as in the D’Hondt method, with the numbers of seats awarded under the initial step taken as the starting point in determining which divisors to apply to each party’s vote total. Since this final step is identical to the late stages of a D’Hondt apportionment, the results are always identical to those which a plain D’Hondt procedure would generate.

Hagenbach-Bischoff quota – (also Newman-Britton quota): a quota (or divisor) for determining the allocation of seats in an assembly, defined as Q = (P/N+1), where Q is the quota, P is the population (or equivalent such as formal vote total), and N is the number of seats available. The formula was proposed by the Swiss mathematician Eduard Hagenbach-Bischoff in the late 19th century. As a divisor, Hagenbach-Bischoff differs from a Droop divisor in that it is lower by precisely 1. One consequence of this difference is the hypothetical possibility that a count of votes finishes with N+1 candidates each holding identical votes, while only N seats are available, requiring some tie-breaking method. In large mass-population elections the risk of such a difference arising between the results generated by a Hagenbach-Bischoff quota and those of a Droop quota is so improbable as to be irrelevant. (See also simple quota).

Hallett’s method – a procedure for dealing with a Condorcet cycle, should it arise during the use of the Condorcet method of determining a majority winner. Proposed by George Hallett in the 1920s (see Hoag and Hallett, Proportional Representation, 1926).

Hamilton method – see largest remainder method.

Hamilton quota – see simple quota.

Hare quota – see simple quota.

Hare-Clark system – a form of single transferable vote (STV) voting developed in Australia in the 1890s and derived from the proposals of Catherine Spence and Andrew Inglis Clark, building on Thomas Hare’s method of STV. Spence had spent many years promoting the adoption of STV, and (similar to Henry Droop in England) proposed the use of small electoral divisions as a practical solution to criticism that nation-wide (or state-wide) STV would be logistically impractical as well as lead to loss of local representation.

In 1896 Clark, then the Attorney-General of the Colony of Tasmania, was able to get Hare’s system of proportional representation adopted – but only on a trial basis – as the method to elect members of the Tasmanian House of Assembly for two divisions covering Hobart (6 seats) and Launceston (4 seats). This Tasmanian Hare-Clark system, as it became known, was used at elections in 1897 and 1900, discontinued in 1901, re-adopted from 1909 for use in five large electoral elections covering the whole State (as Tasmania had by then become), and has remained in use in this form for over a century since. A specific modification to Hare’s original method also introduced by Clark was the introduction of the Gregory method.

The Hare-Clark system was subsequently adopted for use in elections for the New South Wales Legislative Assembly (1922 and 1925, but thereafter repealed), the Australian Senate (1949 –), the New South Wales Legislative Council (1978 –), the South Australian Legislative Council (1985 –), the Legislative Council of Western Australia (1987 –), the Australian Capital Territory Legislative Assembly (1995 –), and the Victorian Legislative Council (2002 –).

The Hare-Clark system is now the standard form of STV voting in use in Australian elections for the national Senate, some State lower and upper houses, many local government councils and in many other systems of election within public, industrial, academic or private bodies. In its modern form the key features of Hare-Clark include fixed-magnitude divisions, use of the Droop quota, the inclusive version of the Gregory method of surplus transfer, and sequential elimination. Preferencing is either optional or compulsory, but is now more commonly the former. In the system’s home state of Tasmania and the ACT, the use of ballot papers printed with Robson rotation is also now a canonical feature of the system.

harmonic mean – one of the three classical Pythagorean means (the others being the standard arithmetic mean and the geometric mean): the harmonic mean of two successive integers is twice the product of the two integers divided by the sum of the two integers, or 2(N(N+1))/(2N+1).

harmonic rounding – the practice of rounding remainders below the harmonic mean of the two integers on either side of the quotient down to the integer below, and rounding remainders at or above that value up to the integer above. The technique is used in Dean’s modified divisor method.

Hatomander – (slang) a change in the design of an electoral system by reducing the magnitude of electoral divisions (ie: the number of members elected in each division) with the deliberate intent of boosting a governing party’s majority at future elections, or of eliminating the representation of smaller parties. The term arises from unsuccessful 1956 proposal of Japanese Prime Minister Hatoyama to reduce divisions to single-member size[xvi]. The reduction in size of divisions for the Tasmanian House of Assembly from 7 to 5 in 1998, which was supported by the two major political parties in the House with the deliberate intention of reducing representation of the Greens minor party, was a similar instance of such voting system design manipulation.

heritage – a criterion for assessing single-choice voting systems. The criterion requires that the removal of non-winning candidates from the list of candidates does not change the winner. Thus if a candidate is elected from a set of candidates under a voting method, and the same voting method is then applied to a subset of those candidates which includes the initial winner, and the initial winner is still elected, that voting method is said to meet this criterion. Preferential methods that use sequential elimination are liable to fail this test.

highest average methods – see escalating divisor methods.

highest consent winner – an alternative way of referring to a candidate who is the Condorcet winner of an election. The emphasis of this alternative term is on the electorate’s collective consent to the candidate being awarded the position, rather than the stronger term support. Hence, the highest-consent candidate is the most consented-to of all the candidates, even if some of that consent is derived from lower-order (and potentially tactical, or reluctant) preferences, undermining a claim that a majority of voters fully support the candidate’s election.

Hill’s system – a hybrid of the STV electoral method and the party list seat allocation system, proposed by Ian Hill in 2011[xvii]. In this system the ballot paper would allow voters not to vote for individual candidates, but instead to record ranked preferences between each of the parties on offer (presumably treating each independent candidate as a party for this purpose). Parties would submit in advance ordered lists of their candidates. The counting process would then proceed in the manner of an STV count, but on the assumption that every 1^st preference vote for a party represented a vote for the highest candidate on that party’s list. Surpluses would be transferred as in STV, on the assumption that the next preference for every ballot for an elected candidate was the next candidate on that party’s list. If at any stage there are positions still to be awarded but no surpluses to transfer, candidates are not eliminated individually, but the whole party (ie: all its candidates) with the lowest vote is eliminated next, and the votes transferred to the various parties that are next preferred by each voter, resulting in increased party vote totals. This method is functionally a variant of STV, except that in effect voters have their choices reduced the two ways: voters are denied the opportunity to rank preferences at the level of individuals across or between parties, and they are denied the opportunity to rank the candidates within each party in any order other than that proposed by the party. Small parties are also disadvantaged compared to ordinary STV by the prospect of bulk elimination of all their candidates.

Huntington series – a type of divisor series used for indirect election systems of seat allocation to parties. The divisor series consists of the square roots of the sums of successive integer pairs: {√(1*2), √(2*3), √(3*4), √(4*5), √(5*6), etc}. Using the Huntington series in an escalating divisor calculation generates the same results as the Huntington modified divisor quotient rounding method.

Huntington’s method – (also known as the Huntington-Hill method and the method of equal proportions); an adjusted divisor form of quotient rounding apportionment method. This method determines an initial divisor by dividing the national total population by the intended total number of seats, rounding all the resulting quotients up or down according to rounding at the geometric mean, and taking the resulting values as the seat allocation to each state. This formula results in outcomes that are less proportional than other formulae, but have the property of assuring every state (or party) of at least one seat. The method is used for allotting seats in the US House of Representatives among the US states. The method was developed in the 1920s and adopted in 1941, and is named after its initial proposer Joseph Hill (Chief Statistician at the US Census Bureau) and its key advocate Samuel Huntington (a prominent US mathematician).

Hyland free riding – [pending]

Imperiali quota – Q = (P/N+1)+2, where Q is the quota, P is the population (or equivalent such as formal vote total), and N is the number of assembly seats available: a formula for determining a quota for the allocation of seats in an assembly. Compared to the more standard Droop quota, the Imperiali formula yields a lower quota, presenting the risk that too many candidates may achieve it compared to the fixed number of seats available. Using an Imperiali quota has some effect to balance the problem of exhaustion when preferencing is optional. Given this limited justification, it is not in use in any actual electoral system. (See also Hare quota).

Imperialli series – a type of escalating divisor series used for indirect election systems of seat allocation to parties (or, unusually, to allocate seats to states or other electoral divisions within a nation). The series is {1, 1.5, 2, 2.5, 3 etc}.

implicit threshold – the impact of the number of seats available in an electoral contest on the minimum vote share needed for a candidate to be elected, even where there is no strict rule that a certain number of votes is needed to be eligible to win a seat.

inclusive Gregory method – the ‘all-ballots’ variant of the Gregory method, a technique for transferring the surplus of votes from a candidate who has won a seat in an STV election in the immediately preceding stage of counting. Inclusive Gregory method transfers require significantly more counting labour that random selection transfers, and would have been logistically difficult in the early decades of the use of STV, but since the advent of computing technology the management of counting and calculations has become straightforward.

independence of irrelevant alternatives – (IIR); a criterion of social choice theory. Applied in relation to voting systems, IIR requires that if a voter initially prefers candidate A to candidate B, then the addition of candidate C to the voter’s list of options must not have the effect of reversing the voter’s expression of preference of A over B. As such, IIR is a desideratum regarding voter behaviour and strategy, rather than a technical voting system rule. If a voting system has other design features, or if the voter votes strategically in the context of predictions about the decisions of other voters such that, with the inclusion of candidate C, there is a strategic imperative for the voter to reverse their original A>B preference, then that system would as a whole be said to fail the IIR criteria. IIR is one of the four criterion used in Arrow’s theorem.

indices of proportionality – any of several statistical formulae for measuring the proportionality between votes and seats won. Examples include the Gallagher Index (based on ‘least-squares’), Rae’s Index, the Loosemore-Hanby Index and the largest deviation index. Each of these indices can be correlated with an apportionment method to which it will have a corresponding mathematical logic, leading to the difficulty that each apportionment method will appear to yield superior proportionality results if it is measured using its corresponding index of proportionality. (See Gallagher: Proportionality, Disproportionality and Electoral Systems, 1991)

indirect voting – any electoral system where the direct object of the votes cast by voters is some entity other than an individual candidate. Voting systems where a separately elected state assembly or an electoral college makes the final decisions about appointments to a national assembly are also types of indirect election. But most commonly the alternative object of voting is political parties, as in the seat allocation election systems commonly known as party list systems. A variant of the party list seat allocation approach, known as open list systems, allows voters to mark votes for individual candidates, not to affect the allocation of seats to parties, but to influence the order in which individual candidates are awarded a party’s allocation of seats.

informal vote – (or invalid vote); a voters’ ballot which is disregarded from the counting of votes due to some inaccuracy or incompleteness in how it is filled out, including ballots which are left blank.

insincere nomination – the practice of nominating candidates who are not genuinely intended to win seats for some politically tactical reason, usually based in the voting method being used and/or to the division of political viewpoints in the electorate.

insincere preferences – in preferential voting systems, a form of insincere voting in which a voter who is aware of the nature of the vote counting method and of the likely prospects of success of various key candidates rank candidates in an order other than in the order of their true preferences, so as to have a particular impact (other than the success of the candidates they personally favour) on the election outcome.

insincere voting – the action of a voter who is aware of the nature of the vote counting method and of the likely prospects of success of various key candidates, to cast their vote (typically a single non-transferable vote) for a candidate other than the one whom they truly prefer, so as to have a particular impact (other than the success of the candidate they personally favour) on the election outcome. (See also strategic voting.)

instant runoff voting – (IRV): a term commonly used in the United States to refer to preferential voting with sequential elimination, and equivalent to the term alternative vote used in the UK. A narrow reading of this term, focussing on the word ‘runoff’, would indicate that the preference distribution should move directly to a comparison between the top two candidates, that is to say that all other candidates would be simultaneously eliminated and the preferences on ballots cast for them redistributed only to the remaining two candidates. On that reading IRV would be more closely equivalent to the contingent vote.

Jefferson’s method – (also greatest divisors method): an adjusted divisor form of quotient rounding apportionment method. This method determines an initial divisor by dividing the national total population by the intended total number of seats, rounding all the resulting quotients down, and taking the resulting values as the seat allocation to each state. The method is mathematically equivalent to the D’Hondt escalating divisors method in that they always give the same results, although the method of calculating the apportionment is different. The method is named after its proponent, US statesman Thomas Jefferson.

joint ticket – the practice of nomination by two or more parties of a common list of candidates for election to multi-member positions: also, a pre-election statement by parties to voters indicating the specific form of ballot which they urge their supporters to cast.

Kemeny-Young method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. The Kemeny-Young method [… pending]

Lakeman’s dictum – the proposition – attributed to British electoral scholar Enid Lakeman – that political parties should only be the focus of election systems to the extent that individual voters wish to make them so.[xviii]

largest remainder method – (the primary form of quota-and-remainder apportionment): a calculation rule for the allocation of a fixed number of seats to a state (based on population) or to a political party (based on the total votes won in an election). The population (or vote) is first divided by the number of seats available to yield a quotient. Each state (or party) is then initially allocated the number of seats equal to the number of whole quotients they have. The remaining available seats are then allocated to states (or parties) in descending order of the size of the remainder in their quotient results. There are various options for the formula to be used to calculate the quotient, of which the most common three are the simple quota (Q=(N/V), the Hagenbach-Bischoff quota (Q=(N/(V+1)), and the Droop quota (Q=(N/(V+1)+1). When applied to allocations to a large number of recipients (such as for example the 50 US States), the method is subject to anomalous effects due to the use of the relativities between the remainders rather than those between the full quotients – see Alabama paradox, population paradox, migration paradox, and new state paradox.

The Niemeyer method is a variant of the largest remainder method, in which the allocation of seats is restricted to only those parties which achieve at least one full quota of votes.

The Hamilton method (or Vinton method) is the largest remainder method coupled with a rule for a minimum of one seat for each state, applied before having resort to the remainders.

least squares index – see Gallagher Index.

levelling seats – (also adjustment seats, compensatory seats): seats in an assembly that are allocated to political parties with the objective of ensuring that the representation of parties in the assembly is in proportion to their national vote shares. Various methods of achieving this aim include the mixed member proportional systems used in Germany, Lesotho and New Zealand, the additional-tier seat allocation systems used in Austria, Denmark, Estonia, Iceland, Norway, South Africa and Sweden, and the British AMS systems used in Scotland, Wales and London.

limited preference voting – a constrained form of preferential voting in which voters are only permitted to indicate a limited number of preferences beyond the 1^st preference regardless of the number of candidates. Voting which is limited to 1^st and 2^nd preferences is called supplementary voting. Limited preferencing has the outcome that a significantly larger proportion of votes can be expected to exhaust. As a result, the final tally of votes for the winner is not assured of being 50% or more of the original total formal vote, meaning that perceptions that the winner has won a majority may not be justified. This method currently used to elect the Parliament of Papua New Guinea is one of limited preference voting allowing 1^st, 2^nd and 3^rd preferences only.

limited transfer by lists – an early proposal by English scholar Walter Baily (see Baily 1869, also Droop 1881, p 32) for ticket-based preference transferring in the single transferable vote (STV) voting system, similar to the above-the-line system currently used in the Australian Senate. In effect voters would not determine their own preferences between candidates at all, but would simply vote for a single candidate as in plurality voting. Each candidate, however, would draw up and publish before the poll a preference order of all candidates other than themselves. Whenever a candidate is elected and has a surplus to transfer, or is eliminated, the lists would be used to determine the candidates to which the relevant votes transfer. The proposal has never been used in actual STV systems.

limited vote – a form of multiple non-transferable voting in which voters in a multi-seat division to elect N members may only vote for a number of candidates fewer than N. While functionally similar to the block vote, this method tends to produce representation of a wider range of voter groups. Limited voting was proposed in England as early as 1831 as a device for making election outcomes more party-proportional, and was adopted in 1867 for use in the small number of 3-member (12) and 4-member (1) House of Commons divisions then existing, with the rule that voters cast one vote less than the number of seats available. (These arrangements continued in effect until the adoption of single-member-only division arrangements in 1948.)

local PR – a proposed Canadian variant of single transferable vote (STV) voting, involving two tiers of electoral division. Voting for candidates would essentially be done at the level of the larger region, but ballot papers would be structured as a matrix in which every local electoral division (‘riding’ in Canadian terminology) is a row and every party is a column, allowing for one candidate for each party in each riding. The STV counting process would proceed on the basis that the number of seats available is the same as the number of ridings. Whenever any candidate achieves a quota of votes, all the remaining un-eliminated candidates nominated in that riding would immediately be eliminated. This process would eventually ensure that one winner was identified from each riding. The system would also – depending on a reasonable number of ridings being grouped into each region – achieve a degree of party-proportionality, both within each region and overall across the whole electorate. However the system would not guarantee that the winning member for each riding was the most popular candidate of the riding, either in plurality or preferential terms. The proposal was developed by academic Byron Weber Becker and was advanced by FairVote Canada and FairVote BC for consideration in the electoral reform process in British Columbia in 2018.

local STV – a proposed Canadian variant of single transferable vote (STV) voting involving two tiers of electoral division, including larger electoral ‘regions’ subdivided into local ‘ridings’ (the Canadian term for local divisions or districts). In the STV process of counting the votes, the elimination rules would be altered so that at least one candidate from each local district was guaranteed of being awarded a seat. The rule would inevitably cause some derogation from the principle that the same quota of votes must be achieved by every successful candidate in order to win a seat. The proposal was put forward by academic Byron Weber Becker as part of the parliamentary inquiry held during 2016 for reform of the Canadian electoral system, and was also proposed for use in electoral reform in the town of Guelph, Ontario (but was not implemented).

local transferable vote – a proposed variant of single transferable vote (STV) voting, put forward by Leonid Elbert of New Brunswick as part of the parliamentary inquiry held during 2016 for reform of the Canadian electoral system. The system would use two tiers of electoral divisions, with local riding subdivisions within larger electoral regions. The nomination of candidates, voting and STV counting processes would apply at the level of the larger region, but the elimination rules would be amended to ensure that at least one candidate from each riding was ultimately elected, by protecting the last remaining candidate from elimination, if necessary even if they did not achieve a quota of votes. In addition the total number of seats for the larger region would be set as greater than the number of ridings (perhaps only requiring 2-3 additional places per region) so that by the end of the STV process of transferring of preferences, the overall set of seat winners would achieve rough party proportionality. However the rule for protecting at least one candidate for each local riding would inevitably cause some derogation from the principle that the same quota of votes could be achieved by every successful candidate in order to win a seat.

Loosemore-Hanby Index – one of the measures of proportionality between votes and seats won in seal allocation systems. The measure involves summing all the raw values for the different vote share percentages and seat share percentages for every party contesting the election, and dividing tot total by 2. The measure, first proposed in 1971, is widely used, but has also been criticised for results being easily distorted, particularly by elections with small seat magnitudes. The measure is a mathematical correlate of the largest remainder method of apportionment, in that seat allocations determined using that method will appear to have their highest proportionality when the Loosemore-Hanby index is used as a measure of disproportionality (see Gallagher: Proportionality, Disproportionality and Electoral Systems, 1991).

lower house – (also popular house): in a bicameral parliament, typically the larger house elected to represent all enfranchised people. (In contrast an ‘upper house’ may also be nationally representative, but is more typically elected to represent the people or governments of states, or is an appointed or hereditary chamber.) In unicameral system the sole house is equivalent to the lower house. Titles include House of Commons (United Kingdom, Canada), House of Representatives (United States, Australia, New Zealand), National Assembly, Assemblée Nationale (French), Cámara de Diputados (‘Chamber of Deputies’, Spanish), Diet (German, used also in Japan), Bundestag (German), -thing (Nordic languages), Duma (Russian), Majlis (Arabic), Lok Sabha (Hindi) and many others.

major fractions method – see Webster’s method.

majority candidate – (also majority preference, most preferred candidate or Condorcet winner) : the candidate who is preferred by a majority of voters to every other candidate after considering all the possible one-on-one comparisons between candidates. Except in the case of a tie, there cannot be more than one such candidate. In unusual cases, it is possible that a loop forms between three leading candidates, such that the voters in an electorate will prefer A over B by majority in a two-candidate comparison, and prefer B over C, but also prefer C over A. This situation is termed a ‘Condorcet cycle’ and in such cases, there is no Condorcet winner, and all the candidates with the highest number of pairwise victories are collectively termed the Smith Set. Condorcet cycles become more common where there are a large number of candidates. The terminology is derived from the electoral methods proposed by Condorcet.

majority consistency – (also majority principle and Condorcet consistency) : an attribute of a voting system that the system guarantees the success of the Condorcet winner. Applied to multi-seat elections, a voting system is ‘majority consistent’ if it guarantees that a party A which wins more votes than party B will have more seats than, or at worst the same number of seats as, party B.

majority judgement – a variant of range voting, in which the successful candidate is that with the highest median vote, rather than the highest mean of all votes. (See Balinski and Laracki)

majority remainder method – a variant of the largest remainder method. The population for each state (or other type of electoral division) is first divided by a quota (usually a simple quota or a Droop quota) to give a quotient. Each state is initially allocated a number of seats equal to the integer of its quotient. Additional seats are then allocated to each state with a remainder of 0.5 or greater. The total number of seats so allocated may differ (usually only by a small number) from the initial targeted total. This method is currently used for the allocation of seats in the Australian House of Representatives among the six Australian states. The method can also be used in actual seat allocation to parties, based on their votes, but no nation currently uses this approach.

majority – the condition that one numeric value in a set has more than half the total of all the numeric values in that set, or in other words 50% or more. For example where four election candidates have vote shares of 52%, 27%, 11% and 10%, the first listed candidate has a majority. Note that majorities of different populations may need to be distinguished, for example a candidate might win election by having a majority of the formal votes cast, but not a majority of the enrolled electors (because some of the enrolled voters have not cast formal votes), or a majority of the enfranchised citizens (because some have not been enrolled).

malapportionment – in comparisons between two or more electoral divisions, a disproportionality between the voter population in each division (normally defined as the number of voters enrolled, but sometimes defined as an estimated total voter population, or even simply the total resident population) and the number of representative seats allocated to each division. Lack of proportionality causes a significant difference in the effective equality of each voter’s voting power – a lack of ‘one vote one value’ status.

Malapportionment between electoral divisions drawn on geographical boundaries is impossible to exclude entirely and difficult to minimise even through good system design and administration. The most common case of malapportionment is in regard to systems made up of single member divisions, where any significant variance in the enrolment numbers for each division away from the mean value will result in significant malapportionment. In federal systems where fixed numbers of seats are allocated first to states, and then to electoral divisions within each state, malapportionment is inevitably present between states even if care is taken to draw boundaries within states. Malapportionment can often worsen the degree of concentration distortion arising from electoral division boundaries (although in specific cases it could also have the effect of reducing it). Minimisation of malapportionment by careful drawing of boundaries achieves its goal better with compulsory enrolment; without full enrolment (and indeed without high rates of voter participation in elections) attempts to eliminate malapportionment can at best only tend towards approximations of voter equality.

Malapportionment is sometimes maintained deliberately, as with the rural area seat weighting (ie: divisions with lower numbers of enrolled voters) that was in effect for many decades in Australia at both national and state levels[xix]. In some countries (including Australia, France and the United States) electoral legislation now requires intra-state similarity of numbers of enrolled voters in each electoral division, although variations of up to 10% from the mean enrollment are often permitted in practice. National electoral laws also generally require regular boundary reviews so as to minimise malapportionment emerging from population growth and migration. The periods for such boundary review laws vary from every three years (Australia[xx]) to a decade (United States) or more.

mandate – the claim of legitimacy for the holding of a political office, or for the adoption of a policy or law, said to arise from the approval of the electorate as revealed through election results. Where a specific policy question or legislative proposal is submitted to (and approved by) the electorate in the form of a plebiscite or referendum, that policy or law can be said to have a mandate. The holder of a single-position office which was submitted to direct election can also be said to have a mandate, although in plurality electoral systems this claim is substantially weaker than in electoral systems using preferential voting or which identify a Condorcet winner. Claims of mandate should ideally be related to the proportion of the whole electorate which appears to support an appointment or policy, rather than merely the proportion of those who have voted: as a result, low rates of participation in elections (that is, turnout) reduce the validity of any claims of mandate. Further, a mandate can only correctly be asserted for the legitimacy of a policy, or the holding of an office, if that policy, or appointment to that office, was the clear and distinct subject of the choice being presented to voters. For instance, the holding of executive office by a premier or ministry in a parliamentary system of government cannot correctly be said to have a mandate, as voters at the election were only selecting individual representatives in individual electoral divisions, not choosing directly (and independently of other questions) between alternative candidates for the premier or ministry. (By contrast, the individual members of assemblies might each be said to hold personal ‘mandates’ to hold each of their parliamentary seats.) Similarly, the fact that a party or president has won executive office does not imply that any specific policy they adhere to has achieved a mandate. Notwithstanding this logical limitation, loose claims to possession of a mandate are routinely made by politicians holding government office as a means of denying the legitimacy of objections to their hold on office or to their policy or legislative program. Finally, an alternative correct use of the term mandate in a parliamentary system would be the claim by a ministry holding the confidence of the relevant assembly that it holds a mandate from the assembly, even if not from the electorate. (Note: ‘mandate’ is also the German word for ‘seat’, as in seat in an assembly.}

margin requirement – a requirement relating to single-candidate elections decided by the plurality rule, whereby a leading candidate does not win the election unless their vote share is superior to that of the second-placed candidate by a specified margin, for example 5% of the vote. If this condition is not met, resort is had to an alternative procedure such as a runoff election between the top two candidates. See also qualified plurality rule.[xxi]

mean-median district vote share difference: a measure of partisan bias between political parties in single-member division electoral systems, putatively caused by gerrymandering of electoral district boundaries. The mean-median district vote share difference compares a political party’s mean district vote share to its median vote share. Comparison of a mean to a median is a standard measure used by statisticians to measure ‘skew’ in outcomes. This measure is not, however, able to take into account natural shifts in demography or in political opinions. The measure was developed during 2015 and 2016 by political scientists Michael McDonald, Robin Best and Sam Wang. See also efficiency gap and seats-to-votes curve, which are other attempts to measure partisan bias in district boundary-making.

A detailed discussion of the measure is provided in Extreme Maps, Laura Royden and Michael Li, Brennan Centre for Justice, 2017.

See also partisan symmetry, the efficiency gap and the seats-to-votes curve, which are other methods to attempt to measure partisan bias in district boundary-making.

mechanical effect – the impact of the vote counting method of an electoral system on the outcome resulting from the votes cast, in regard either to determining the winning candidate (for single-seat elections) or else in regard to the degree of representation and proportionality achieved in a multi-member election. The term was first used in the work of Maurice Duverger.

median voter – a concept arising from the spatial voting theorem in social choice theory. The theorem assumes the existence of an electorate in which political views lie along a one-dimensional spectrum, and predicts that the outcome of a decision taken by that electorate will match the view of the voter whose position is mid-way along that spectrum in terms of numbers of voters (that is, there are as many other voters to ‘the left’ of the median voter as there are to ‘the right’, whatever the political question may happen to be). The theorem predicts that if the voting system for decisions (including for electing candidates) is a single-winner method, then candidates will have a substantial incentive to identify the median voter and adopt this voter’s viewpoint as their policy, and candidates who hold to such a policy are more likely to be elected than other candidates. The theorem will likely prove true (as a predictor and explanation of results) where there is a ‘single-peak’ profile to the electorate’s spectrum of views, but may fail if there is, say, a political profile with two strong partisan forces and also one of more moderate candidacies occupying the political ‘centre’. Note also that the theory is conceptually only relevant to a ‘single question’ of political choice (or at best, to a set of very similar questions), and is more likely to fail (as a predictor and explanation of results) in a complex political environment where voters must weigh up multiple issues (and thus have preference positions at different points on multiple spectrums) such as is the case in a real election. The median voter theorem was first stated by Duncan Black in 1948-49, and was later clarified by Downs (1957)[xxii].

Meek’s method – a detailed method of managing the vote transfer calculations in STV elections, developed in 1969 for early computer era coding technologies by French mathematician Brian Meek.

member – (also deputy, representative, delegate): any individual member of an assembly.

migration paradox – one of a number of controversial results generated when using the largest remainder method of quotient rounding to carry out an apportionment of assembly seats. This paradox occurs where the total population and the total number of seats remain unchanged, but changes in the population of other states (ie: migration between them) have the effect of changing the allocation of seats to a third state, even though the latter state’s population did not change. The effect occurs due to the use of the relativities between the remainders rather than those between the full quotients. (See also Alabama paradox, population paradox and new state paradox.)

minimum seat rule – a rule used in quotient rounding apportionment procedures under which there is some minimum seat allocation to each state. The most common manifestation of such a rule is that every state must be allocated at least 1 seat. The usual method of application of this rule is that after the initial allocation of seats based on whole quotients, the remaining available seats are first used up awarding 1 seat each to any state with less than one whole quotient. Once any such allocations are made, the allocation of remaining seats (if any) proceeds according to remainder method being used.

mixed member majoritarian – (MMM); a method of electing a composite assembly in which some members are elected in local divisions (usually by the plurality voting method), and some through regional or national party list seat allocation, but without any mechanism to ensure that the total composition of the assembly is party-proportional to national vote totals for each party, as is the case with mixed member proportional or other levelling systems. MMM systems are a major subset of parallel voting systems.

mixed member proportional – (MMP); a method of electing a composite assembly initially developed in Denmark, later adopted for the German Bundestag from 1953, and since used in variant forms in other jurisdictions including Hungary, Iceland, New Zealand (from 1998) and Norway. One part of the assembly will be directly elected using local single member divisions and the plurality voting method, and the remaining seats (sometimes called levelling seats) will be allocated to parties in the numbers required so that the party composition of the whole assembly is proportional to the national vote totals of each party. The system may either re-use ballots cast for local candidates as the basis for the national proportional allocation of seats, or it may require voters to cast two votes, one for the local member, and a separate vote for parties that is used in the national calculation of proportionality. Minimum thresholds, typically of 3% or 5% of the national or divisional total of votes, are often imposed for parties to be eligible to be allocated any levelling seats. The levelling seats may be determined either in regional electoral divisions, or for the nation as a whole. The system can sometimes result in members being initially elected in local divisions in numbers greater than the proportional target for the party concerned, in which case the standard German practice is to increase the size of the assembly by a number of compensation seats necessary to ensure that each eligible party has seat numbers in relative proportion to one other’s vote shares.

A modern variant of the MMP approach is the British AMS system (used for elections for the Scottish Parliament and Welsh Assembly), which directly links the winning of seats in local electoral divisions with the counting process for allocating seats to parties.

mixed system – a common general term for any electoral system using two or more electoral methods to constitute a composite assembly.

modified Borda count – a variant of the Borda count voting method, in which the number of points given for a voter’s first and subsequent preferences is set by the number of candidates they have ranked, rather than the number standing. Thus if there are ten candidates but a voter ranks only five, then their first preference will receive only five points: their second preference 4 points, and so on. This method penalises voters who do not rank a full ballot, by diminishing the number of points their vote distributes among candidates. The method thus penalises voters to fail to submit a fully marked ballot. The method was first proposed by Peter Emerson of the De Borda Institute, and has been used in internal elections by the Irish Green Party.

modified Sainte-Laguë – a type of escalating divisor series used for indirect election systems of seat allocation to parties. The modified series is identical to the Sainte-Laguë series except that the first divisor is set at a value of 1.4, not 1. This change has the direct effect of presenting parties with smaller vote support with a higher threshold to secure their first (and normally only) seat, and thus acts as a raised vote threshold for parties to win representation. This formula is used to elect the Swedish Riksdagen and the Polish Sejm.

monotonicity – a criterion for assessing single-choice voting systems. A voting method is said to fail the criterion if under a given voting rule an improvement in the ranking (in ranked voting systems) or the rate or score (in rate systems) of a candidate would, for any reason, have the effect of reducing the likelihood of that candidate being elected. Failure of the monotonicity criterion occurs in relation to various elimination methods used in ranked systems and the two-round runoff voting method, most notably sequential elimination.[xxiii]

multi-member division – an electoral division for which 2 or more members are elected (by voting methods such as SNTV or STV) or allocated among parties (by some form of party list seat allocation system).

multiple non-transferable vote – (MNTV) : a form of rate voting where each voter casts multiple single rate votes (usually in the form of multiple ‘X’s on a ballot paper) for the election of multiple members. Variants include block voting (where each voter casts a number of votes equal to the number of candidates to be elected), threshold MNTV, and limited voting (in which voters have cast some lesser number of votes).

N – standard shorthand for a reference to the number of seats in an electoral division: used in the statement of a formula, for example the determination of an STV quota such as “Q = (V/(N+1))+1.

Nanson’s method – a hybrid of the Borda count system which results in the identification of the Condorcet winner of a preferential election. The method, proposed by Edward Nanson in 1880[xxiv], proceeds as with a Borda count (using votes which appear to be ordinal preferences but which are tallied as rate votes at each counting stage). The process does not declare a winner on the plurality of Borda count scores at the first round of counting, but instead eliminates all those candidates with a total vote tally which is at or below the average tally for all candidates. The ballots are then renumbered and recounted on the basis that only the un-eliminated candidates remain in the election. This procedure is repeated as many times as necessary until fewer than 3 candidates remains at the end of an elimination, in which case the sole remaining candidate, or the one with the highest vote tally, is the winner. Nanson’s method is one of the ‘recursive elimination’ variants of the Borda count, others including the Baldwin and Rouse methods.

Nash method – a form of rate voting and a variation on the preference-points approach used in a Borda count. As in Borda count, voters mark a range of scores for each candidate ranging from 1 to the number of candidates. Instead of being limited to the strict use of integers, however, voters may rate candidates using specific decimal scores (for example 1.5, 3.7, etc). Leaving aside the possibility of tied scores, the voter’s order of preference is immediately apparent, but the score ratings for each candidate have a more fine-grained quality. The method was first proposed by Nobel Prize winning mathematician John Nash, who further proposed that the vote scores for each candidate should be multiplied together (in a manner not specified) to determine the winner. Riker (1982) formalised this approach and named it after Nash.

Negative vote weight – (German: negatives Stimmgewicht) a phenomenon which may occur  in mixed-member proportional electoral systems, whereby hypothetical additional votes for a party in a specific regional electoral division would have the effect of resulting in fewer seats being won nationally. The phenomenon manifests where…Since it can be shown that the phenomenon consistently operates to the disadvantage of small parties as against large parties, in 2008 the German constitutional court invalidated the relevant electoral law on the basis that the phenomenon breached the equality of voter influence principle in the German Constitution.

new state paradox – one of a number of controversial results generated when using the largest remainder method of quotient rounding to carry out an apportionment of assembly seats. This paradox occurs where the total number of seats and also the populations of the existing states are all unchanged, but the addition of a new state into the system causes the seat allocations of other states are altered in numbers greater than would be expected simply to accommodate the new state, including the prospect that some other states may in fact gain one seat. The effect occurs due to the use of the relativities between the remainders rather than those between the full quotients. (See also Alabama paradox, population paradox and migration paradox.)

Niemeyer method – (or Hare-Niemeyer method) a form of largest remainder method for seat allocation. The method differs from the standard largest remainder method in that that the Niemeyer method stipulates that only parties that have achieved ‘eligibility’ to win seats, by winning at least one quota of votes, are involved in the allocation calculation. The method calculates the seat allocations using the relative votes only of the eligible parties (that is, all votes cast for the small non-eligible parties are disregarded altogether in the calculation), and the quota value for each party is therefore the party’s vote divided by the total of the votes of all seat-eligible parties.

nomination – the administrative act of registering a person as a candidate for election.

non-dictatorship – (also anonymity), a criterion of social choice theory. In regard to voting systems, this criterion requires that no voter has any special advantage over others. The criterion fails if there exists a voter whose preferences will prevail in every situation. The illustration of a single decisive swing voter between two equally sized populations of partisans describes the existence of a ‘dictator’ in a casual way. (The illustration is not strictly correct in that any voter could potentially change their vote, but the illustration could be said to be provisionally correct for precisely that scenario/ballot, taking the fixed preferences of the partisans as given.) Non-dictatorship is one of the four criterion used in Arrow’s theorem.

Oklahoma primary method – a variant of the descending fractions voting method, used briefly in party primary elections in Oklahoma in the 1920s. The variations included that preferencing was limited, and that there were special victory conditions applied at the conclusion of the initial tally of vote scores (the ‘first count’), and also after a second count (which count, if required, was then conclusive).

one person, one vote – the principle (or judicial doctrine) that all voters should have equal power at elections, and specifically in the sense that the voter populations of all electoral divisions should be the same in size. If divisions elect differing numbers of members, the doctrine would require that there be proportionality between elector populations and division magnitudes to achieve the same result. The doctrine is effectively a rejection of malapportionment. In practice even in countries where this doctrine is currently applied elector population sizes are allowed to vary within a tolerated margin, perhaps up to 10% from the mean. The doctrine won favour in United States Supreme Court decisions in the 1960s, and also came to be generally accepted in Australian in the 1980s, with the current electoral law in both nations covering both state and national elections. The doctrine does not prevent forms of concentration distortion other than malapportionment.

open list – a feature of some seat allocation election methods in which the ballot paper provides an opportunity for voters to indicate ‘preferences’ for one or more of the individual candidates on the party list. The ballots are first used to determine the party’s total votes (derived from the sum of all votes cast for individuals from each party) and using those totals the allocation of seats to parties proceeds according to one of the various seat allocation methods. The seats won by a party are then allotted to individual candidates with reference to the individual vote tallies of candidates.

In ‘unconditional’ or ‘fully’ open list systems, the votes received by each individual within a party list will be tallied as in an SNTV count, and the seats available to that party will be allocated to individuals in the order of that vote tally. However some open list systems are ‘conditional’ or ‘partially open’ in that candidates must first achieve a predetermined fraction of the allocation quota (for example 3% in Slovakia, 8% in Sweden), and those who do so are moved to the top of the list (in the order of their personal vote). This revised order of candidates is then used to allocate the seats.

Open list systems allow the potential for an individual candidate with strong voter support to be chosen ahead of others in the order stated in the party list. There is thus some similarity of outcomes between direct voting systems which will have the effect of allocating seats to party candidates in general proportionality to the party total vote (such as STV) and open list seat allocation systems. However in open list systems, an individual candidate still cannot be elected outside of the party-based seat allocation calculations, and thus these systems do not constitute true direct voting.

open signalling – (see Mclean and Shapley, 2004)

openness (of voting) – the degree of access enjoyed by voters (through their ballot papers) to a broad range of individual candidates to support in an election. A range of electoral system features can reduce the range of candidates available to voters, including any formal limitations (such as payment, or onerous signature requirements) placed on individuals seeking to nominate for election, the elimination of potential candidates through primary elections and other forms of political party preselection practices, the use of electoral divisions (especially single member divisions), and the adoption of indirect party list seat allocation systems. The most open voting approaches will therefore have minimal formal nomination requirements, minimal party preselection constraints, minimal or no use of electoral divisions across a nation, and only use direct election methods.

optional preferencing – in relation to a preferential voting method, a rule than voters are free to mark preferences on the ballot paper for only as many candidates as they wish to. Where sequential elimination is used, one result of optional preferencing is that some ballots will become exhausted. It may also result in the person or persons winning seats doing so without achieving 50%, or other relevant quota, of the total number of formal votes. Optional preferencing permits voters to more accurately convey their true preferences, especially where they have no preference to indicate for some candidates. By contrast, compulsory preferencing requires many voters to record insincere or random preferences on their ballot.

Ostrogorsky’s Paradox – a possible phenomenon, apparently undemocratic in nature, arising from the simultaneous election of an assembly expected to decide upon three or more known policy choices, at the same time as allowing the elected assembly members to choose a majority government from political parties which have positions on those issues. Simplified, the scenario is as follows. Suppose that there are two parties, the ‘Yes’ party and the ‘No, party, and three key issues in the election (A, B and C), for each of which there is a ‘yes’ and a ‘no’ position. Suppose also that one third of the electorate supports the ‘yes’ position on issues A and B, but – with greater intensity – supports the ‘no’ position on issue C, as a result of which they decide to vote for the No party at the election. Another third of the electorate likewise supports ‘yes’ on issues A and C, but more strongly supports the ‘no’ position on issue B. Similarly the final third of the electorate supports ‘yes’ on B and C, but is strongly ‘no’ on A. At the election, whilst two-thirds of the electorate supports the ‘yes’ position on all three issues, the No party wins election in a landslide, committed to opposing the will of the electorate on every issue. The paradox is less likely to manifest the more complex the number of parties, the range of positions offered by those parties, and the number of issues involved. In addition, the division of the community into the uniform blocks of opinion needed to generate the paradox is fairly abstract. The articulator of the problem, Moisey Ostrogorsky, conceived his theory watching English and US two-party politics in the late 19th century, when electoral analysis – especially in the UK – tended to regarded each election as being concerned with a limited number of current issues, rather than the large number of issues present in most modern elections.

overhang and compensation seats – (German: Überhangsmandate and Ausgleichsmandate); terms used in mixed member proportional electoral systems, as developed for the German Bundestag and German state elections. Such systems are based on the premise that all eligible political parties (usually defined as those winning 5% or more of the vote across the whole electorate) will ultimately be allocated numbers of seats proportional to one another’s vote totals.

For these systems a fixed number of local electoral divisions will be established, and a nominal number of additional seats to be allocated to reach the proportional situation will also be set, therefore defining a nominal grand total number of seats. However it is common for the leading political party to win more local electoral division seats than the total number of seats which their proportional share of the vote in the region would nominally entitle them to. The excess number of seats is termed an Überhang ‘overhang’ (but note than no local seat is itself individually referred to as an ‘overhang seat’). The greater the share of the total nominal number of seats are established as local electoral divisions, the more likely this result is to occur.

In addition, the greater the vote share lead of the largest party, the more likely it is to occur. If both leading parties have fairly low vote shares and several minor parties perform well, it is possible that both major parties could have overhangs. To restore the situation to party-proportional balance the German solution is simply to allow the expansion of the total size of the assembly to the point where each party has seat allocations in the correct proportion relative to one another. The additional seats so awarded are termed Ausgleich (compensation) seats. In practice the number of local seats won by the largest party will form the point of reference for such a calculation, although if that party itself has an overhang, the expansion will be larger still.

In German elections in recent decades the two largest political parties (CDU and SPD) have generally been receiving decreasing shares of the total vote, and up to 4 minor parties have frequently been passing the 5% vote threshold to be eligible to win seats; meanwhile the two major parties have continued to win all the local electoral division seats. This has caused the number of compensation seats awarded to rise. The German constitutional court has also ruled that in order to maintain equality of voter influence, the system of compensation seats must be guaranteed (as least with respect of the threshold-achieving parties) over and above any other electoral rule which seeks to set the number of members for the relevant assemblies.

packing – a term used in relation to gerrymandering, in which those drawing manipulative electoral division boundaries combine two or more adjoining geographic populations of voters with a predictable distinct voting trend (ie: high concentrations of Democratic party supporters) into one electoral district, effectively ensuring that a candidate of the preferred party will win that electoral division easily, but removing many of those voters from having influence in one or more neighboring divisions, with the result that a rival party will be successful in electing representatives in the latter.

pairwise comparison – any of several proposed winner determination rules, designed to declare a winner where no Condorcet winner is present. Prominent variations are the Dodgson, Simpson, Copeland, Schulze and Tideman methods.

panachage – (also free list) : a practice combining the open party list type of seat allocation system with elements of multiple non-transferable voting. Voters are allocated a set number of ‘preferences’ (which are in fact non-transferable rate votes, and should not be confused with ‘preferences’ in the sense of preferential or ranked voting) which they may mark on their ballots in favour of any individual candidates on any party lists. Optionally, each voter may be permitted to allocate more than one of their votes to the same candidate – for example in Switzerland, up to 2 votes may be given to a candidate, a practice referred to as cumulation. The total number of these preferences, however cast, are then used as in open list party list systems – firstly to provide the vote totals for each party which determine how many seats each party is allocated, and secondly to reorder the candidates within each party’s list so as to determine which individuals are awarded the seats allocated to each party. Panachage is used for national elections in Switzerland and Luxembourg.

parallel voting – any of several possible forms of electoral system for constituting a composite assembly by means of separate ballot methods to elect the members in two or more groups. In a typical case some of the members of the assembly are elected directly (for example in single member divisions using plurality voting), and other seats are filled indirectly using a party list seat allocation method. Voters will cast separate ballots for each of the two component parts of the assembly. The indirectly elected members may be allocated nationally in one vote, or at the level of several electoral divisions (such as states or regions). The two voting methods are conducted entirely separately, and no formal method is used to ensure that the overall final composition of the assembly has parties represented in numbers proportional to their national vote totals. Most of these systems can also be described by the term mixed member majoritarian (or ‘MMM’). Parallel systems can be distinguished from mixed member proportional (or ‘MMP’) or ‘British AMS’ systems, which are designed to ensure that the overall final composition of the assembly has parties represented in numbers proportional to their national vote totals. The electoral systems used to elect the lower houses of Bulgaria, DR Congo, Egypt, Japan, Lithuania, Mexico, Palestine, the Philippines, Romania, Russia, South Korea, Taiwan, and Thailand are varieties of parallel voting or MMM voting systems.

Pareto efficiency – an economic and engineering concept, also employed as a criterion of social choice theory. A ‘Pareto efficient’ situation is one in which no actor can become ‘better off’ without one (or more) other actors becoming equivalently ‘worse off’. In regard to voting systems, this criterion fails for any voting method where a voter can change their preferences so as to give any single candidate a stronger preference ranking without simultaneously reducing the strength of the rankings given to one (or more) other candidate. Pareto efficiency is one of the four criterion used in Arrow’s theorem.

parliament – (also congress, national assembly, parlement (French), diet (German), congresso (Spanish), -thing (Nordic languages), duma (Russian), majlis (Arabic), Dàhuì (pinyin Chinese) and many others) : a composite political institution consisting of one or two houses (or ‘chambers’). In most cases a monarch or executive president must also give approval to bills as part of the legislative process, and that office is thus sometimes defined to be a component part of the parliament.

parliamentary system – a broad class of governance system for installing an executive government in which the executive, typically led by a prime minister, is appointed and holds office on the basis of continuing to hold the confidence of an elected assembly, normally the lower house of the national parliament. Such an executive, and the full ministry they lead, can usually be removed from office at any time (although there may be procedural or timing constraints in place) if the confidence of the assembly changes. Depending on the electoral system used to elect the assembly, this category can be subdivided into the plurality-parliamentary and representative-parliamentary systems. (Contrast with presidential system).

participation (rate) – (also turnout): the proportion of the enrolled electorate which takes part in voting in an election. Rates of participation around the world vary broadly from below 50% to as high as 95%. Some participating voters submit informal ballots (deliberately or inadvertently), so the full participation rate and the formal participation rate will be slightly different statistics. In systems with compulsory voting, participation statistics are naturally higher (averaging around 94% in Australia).

partisan bias – a term used in the US in regard to gerrymandering, where it generally refers to the partisan slant of any given ‘map’ of electoral division boundaries in a jurisdiction. Such bias may be measured by several tests. More specifically, in relation to the concept of partisan symmetry, partisan ‘bias’ means the converse of partisan ‘symmetry’.

partisan symmetry – a measure of the extent of gerrymandering in the electoral districts of a jurisdiction that is divided into single member electoral districts. The idea was developed in – and is referred to only in – the United States, which is the nation most actively grappling with issues of gerrymandering, and which features two very dominant political parties.

The proposed symmetry is best illustrated by example. If the electoral district map of a jurisdiction is such that were major party A to win (say) 55% of the votes across the jurisdiction it would win 60% of the seats, and alternatively if Party B were to win 55% of the votes it would also win 60% of the seats, then the jurisdiction’s electoral may may be said to have ‘partisan symmetry’ (or alternatively, it would not show ‘partisan bias’). It is not necessary that the electoral system ensure that 55% of votes translates into 55% of seats, or produce other similar proportional correspondences, since such exaggerations of seat shares are in fact typical of single-member division systems. All that is required is that the particular map of districts adopted for the jurisdiction not result in different outcomes for each of the parties.

The concept does not address the problem that for any given electoral map, symmetry may be satisfied at certain votes-and-seats values (eg 55%/60%, as in the example above), but not be satisfied at other values (eg 60%/70%).

The concept also does not guarantee that for any given jurisdiction, a symmetrical map of the jurisdiction must necessarily exist. It is possible that uneven geographic spread of partisan voting habits may render a jurisdiction incapable of achieving partisan symmetry through any map (as would seem to apply in the Australian state of South Australia).

Nor does a finding of partisan symmetry in a given electoral map necessarily guarantee a ‘majority win’ result in any given election in that jurisdiction (ie: the assurance that a party winning a majority of the votes will necessarily win a majority of the seats). Particularly in close elections, all SMD-based systems are subject to a possibility of an ‘inverted’ result where the party with the minority of votes manages to win a majority of the seats. All that the symmetry concept requires is that, leaving aside issues of probability in close elections, the electoral map does not inherently feature a non-symmetrical partisan bias between the two parties.

The theory has very reduced application to any jurisdiction which does not feature two very dominant major parties, as occurs in all the states of the United States.

Finally, the manner in which the measurements of symmetry are made must be based on data at specific points in time, including the initial point at which the electoral map is crafted (which can be evaluated using past election result data) and also at any subsequent elections. The extent of symmetry may be different for each different set of voting data discovered over time.

The concept of ‘symmetry’ as a measure of partisan bias in electoral districting was developed as early as 1994 by US political scientists Andrew Gelman and Gary King, later joined by Bernard Grofman and Jonathan Katz. In 2004 the divided opinions of a number of justices of the US Supreme Court (Veith v Jubilirer) caused political scientists (as well as lawyers and judges) to begin searching for a ‘judicially manageable standard’ the could allow objective measurement of the presence and extent of gerrymandering in any given state map of electoral districts. In 2005 the concepts authors promoted the idea to the US Supreme Court in an Amicus brief in the case of LULAC v Perry. In 2016 Andrew McGann and others promoted the utility of the concept in their book Gerrymandering in America. However in recent years three alternative means of measuring gerrymandering – the efficiency gap, mean-median district vote share difference and the seats-to-votes curve – have come more to the fore in US litigation on the subject.

party block vote – (also slate vote): a variant of the block vote in which voters cast simple rate votes between parties, in effect choosing between party tickets. The party ticket with the plurality of votes is awarded all the available seats.

party configuration – (also, especially in the US, party system): a reference to a consistent pattern in the proportional strength of voter support for the political parties in a nation’s electoral results. A two-party system is one where two dominant parties between them receive the bulk of voter support (generally over 80%) at several successive elections, resulting in these parties tending to take turns forming governments. The United States and Australia are examples. A multi-party system is one where three or more parties share substantial proportions of the vote, without any party achieving a majority of votes or seats on any regular basis. Almost all European counties, and the majority of world nations, are of this kind. The United Kingdom has most of the features of a two party system, but also has a significant long-term third party and also strong regional parties in Wales, Scotland and Northern Ireland. Canada has developed similar characteristics in recent decades. Alternatively, some parliamentary democracies have one-party-systems, where a single party is continually dominant over others – examples include Singapore, Tanzania, Mexico for many decades until recently, South Africa since 1994, and Russia in recent elections. There is a substantial debate around the idea that party configurations – together with other factors such as party expectations of future success as well as class, economic and industrial interests – influences electoral system design choices by legislatures (Colomer, Cusack et al, Boix, Kreuzer, etc, 1990s–). Conversely there is also a long-standing literature (Duverger, Lijphardt, 1950s–) on the idea that electoral system design influences the emergence and continuation of specific party configurations.

party list system – any of the indirect election systems providing for the allocation of seats to political parties on the basis of relative voter support for each party. There are closed list and open list variants of such systems.

party ticket – a template vote, possibly including a series of preferences, determined by a political party and intended for adoption by individual voters.

pendulum – a visual model for presenting the results of elections in parliamentary systems of government using single-member electoral divisions which also have a strong two-party political culture. In such diagrams, which usually have a U-shaped form, the safest seats for one party begin at one end of the “U”, and the safest seats of the other party are at the other end, with the ‘marginal, ‘contestable’ or ‘swing’ seats located in the middle. Prior to a looming election the model is used to present the seat position at the previous election, thus displaying the swing towards the opposition party that would be required to won a majority of seats and thus bring about a change of government.In the 1950s British elections commentator David Butler began using such a model, termed the swingometer. In Australia in the 1970s elections commentator Malcolm Mackerras began using a similar presentation, termed the pendulum. The use of the model remains common in Australia where a strong two-party system continues to prevail, although specific seats where a minor party or independent candidate places second fit anomalously into such a model. In Britain the rise in voter support since the 1970s of the Liberal Democrat party, the regionalist parties and more recently the UK Independence Party has reduced the utility of the model. In Canada, with a long-established three-party system, the model has not appeared in common use. Nor has a pendulum been used in the United States, despite the very strong two-party system there, because executive governments are formed from direct election of the president and governors, not from parliamentary majorities.

Penrose Law – a mathematical formula for analysing the distribution of the voting power among delegations, blocks or parties in an assembly (or other body). It states that the voting power of any voting bloc in an assembly is inversely proportional to the square root of the size of the block. This result was used to design the Penrose method for allocating voting weights in assemblies so as to represent several constituencies with differing populations. The law was originally formulated by British mathematician Lionel Penrose in 1946[xxv].

Penrose method – a mathematical formula for allocating numbers of seats (or by allotting voting to delegates or delegations) of representatives in a body (such as an assembly) to represent several constituencies with differing populations. The rule is that the voting weight of each constituency should be inversely proportional to the square root of the population of that constituency.

personal vote – in European political science terminology, a reference to votes cast for individual candidates, as opposed to votes cast for political parties. Votes cast in direct voting systems are thus all personal votes. In party list seat allocation systems which use some form of open list method, the term personal vote refers to the votes cast for individual candidates within those lists.

PLACE voting: a proposed hybrid seat allocation voting system using some elements of the STV voting method as well as local single-member electoral divisions, put forward by US statistician and electoral reform advocate Jameson Quinn in 2017.

Under this system the electorate would be divided into local electoral divisions, each of which will ultimately be allocated a single winning candidate. Candidates nominate in local electoral divisions. Voters select a single preferred candidate in their electoral divisions, but they have the option of writing in a vote for a single candidate nominated in another electoral division (or at least, from within a broader electoral region, if not from the whole national jurisdiction, depending on the chosen geographical electoral design).

Prior to the election all candidates would declare that they either ‘endorse’ or not every other candidate across all other electoral divisions (or at least, all the candidates in those local divisions within one larger electoral region, if not the whole national electorate). This they may do regardless of the party affiliation of other candidates. From these declarations for every candidate there is generated a preference ranking of all the other candidates initially in four groups: (i) same-party endorsed; (ii) same-party not endorsed; (iii) other-party endorsed; and (iv) other-party not endorsed. After the election, when the tally of individual votes for every candidate is known, the order of candidates within each group is then further refined by ranking the candidates within each group in descending order of their direct votes (including both local and nonlocal votes).

The end result of these steps is that for each candidate, a complete ordering of all the candidates across the region is generated. From this point on every ballot is treated as a preferential ballot indicating a full ranking of the region’s candidates, starting with the primary candidate as the first preference recipient, followed by all other candidates in the order determined by the above rules.

Hereafter the counting of ballots proceeds in a manner based on the counting system for single transferable vote (STV) voting, including the elimination of low-vote candidates and the transfer of the vote surplus of candidates achieving the vote quota, with the broader electoral region constituting the electoral division for the purposes of the counting.

An initial step is taken of eliminating all candidates who did not win at least 25% of the vote in their home local electoral division.

During the counting and transferring of votes, two rules are used to ensure that the end result will be one successful candidate for each electoral district. Firstly, as each successful candidate reaches the required vote quota and is elected, all other remaining candidates originally nominated in the same local electoral division are eliminated (and their ballots transferred). Secondly, the selection of a candidate to eliminate at each stage is done not using each candidate’s current vote total (as in standard STV) but rather on the difference between their vote total and the highest vote total of a candidate nominated in their district. This latter rule ensures that the last remaining candidate in a district will never be eliminated.

plebiscite – the term normally used to refer to a vote of the whole electorate, similar to a referendum, but without the legal consequence of adopting a proposed law or constitutional amendment. For example, in Australia plebiscites have been held regarding the national policy on conscription (1916 and 1917), and on the adoption of a national song (1977) : these votes did not have direct legislative consequences.

plumping – (also bullet voting): the practice of voters placing all their support solely behind a single candidate, even if the ballot paper permits them to support multiple candidates, to indicate preferences, or to allot multiple votes or points among more than one candidate. When used in optional preferential voting method, the effect is to shift the election results towards those characteristic of the plurality method.

plural voting – systems in which some electors have more than one vote, or are entitled to vote in each of more than one electoral division. Such entitlements were largely a feature of electoral systems where the franchise was based on the local ownership of property, and were largely abolished by the early 20^th century.

plurality-parliamentary system – one of the categories of system of government by which nations provide for the exercise of the executive power of government. In these systems executive power is exercised by a prime minister on the basis of the continuing confidence of a majority of members in the national assembly, which is elected through a pluralitarian electoral system (or perhaps a majoritarian one, as in Australia), rather than a more broadly representative electoral system. Major nations using this system include India, the United Kingdom, Canada, Italy, Japan, Bangladesh and Pakistan.

Distribution of world nations using the plurality-parliamentary system of government

plurality voting – a form of rate voting, in which seat(s) are won by the candidate(s) winning the highest number of votes. The rule is normally associated with single member divisions, where it is also referred to as the ‘first-past-the-post’ system. The multi-member division form of plurality voting is referred to as single non-transferable voting. A variant involving a qualified plurality rule or minimum threshold (usually of 40% or 45% of the formal vote) or perhaps of a margin requirement (or ‘lead’) of (for example) 10% over the next highest candidate, is sometimes used. In such cases if the lead candidate fails the additional requirement then a second round of voting is held between the two leading candidates, as in two-round runoff preferential elections. Argentina uses this threshold method to elect its President.

plurality – (also relative majority): the largest in any set of three or more numbers (the largest of two numbers would simply be a majority): for example, where four election candidates have vote shares of 31%, 27%, 22% and 20%, the first listed candidate has won a ‘plurality’.

pooling – in relation to open list variants of party list seat allocation systems, the technique of combining the personal votes (or ‘preferences’) cast for various individual candidates from the same party to form the party vote total that is then used in the seat allocation calculation

Popper-Polsby compactness score – a measure of the geographical compactness of electoral divisions based on their map outlines, relevant to US disputes about whether electoral divisions have been gerrymandered. The score is calculated by first measuring the length of each district’s perimeter and deriving from it a hypothetical circle with that same perimeter. The ratio of the particular district’s area to the area of the hypothetical circle is its Popper-Polsby Compactness Score, and the higher the score, the greater the geographic compactness.

popular legislative support – the combined vote share of a legislative coalition (See Nagel 2000).

population paradox – one of a number of controversial results generated when using the largest remainder method of quotient rounding to carry out an apportionment of assembly seats. This paradox occurs where two states each have population increased, but the resulting change to the remainders of the two states differs, causing one of those states to lose a seat to the other, even though the first state’s population increases. The effect occurs due to the use of the relativities between the remainders rather than those between the full quotients. (See also Alabama paradox, migration paradox, and new state paradox.)

preference deal – (also exchanging preferences) : any agreement between election candidates (or their parties) to bring about cross-preferencing.

preference schedule – any list of the rank order of a voters preferences. A ballot paper used in a preferential vote or in a Borda count is thus a form of preference schedule.

preferences – (1) in relation to preferential voting – the ordinal listing of candidates used in preferential voting: (2) in relation to open party list variants of seat allocation electoral systems – the indication of votes for individual candidates.

preferential block vote – (also multiple majority preferential, multiple count preferential): a form of preferential voting used to elect multiple members. The voting method is identical to simple preferential voting with sequential elimination for a single seat, but in this system the same ballots are fully recounted multiple times to elect the required N candidates, with each new count simply disregarding the presence of candidates on the ballot that have already won seats (with votes being transferred to the next most preferred unelected candidate). In practice, the result is highly disproportional as between parties, and is similar in nature to block voting in that candidates of the same party (assuming sufficient candidates are on the ballot) will tend to win all the seats, the only difference being that the successful party will be the one which has preferential majority support in the electorate, not merely the party with plurality support. The system was used for elections to the Australian Senate from 1919 to 1946, and in 55 of the 60 such elections (in 6 states over 10 elections), one party won all 3 of the available seats.

preferential ridings proportional voting – a proposed Canadian hybrid electoral system using two tiers of electoral divisions. In local divisions a single candidate would be directly elected using the Condorcet method. The upper tier would consist of regions formed from multiple local divisions. In these regions seats would be allocated to parties proportionally by either of two party vote total values: (1) the aggregate by party affiliation of all ballots which had not seen their first-preferred candidate win a local district seat; of (2) according to party totals recorded on a second ballot, as in MMP voting. The system was proposed by Yukon retired electoral administrator Dave Brekke in 2016.

preferential voting – (also in the 19th and early 20th century sometimes termed contingent voting (Hare 1857, 1860), uncommonly successive voting (Droop 1869), transferable voting, and in more modern usage alternative voting and instant runoff voting): a general term for rank voting, that is voting by ordinal ranking of candidates. The term is associated with all voting systems which list preferences and then transfer votes between candidates, by a variety of possible vote transfer methods, to determine one or more winners.

Voting in this manner to fill single-office positions using sequential elimination is referred to as ‘preferential voting’ in Australia, the nation where its use is most widespread, as ‘instant runoff voting’ in the United States, and as the ‘alternative vote’ in Britain and Canada.

Vote counting systems which compare all pairs of ranked candidates to determine the Condorcet winner use the same source preference information (ie: orderings of the candidates on ballots), but are not generally referred to by the term preferential voting.

premier-presidential system – one of the categories of system of government by which nations provide for the exercise of the executive power of government. In these systems executive power is exercised jointly by a president and a prime minister (or ‘premier’), although the two may or may not be politically allied. The prime minister and ministry are usually the dominant players in regard to most issues of domestic government, with the president typically responsible for national leadership, international representation, foreign relations and defence matters. The president is typically directly elected. The prime minister is typically chosen on the basis of the continuing confidence of a majority in the national assembly, although in some cases the president may be able to nominate the prime minister in the face of assembly discontent. The quality of representativeness of the national assembly may range from being broadly representative to being constituted by a pluralitarian or majoritarian electoral system. Major nations using this system include France, Russia, Indonesia, Nigeria, the Philippines, and (until 2019) Turkey.

preselection – any of a range of practices by which a political party selects one or more persons to be its candidate(s) for an election. Primary elections are a form of preselection.

president – the head of state, and often also the head of government, in presidential political systems.

presidential system – one of the categories of system of government by which nations provide for the exercise of the executive power of government. In these systems executive power is exercised by a president subject to only limited constraints from an assembly. The president must typically seek legislative enactments and budget approval through the national parliament. The president is typically directly elected, but might alternatively by selected by the parliament, selected by a special form of national assembly, or indirectly elected through an electoral college. Major nations using this system include the United States, Brazil, Egypt and Mexico. A variant of this form of government in which a distinct executive government led by a prime minister is formed in a national assembly, with or without the cooperation of the president, is termed the premier-presidential system. (Contrast with parliamentary system).

primary election – a form of preliminary preselection process, resembling a public election, conducted by or for a political party by which a person is chosen to be the candidate for that party for an upcoming election. A primary may be open to all voters of the electorate, or limited to registered party supporters (however defined) or to party members.

prime minister – (also premier): the chief officer of the executive government, other than a president if the political system is a presidential one where the president has executive (as opposed to constitutional) responsibilities.

proportional-preferential-personalized vote: (P3); a hybrid voting system, combining elements of the single transferable vote (STV) system with open party list seat allocation, proposed by Canadian politician Stéphane Dion in 2012 for possible use in electing the Canadian House of Commons (see Federal News, vol 3(4), April 2012). Within each multi-member electoral division (Dion proposed division sizes ranging from 3 to 5 seats) voter ballots would preferentially rank the contesting political parties, not the individual candidates. Voters would also mark a preference for one individual candidate among those nominated by their first-preferred party.

Within each division an STV-style quota would be used to determine how many seats each party was allocated. Ballots that preferred parties with a vote tally below a quota would be successively eliminated, and such votes transferred to the tallies of each ballot’s next-preferred other party, until there remained only parties which had won at least one seat. Dion did not specify whether the quota in question was a Droop quota (as is used in standard STV systems) or a higher simple (Hare) quota.

Dion wrote that this between-parties transfer process would “produce the percentages of votes that determine the number of seats obtained by the various parties”. It is not clear what Dion meant here. One possibility is that he meant that elimination of minor parties would proceed until – by analogy with individual-candidate STV – the number of quotas held by parties matched the available number of seats. This might possibly also include one or more ‘below-quota’ seat wins if any such below-quota tallies were greater than the surpluses held by more successful parties. Dion did not mention any process of transferring surpluses from parties with more than a quota in their vote tally at any point in this process. Alternatively, Dion may have contemplated a process of continued elimination and transfer of minor parties until there were no remaining parties with less than a quota of votes, at which point some seat allocation formula (either largest remainder or a divisor formula) would be applied to the vote tallies of the remaining parties to generate a seat allocation.

In any case, once the seat allocation numbers for each party within a division were known, the individual preference votes on all ballots cast for successful parties would be examined, and the best placed candidate(s) would be chosen as the individuals allotted the seats won by each party. The ballots marked for eliminated parties would, obviously, not be referred to in this final procedure.

proportionality – the extent to which the relative prevalence of different points of view across an electorate is matched by the relative numbers of members representing those points of view in an assembly. Since the early 19^th century, proportionality has been considered one of the most important measures of fairness for an electoral system, although some electoral systems still in widespread use, such as single member division plurality voting, make no attempt to achieve proportionality. Proportionality is most typically defined in terms of party-proportionality – the relative share of assembly seats held by political parties in comparison to the proportion of voters who support those parties.

The desire to achieve party-proportional outcomes was a key motivation for the development, in the period from the late 1850s to the end of the 19^th century, of alternatives to traditional rate voting systems, leading to the development of multi-member single transferable voting (STV) and also the party-based seat allocation systems.

In direct voting systems (ie: those in which voters vote for individual candidates, not for parties), party-proportionality is normally estimated by using the aggregate total of the votes for all the candidates of each party as a proxy for the ‘vote for’ each party. (For example, in multi-member division elections, if there are multiple candidates nominated for a party, the party vote is taken to be the sum of the votes cast for all the candidates of that party.) In preferential voting systems, party support is typically measured in terms of the first preference votes. Proportionality can be measured at the level of individual electoral divisions, or for an assembly as a whole. The optional use of thresholds for parties to be eligible to win seats decreases the overall degree of proportionality achieved.

Actual proportionality of results varies from election to election depending on the choice of electoral method and also the relative vote shares cast for each party. Proportionality can be measured by various statistical indices, ranging from a simple measure of the largest individual deviation between vote share and seat share (which is typically the overrepresentation of the largest party), to broader statistical indices covering variation for all parties, such as the Gallagher Index.

Electoral systems based on single member divisions generally achieve much lower degrees of proportionality than are achieved by the STV or seat allocation electoral systems. The party seat allocation systems only provide a means of providing proportionality as between parties, whereas STV systems can reveal proportions of support between a wider range of candidates or policy interests within the electorate.

psychological effect – the impact of the design features of a voting system on the decisions of voters as to which candidate or party to vote for. The classic instance of a psychological effect is with plurality systems, where a voter whose primary preference would be to support a candidate who is known to have little electoral support chooses instead to assess which two candidates are likely to be those receiving the largest two vote totals, and cast their vote for the candidate among those two which the voter most prefers. The term was first used in the work of Maurice Duverger.

Q – standard shorthand for a reference to the quota of votes needed to be elected (or for the allocation of seats to parties, or between states): used in the statement of a formula, for example the determination of a STV quota such as “Q = (V/(N+1))+1.

qualified plurality rule – a requirement relating to plurality, single-candidate elections in which a leading candidate does not win the election unless their vote share exceeds a specified minimum threshold, for example 40%. If this condition is not met, resort is had to an alternative procedure such as a runoff election between the top two candidates.[xxvi]

quota – the number of votes necessary for a candidate to be elected, for example in an STV voting system. Examples of specific quota formulae include the Hare quota, Droop quota, Hagenbach-Bischoff quota, and Imperialli quota.)

quota-Borda system – (QBS) : a hybrid of seat allocation by quotas and the Borda count voting methods for use with multi-member divisions. The method unfolds in four stages, of which the first three use a Droop quota as the primary criterion for election. The four stages are: (1) any individual candidate achieving the quota is awarded a seat. (2) any pair of candidates (including any who are awarded a seat at step 1) which together are the first and second preference on a set of ballot papers (in either order) and which collectively win two quotas of votes are both awarded seats: (3) among any pair of candidates (excluding any who are awarded a seat at steps 1 or 2) which together are the first and second preference on a set of ballots (in either order) and which collectively win one quota of votes are awarded one seat, with the member of the pair having the higher modified Borda count score receiving the seat: and finally (4) any remaining seats are awarded to the remaining candidates with the highest modified Borda count scores. Logically, at steps 2 and 3 no candidate may be a member of more than one set for those two stages (taken together). If the available seats are fully allocated at any of the first 3 stages then the later stages are disregarded. Note that the association with STV is strictly inaccurate – at no point do preference distributions occur as is the case with STV. The method was devised by the British philosopher Michael Dummett and published in Voting Procedures (1984) and again in his Principles of Electoral Reform (1997). The association arises because the QBS system was developed by supporters of STV seeking to address the problems arising from sequential elimination in standard STV.

quotient rounding methods – a family of apportionment methods used to allocate a predetermined number of seats between electoral divisions such as states or provinces before elections (based on population figures), or to political parties at elections (based on votes cast at an election). The simplest option within this class is the simple rounding approach. Other more advanced methods which use adjusted divisors to achieve an exact number of seats, and more precisely proportional results, include Webster’s method, Jefferson’s method, Adams’ method, Dean’s method, and the Huntington-Hill method.

quotient – an element of quotient rounding apportionment procedures, derived by dividing the population of each state by the common divisor. Each state’s quotient is then dealt with according to some quotient rounding method to generate the resulting numbers of seats allocated to each state.

R (or ‘r’) – a measure of reduction in proportionality (see Taagepera and Shugart 1989, p.273).

Rae’s index – one of the measures of proportionality between votes and seats won in seal allocation systems. The measure involves summing all the raw values for the different vote share percentages and seat share percentages for every party which wins more than 0.5% of the vote at the election, and dividing the total by the number of parties meeting that criterion. The measure, first proposed in 1971 as an alternative to the Loosemore-Hanby Index, tends to produce distorter results in elections with a large number of micro parties which exceed the 0.5% threshold. (See Gallagher: Proportionality, Disproportionality and Electoral Systems, 1991).

range voting – a form of rate voting in which each voter gives each available candidate a score out of a set range, for example a score from 1 to 10. The votes allocated to each candidate on the ballot are then tallied as averages (or as total votes/points – the result is mathematically identical), and the candidate with the highest average (or total) is the winner. The method can also be used to award multiple seats to the N highest-rating candidates.

rank voting – (also ordinal voting): one of the two fundamental classes of methods of casting votes (the other being rate voting). Rank voting is a class of voting methods including various forms of preferential voting for single positions (as in executive officers, or with single member divisions in the election of assemblies), including two-round runoff systems, as well as various forms of single transferable voting (STV) for multi-member divisions.

ranked choice voting – a term coming into common usage in the United States for preferential voting (using sequential elimination). The term ranked choice voting encompasses both elections for single-member districts (termed the alternative vote in the UK and Canada) and in multi-member districts (where the term means the same as single-transferable vote (STV) voting). The equivalent term instant runoff voting is also used in the US, primarily to refer to use in single-member divisions.

rate voting – (also categorical voting): one of the two fundamental classes of methods of casting votes (the other being rank voting). Rate voting is a class of voting methods including the single non-transferable vote (‘SNTV’), plurality voting (also ‘FPTP’), multiple non-transferable voting (including limited voting and block voting), range voting, approval voting and others.

Raynaud’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. Raynaud’s method is an elimination method based on Simpson’s Method. The candidate with the largest single pairwise defeat among remaining candidates is eliminated, and this step is repeated until only one candidate remains.

redistribution – (also redistricting, delimitation, or boundary review): the practice of redrawing the boundaries of electoral divisions, ideally with regard to specific criterion of electoral fairness, and renewed regularly. Redistributions are normally carried out at frequent intervals to minimise malapportionment emerging from demographic changes. The Venice Commission (a body established by the Council of Europe to assist with constitutional development and institutions of democracy) has proposed guidelines for good practice in regard to redistributions[xxvii], as has the International IDEA organisation[xxviii].

Reock compactness score – a measure of the geographical compactness of electoral divisions based on their map outlines, relevant to US disputes about whether electoral divisions have been gerrymandered. The score is calculated as the ratio a particular district’s area to the area of the smallest bounding circle that can be drawn to completely contain the district. The higher the score, the more compact the district.

referendum – a constitutional procedure by which the passage of a law is approved through a vote of the whole electorate. Very few national constitutions (those of Switzerland and New Zealand being examples) provide for the use of referenda for the passage of ordinary laws: the normal usage of referenda is only to approve amendments to a national constitution.

reinforced majority – the practice of awarding additional seats to the party (or coalition of parties) which has ‘won’ an election in the sense of recording the highest national vote total, or plurality. The purpose of this award is nominally to ensure that the most favoured party is assured of having a majority in the resulting assembly. Reinforced majority seats are by definition not directly elected, and are drawn from some form of party list. Seat reinforcement obviously makes an election result less proportional in terms of the ratio between the party numbers in the assembly and the election vote totals for each party. Variants include an option for additional seats awarded where the party with the most votes does not win the most directly elected seats (as in Malta), a straight bonus of seats to the national vote winner (as in Greece, where 40 seats are added to the 260 already allocated by a party list seat allotment method), or a modification during the counting process of the proportion of seats allocated to the party or coalition with the largest national vote (as in Italy, where the party list seat allocation is subject to the rule that the largest coalition is guaranteed at least 55% of the seats regardless of its vote share). Seat reinforcement can also take place at a regional electoral division level, rather than a national level (as occurs with the upper house elections in Italy).

representative-parliamentary system – one of the categories of system of government by which nations provide for the exercise of the executive power of government. In these systems executive power is exercised by a prime minister on the basis of the continuing confidence of a majority of members in an assembly which is elected through a broadly representative electoral system, rather than a plurality-elected system. Major nations using this system include Germany, the Netherlands, Poland and Spain.

Roberts’ Rule – a method for choosing between multiple alternative proposals, usually in the form of competing motions (including amended variants) at a meeting. The rule is essentially equivalent to identifying the Condorcet or majority candidate. Once all the competing propositions are identified, comparison votes between pairs are conducted (in any order), and any loser of a comparison is eliminated. New pairings (comparisons) of the remaining options continue until only two options remain, when a final vote determines the most favoured outcome. Unlike Condorcet votes, not all pairings need to be examined, as each losing proposition from any comparison can be eliminated as it is clearly not the majority favoured result.

Robson rotation – the practice of printing multiple versions of ballot papers with the candidates’ names appearing in different orders, such that the random donkey vote benefit of candidate names appearing first on all papers is eliminated. The number of ballot paper variants, and the numbers issued, is usually planned such that each candidate appears first in equal numbers of papers. With multi-member division ballot papers the method may be applied in multiple ways, such as by both randomising the order of party columns across a paper, and by randomising the order of candidates down the party columns. Randomisation may also be applied within each variant paper, so that the appearance of the next candidate following each candidate which appears at the top of the paper is further randomised. Obviously such additional sophistication multiplies the number of variants to be printed, which may present logistical challenges depending on the available printing facilities. The first legislative adoption of the method – in Tasmania in 1979 – was championed by a sitting member of the Tasmanian House of Assembly, Neil Robson MHA, after whom the technique came to be named.

Rouse’s method – one of the ‘recursive elimination’ variants of the Borda count, others including the Baldwin and Nansen methods. A Rouse count proceeds as in a Borda count, except that a winner is not declared after the initial count, but after each such count a two-stage process of candidate elimination ensues. In the first stage, the candidate with the highest Borda Score is temporarily eliminated from this stage of counting, and the Borda count re-conducted for the remaining pool of candidates (using a reduced array of scores aligned to each ballot’s order of preferences without the eliminated candidates). This process of counting continues until only one candidate remains, at which point this last candidate is identified as the ‘weakest’ candidate in the pool, and is permanently eliminated from the main count. The Borda count is then re-conducted for the remaining pool of candidates (again, using a reduced array of scores aligned to each ballot’s order of preferences without the eliminated candidates). The cycle of temporary eliminations of strongest candidates to reveal the weakest remaining candidate is then repeated. This two-stage process of identifying ‘weakest’ candidates for elimination continues until only two candidates remain, at which point the one with the higher Borda score (now simply equivalent to having a majority of the two remaining preferences) is the winner.

rural-urban PR – see flexible district PR.

top-up seats – a reference to additional seats allocated to political parties at an ‘upper tier’ regional electoral division as part of an MMP, British AMS electoral or similar electoral system. The term was used in Britain from the electoral design period of the late 1990s onward, and has also been used by Canadian electoral system designers.

Sainte-Laguë Index – an index for measuring the proportionality of an electoral outcome.

Sainte-Laguë series – (in Germany also termed the Schepers method or Sainte-Laguë/Schepers method): first proposed by Andre Sainte-Laguë in 1910, a divisor series used in the escalating divisor method for seat allocation to parties (or, unusually, to allocate seats to states or other electoral divisions within a nation). A reference to the Sainte-Laguë method or the Sainte-Laguë formula is thus a reference to seat apportionment by the escalating divisors method using this series. The Sainte-Laguë series requires that the number of votes for each party (or the population of each state or division) be divided by successive odd integers {1, 3, 5, 7, 9 etc} as the divisors. Sainte-Laguë is used in elections in Germany, Iraq, Latvia, New Zealand, Norway, Palestine and Sweden. The Sainte-Laguë formula escalating divisor method generates the same election results as the Webster modified divisor quotient rounding method. This formula does not inherently favour larger parties over smaller ones and is thus more proportional than use of the D’Hondt series, for which reason Sainte-Laguë is often considered the fairest of the apportionment methods. (See also the modified Sainte-Laguë series.)

salience – a measure of the real effective value of a voters’ vote, in terms of influence in determining which representatives are elected. A voter’s salience can be defined by reference to the size of the pool of voters with whom the voter must combine in order to change an election outcome. The mean value of all such pool sizes is the fair salience: voters whose pool size is larger than the mean have lower salience, and vice versa. Salience is directly affected (increasing the salience of the votes of some voters, while decreasing that of others) by phenomena such as concentration distortion (including gerrymandering) and malapportionment, but also relates to the extent to which a voter has any propensity to change her vote.

Scelba Law – an electoral method legislated in Italy in the early 1950s for use in elections for the nation’s national legislature. The system provided for distorted majority election results despite the electoral system nominally being one of party-proportional seat allocation. The law provided that if one party, or one allied group (using apparentment) of parties, won at least 50% of the total national vote, it was awarded (a minimum of) 65% of the available seats – 385 out of 590, with all other parties having their relative seat allocations reduced accordingly. Should a party or alliance secure the premium minimum of seats, the system required complex to allocate the seats among both the parties and the regional electoral divisions, causing anomalous results in terms of equal representation of the people per vote. The system worked strongly against the electoral interests of the nation’s less-supported parties, specifically the Communist party, and this was indeed the motivation for the adoption of the legislation by a combination of centre-left, centrist and centre-right political parties. The system did not, however, prevent the ill-discipline, party fragmentation and party mutation that was to follow in Italy in subsequent decades. Indeed, by creating more complicated opportunities for electoral gamesmanship, it may have led to complication of the political landscape. The technique of deliberate proportionality distortion has continued in use in Italy in various forms to this day, despite several modifications of the nation’s seat allocation electoral system. The only other nation to adopted any related form of deliberate proportionality-distortion electoral system is Greece.

Schulze STV – [pending]

Schulze’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. German mathematician Marcus Schulze’s method, proposed in the 1990s, uses the concept of beatpaths to resolve pairwise cycles. A beatpath is a sequence of pairwise victories linking one candidate to another, for example the candidate sequence A>D>B>E>C. Each beatpath has a strength equal to the strength of the vote margin at weakest victory it includes. The candidate who always has a stronger beatpath to any other candidate is the Schulze winner. Schulze’s method results usually agree with those of the Simpson Method, but Schulze’s method avoids certain paradoxical results occurring with Simpson. (The Dodgson and Simpson methods very rarely choose a Condorcet loser, but the Copeland, Schulze and Tideman methods never do.)

Schwartz set – a variant of the Smith set, a subset of a field of candidates each of whom would defeat any other candidate in the field. The Schwartz set expands slightly on the Smith set by including any candidates who would be undefeated by any candidate outside the set, rather than requiring strict victories, this allowing tied results to permit a candidate to be included in the set.

Scorporo – a form of mixed-member proportional (MMP) composite assembly system, combining a directly election component and a seat allocation component, with the special feature that the numbers of votes used to calculate the seat allocation to parties in the allocated component of the assembly involves deducting some or all of the votes which have contributed to the successful election of members to seats in the directly elected component. The amount of deduction to be used is optional; examples include the whole of the vote for winning candidates (as currently used in Austria), or an amount equal to the second place candidate’s vote total +1 (as in the ‘Mattarellum’ system formerly used in Italy). A scorporo system  necessarily uses the single-vote, not a dual-vote, version of MMP. Issues such as party list form, any minimum threshold for eligibility to receive sets, and the formula for seat allocation calculations can be selected from any of the options otherwise used in seat allocation systems. The purpose of the system is to re-use the ‘wasted votes’ from the directly elected component to secure a form of proportional representation for minority parties.

Scorporo has been used to elect national lower houses in Italy, Austria and Hungary in recent decades, but is now used only in Austria. When used in Italy’s 2001 election a controversial defect in the rules, which allowed district candidates to be associated with ‘decoy’ parties so that their votes could be re-used in the later proportional allocation of seats, resulted in perverse outcomes which ultimately led to the system being abandoned.

seat allocation – a process for allocating numbers of assembly seats either (i) among states or other electoral divisions within a nation; or (ii) to political parties, as part of an indirect form of election for an assembly. Both exercised require statistics of the several populations (for states or electoral divisions) or tallies of votes cast (for parties), expressed either as absolute numbers or percentages.

One of several possible methods of apportionment is then applied to the population or vote numbers to generate seat outcomes. The simplest formulae available is the simple quota and largest remainder method. This method uses a divisor, normally calculated with reference to enrolment or formal vote totals, and a specific quota formula, of which the simple quota or the Droop quota are by far the most common. These systems use a remainder technique, generally either largest remainder or majority remainder, but simply rounding down to the integer below is also an option.

Alternatively, the apportionment calculations can be done in a single step through either a modified divisor or an escalating divisor approach.

When used as an indirect election system, seat allocation is a common means of awarding assembly seats to political parties based on the electorate voting to indicate support for those parties, rather than voting for individual candidates. Such elections, usually known as party list electoral systems, are thereby distinct from methods of direct voting for individual candidates (see rate voting, rank voting, and the various specific methods of voting which belong to those two classes). The means of expression of voter support for parties is, however, normally itself a form of voting by ballot, and in all elections for national assemblies currently in use the specific voting method used is equivalent to the single non-transferable vote. Such voting yields a set of vote totals (or vote share percentages) for the parties, which is then used in seat allocation calculations such as those described above.

Seat allocation of this kind was used for allocating seats in the US House of Representatives among the states as early as 1788, and the comparative virtues of the largest remainder method, as well as various modified divisor methods, were well understood by around the 1840s.

The use of seat allocation for allocating seats to political parties was being contemplated by the mid 19th century, and was discussed by US and British scholars including Thomas Hare (1860) and Henry Droop (1869). However the first serious proposal for its use was by Victor D’Hondt in Belgium in 1880, at which point the first escalating divisor methods came into view. D’Hondt’s method was first actually used in Belgium in 1899.

seats-to-votes curve – a measure of partisan bias between political parties in elections, putatively caused by gerrymandering of electoral district boundaries, primarily used in the United States to analyze election results at a state level. The measure compares the share of seats won by a party in a specific election to historical averages based on that party’s vote share across the whole electorate. While the electorate in question is typically a state, calculations could also be applied to a nation as a whole. A relationship between a party’s average share of the statewide vote and its share of seats won can be modeled by drawing a curve from the past election result data. Vote shares from subsequent elections can then be mapped onto the curve to yield an expected seat share. Finally, comparing the expected seat share to the actual seat share can reveal the degree to which the outcomes arising from new electoral district boundary maps deviate from the results arising from historical predecessors. Significant changes in seat outcomes can thus provide evidence of recent deliberate gerrymandering of the district boundaries. The measure is not, however, able to take into account natural shifts in demography or in political opinions. The measure has been referred to in academic literature since at least 1973. See also efficiency gap and mean-median district vote share difference, which are other attempts to measure partisan bias in district boundary-making.

A detailed discussion of the measure is provided in Extreme Maps, Laura Royden and Michael Li, Brennan Centre for Justice, 2017.

See also partisan symmetry, the efficiency gap and the mean-median district vote share difference, which are other methods to attempt to measure partisan bias in district boundary-making.

seats/votes ratio – in political science, estimates of party proportionality use the ratio of seats won to votes won by parties as a key metric. Electoral systems where these ratios are significantly different from 1:00, either for key major parties or as averages across all parties, are said to be ‘disproportional’ electoral systems. There are various indices used for measuring the degree of disproportionality.

secret ballot – any procedure for conducting elections in a manner where each voter’s vote is confidential. Through the development of parliamentary systems of democracy in recent centuries, early election systems often involved procedures lacking confidentiality for individual voters , such as voice votes in public places or the recording of votes in a register. By contrast the secret ballot procedure typically utilizes pre-printed ballot papers, which may be blank ballots prepared officially or may take the form of partisan papers distributed to candidates by parties or candidates. The printed secret ballot practice was first utilized in the Australian state of Victoria in 1852, and was promptly adopted throughout the (then) five Australian colonies within a decade. Following this origin, in the United States the secret ballot practice was initially termed the ‘Australian ballot’. By the start of the 20^th century the practice was near-universal throughout electoral democracies, and all modern electoral systems formally provide for secret balloting in elections.

segmented distribution (of preferences) – a technical practice used in counting of votes under the single transferable vote (STV) voting system, particularly in Australian elections. Party-way through a count – especially towards the end – a number of candidates will have already been elected by reaching the vote quota. As a result, the surpluses from their final vote tallies will have been redistributed, causing the ballots transferred from the successful candidates(s) to have a vote transfer value lower than 1 vote. The remaining pool of ballots in the count therefore becomes a mix of ballots weighted at 1 vote and ballots weighted at a lower value(s). When each subsequent candidate is elected and has a surplus to hand on, the practice of ‘segmentation’ means that the surplus are now distributed in two or more seperate transfer calculations, giving priority to all ballots that still have a transfer value of 1 vote (which means that the just-elected candidate was the first successful candidate that these ballots had supported), followed by the transfer of all ballots at the next highest remaining transfer value, then the ballots with the next highest transfer value, and so on. After each such transfer a re-tallying of the new total of votes for the remaining unelected candidates is done. The specific result of this careful segmentation is that where two (or more) candidates are elected as a result of an elected candidates surplus transfer, it may be possible to determining which election came ‘first’ by reference to the seperate sub-stages of counting. That determines the order of election of the candidates and therefore the order in which their surplus transfers are conducted, which in unusually circumstances can have an impact on which candidates are elected in the late stages of the count.

sequential STV – [pending]

sequential elimination – (also Hare elimination or Ware elimination, both archaic): in preferential voting, the staged vote-counting practice of eliminating the candidate with the lowest vote total and transferring their votes to other un-eliminated candidates, and repeating this step until only two candidates remain. If a candidate has won 50% of the formal vote at the first count of votes, or if after any stage of the counting they achieve that target (or at least achieve 50% of un-exhausted votes if the optional preferential method is being used), they can be declared elected and further eliminations are strictly unnecessary, although vote counting often proceeds to determine a final two-candidate tally. The process is also used in the single transferable vote (STV) voting system, where candidates are eliminated until the required number of votes are allocated to successful candidates in full quotas (or alternatively, until the number of remaining candidates is equal to the number of seats to fill). The concept of sequential elimination was proposed by Thomas Hare in regard to the STV system he publicised in 1857 and 1873, and was proposed by William Ware for use in single-member elections in 1870.

signalling – a tactical action by voters in a two-round runoff voting system where, satisfied that their preferred candidate is assured of being among the best-placed two candidates and therefore will be able to contest a second-round runoff, votes in the first round for a selected minor candidate with the intention of signalling to their primary candidate their support for specific policy positions which the minor candidate represents. As many as 8.5% of French voters are estimated to engage in signalling in the first round of National Assembly elections.[xxx]

Simple quota – (also Hamilton quota or Hare quota): the most basic form of quota (or divisor) for determining the allocation of seats in an assembly, defined as Q = (P/N), where Q is the quota, P is the population (or equivalent such as formal vote total), and N is the number of seats available. The quota can be used to allocate a nation’s total pool of seats among states or provinces (or other electoral divisions) on the basis of their population, or to parties on the basis of votes won. (See also Droop quota, Hagenbach-Bischoff quota and Imperiali quota.)

Simpson’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. Simpson’s method chooses as winner the candidate with the smallest maximum pairwise defeat. Simpson can be thought of as successively ignoring the smallest defeat until one candidate is unbeaten. (The Dodgson and Simpson methods very rarely choose a Condorcet loser, but Copeland, Schulze and Tideman never do.)[xxxi]

simultaneous elimination – in a preferential voting system in which candidates are eliminated during the counting process, any round at which multiple candidates are eliminated together on the basis of some rule, for example a threshold minimum of votes. The STV system used for the City Council of Cambridge, Massachusetts uses such a rule, known as the ‘simultaneous drop’. Logically, systems of instant runoff voting in which all but the highest-placed two candidates are all eliminated forms a specific instance of simultaneous elimination, although the term is not used in regard to these systems.

sincere voting – the genuine preference of a voter in regard to the casting of their vote, in terms of rankings (or of rate voting) for the available candidates. Many voting systems carry incentives (including either ‘mechanical effects’ or ‘psychological effects’) for voters to abandon sincere voting and instead engage in strategic voting.

single choice vote – a generic term for the selection of a single candidate to a position, or (with regard to referenda) the selection of a single policy proposition from amongst rival propositions.

single non-transferable vote – (SNTV); a form or rate voting in which each voter expresses support (typically by marking an ‘X’ on the ballot paper) for just one candidate, but from which multiple members are elected, with the seats being awarded to the N candidates with the highest tallies of votes. Strictly, an election for a single position can also be classed as an SNTV form of voting, but such elections are more normally referred to by the terms plurality voting or ‘first-past-the-post’, while the term SNTV is used in relation to multi-member elections.

single transferable vote – (STV); in the US and Canada also recently termed ranked choice voting (although in the US that term refers to preferential voting generally and thus also includes reference to such voting in single-member divisions); in the US also sometimes termed choice voting; in Australia often termed the Hare-Clark system, and also sometimes termed PR-STV; also known to specialists as the quota-preferential method of proportional representation).

Whilst in the broadest sense a reference to a ‘single vote’ that is ‘transferable’ might refer to any preferential voting system, the term STV universally refers to the electoral method of preferential voting for candidates in multi-member divisions based on quotas for election, transfers of surplus (ie: above-quota) votes from elected candidates to other candidates (according to each voter’s ranked preferences), and likewise the transfer of votes following the sequential elimination of the least successful candidates.

STV was the origin of the term ‘proportional representation’, which initially meant that the influence of every individual voter’s vote was equal to (or ‘proportionate’ to) every other vote. The term later came to acquire a second meaning in relation to political parties, this second meaning being concerned with whether the numbers of seats that parties held in an assembly after an election was mathematically proportional to their aggregate voter support at the election.

STV has been highly popular with electoral system specialists since the late 19th century, and has been the official policy of the UK and Australian electoral reform and proportional representation societies throughout their existence.

STV forms a family of methods rather than a single uniform system. The main elements of the method were invented and/or promoted in the 19th century by Thomas Hill (Birmingham, 1819-21), Rowland Hill (London, but for use in Adelaide, 1840), Carl Andrae (Denmark, 1855), Thomas Hare (London, 1857-73), and John Stuart Mill (London, 1861 onward).

Thereafter Catherine Spence (Adelaide, 1861 onward), Henry Droop (London, 1868-84) and Charles Dodgson (London, 1884) promoted the use of the system in defined multi-member electoral divisions only. John Gregory (Melbourne, 1880s) provided enhanced counting approaches for use in the transfer of preferences. Enhancements in the 20th century include conceptually neater counting methods (made possible by computing technology) proposed by Brian Meek (Britain and France, 1969) and later refined by others. Ballot paper design choices that avoid favouring any candidates unfairly were proposed by Neil Robson (Tasmania, 1979).

The method is used in Eire, Northern Ireland, Malta, for the Australian Senate and various Australian State assemblies, for local government elections in Australia and Scotland, and in many US municipalities. STV was used to elect part of the Legislative Assemblies of Alberta and Saskatchewan from the 1920s to the 1950s, and the whole of the Legislative Assembly of New South Wales in the 1920s.

See also Hare-Clark system.

single-member division – an electoral division for which 1 member is elected, typically by the plurality voting method or by one of the forms of preferential voting.

slate vote – see party block vote.

Small’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. Small’s method calculates scores for the candidates as in Copeland’s method. If more than one candidate ties with the best Copeland score, the other candidates are eliminated and the scores are recalculated. This step is repeated until no more candidates can be eliminated.

smallest divisors method – see Adams’ method.

Smith set – in the examination of a preferential ballot to determine the Condorcet winner, the set of all candidates who achieve the highest number of pairwise victories over other candidates. In most contests the ‘set’ will consist only of the Condorcet winner, but if there is a Condorcet cycle there may be 3 (and hypothetically even more) candidates in the set. Cases of 2 members of the set are logically possible, but require the extraordinary occurrence of two distinct Condorcet cycles within the same pool of votes. Named for American mathematician John H Smith, who defined the corresponding ‘Smith criterion’ in 1973.[xxxii]

social choice theory – a theoretical system, originating in economic theory (especially welfare economics) and blended with political and electoral science concepts, which proposes ways of measuring individual interests, values, or welfare outcomes in the course of collective decision-making (typically democratic elections, although social choice methodologies also apply to small-scale decision-making procedures). Kenneth Arrow’s paper Social Choice and Individual Values (1951) is generally credited with the initiation of the modern social choice theory, although thinking as early as that of Condorcet in the 18^th century is conceptually related. A range of theorems and propositions developed by social choice theorists (including Arrow’s theorem and its derivatives) propose conclusions about the suitability of various voting systems to achieve any of a variety of proposed criteria of merit. A range of voting power indices developed by social choice theorists (including the Banzhof index) attempt to present estimates of the relative political strength of parties according to their numbers in a decision-making assembly.

sortition – the practice of appointing political representatives by random selection, rather than by election, as in selection of juries. Sortition was dudes in ancient Greek political systems, and has recently come back into favour with some political reformers, and been used in some countries for the selection of community representatives to debate specific public issues.

spatial theory of voting – the theoretical premise that the political views in an electorate lie along a one-dimensional spectrum. The premise gives rise to various axioms in social choice theory, including that of the median voter.

standard rounding – (also arithmetic rounding and as ‘rounding using the arithmetic mean’) : the practice of rounding quotients with remainders below 0.5 down to the integer below, and rounding quotients with remainders of 0.5 and higher up to the integer above.

STAR voting – a possible hybrid voting system for electing individuals to single-winner positions put forward by US statistician and electoral reform advocate Jameson Quinn in 2017. The system would initially resemble an exercise in range voting, in which each candidate would be given a score of between (for example) 0 and 5. The average score that voters gave for each candidate would then be calculated, and the two highest-placed candidates would become ‘finalists’. The ballots would again be used as if in a simple two-candidate runoff, with the finalist rated highest on each ballot receiving the vote (unless both finalists were equally rated; such ballots could simply be set aside in the final comparison). The method is therefore comparable with ordinary preferential or ranked voting in that it provides an ‘instant runoff’ result. However unlike those approaches STAR voting would have the defect that the initial range scores are subject to gaming through political parties urging their supporters to give artificially low rankings (such as 0) to their key opponents, and artificially high ratings to weaker opponents whom they would prefer to be compared to in the final two-candidate comparison.

state – (also province, region, district, département or canton (French), land (German) and others): the primary (at least for electoral purposes) administrative subdivision within a nation.

strategic voting – any practice by voters of altering their ‘true’ preferences between candidates, and casting votes in some other way than they genuinely wish to, in reaction to either the design of the voting system (the ‘mechanical effect’) or to the anticipated voting decisions of other voters (the ‘psychological effect’). The causes of strategic voting can include various factors such as the electoral method in use, the current party configuration, the available candidates, or the expectations of the voter or the general public about the results of the election. Strategic voting is a common occurrence in elections conducted using plurality voting systems, where the nomination process and the established party configuration strongly suggest that only the candidates of the two most popular parties have a prospect of success, leading supporters of minor candidates to shift from their preferred vote. (Plurality voting systems tend to reinforce this problem by maintaining strong two-party configurations – see Duverger’s Law).

An example of strategic voting occurring in a party list seat allocation system is the tactic of New Zealand’s National Party voters in the 2011 elections (on the advice of the leaders of the party) supporting a candidate of the minor ACT NZ party for one local electorate (where the result was to be determined by plurality voting) in order that the party win a single divisional seat and thereby achieve eligibility to win party list seats at the national level (which its national vote would be insufficient to achieve), and thus be present in larger numbers to support the major party in the resulting Parliament.

strongest path – [pending]

STV-Borda – a hybrid of STV and the Borda count voting method. The system is identical to STV except that in place of sequential elimination as the method of eliminating a candidate (when such a step is necessary), the candidate selected for elimination is the one with the lowest Borda score. The Borda scores for each candidate for this purpose are determined at the beginning of the process prior to the STV counting getting underway. The method was proposed by Australian economist Chris Geller (see Geller 2002).

sufficient divisors – in a process of apportionment for seat allocation, any value within a range (or ‘interval’ in mathematical terminology) which, when used as the divisor in a modified divisor method of quotient rounding, will result in precisely the pre-determined total number of seats.

supplementary vote – a particular form of limited preference voting in which voters are only permitted to indicate preferences for 2 candidates regardless of the total number of candidates. This method is currently used to elect the Lord Mayor of London.

surplus transfer – (also surplus distribution): the practice in single transferable vote (STV) voting systems of transferring a number of votes credited to an elected candidate (which can include first preference votes, and/or votes received through previous surplus transfers from other elected candidates, and/or votes received through transfers upon the elimination of other candidates) to the ballot paper’s next most preferred (un-eliminated) candidate, so as to leave the first-mentioned candidate with precisely one quota of votes.

Under STV counting rules the first candidate (referred to here as the ‘elected candidate’ for clarity) is said to be ‘declared elected’ at the counting stage at which they accumulate a quota or more of votes, and the next stage of counting must always be the transfer of the surplus, prior to any further candidate eliminations. (If at the first preferences stage, or at any later counting stage, two or more candidates simultaneously achieve a quota, the surpluses are distributed from such candidates in turn in descending order of their vote totals.) The technique is integral to the process by which STV counting distributes votes among candidates to achieve proportional representation.

Various techniques can be employed in relation to which ballot papers are examined as part of the transfer, and what weight of votes are allotted to them. Firstly, the ballots to be used for the transfer might be based on random selection, as was proposed by Thomas Hare in his original model. A variant form of random selection is the ‘Cincinnati’ random approach, in which every 11^th ballot in shuffled piles of ballots is selected, until the required number is selected. Alternatively, in determining the recipients of the transferred votes the scrutiny may examine all the ballots, or may examine some pre-defined subset of the ballots, which will require some form of weighting to ensure that the total value of votes transferred is the correct number. The most basic option, generally referred to as ‘inclusive’ approach, is to examine all the ballots credited to the elected candidate, regardless of which stage of votes or preference transfers they were received. A convenient alternative is to use the ‘last parcel’ of votes which – when received by the elected candidate – raised their vote tally to exceed the quota.

There are variant ways of weighting the ballots in inclusive or last parcel procedures. The primary method is the fractional method, which simply applies a fractional value to each ballot equal to the ratio of the surplus to the whole pool of votes concerned: by this means the ‘value’ of votes equal to the quota ‘remains’ with the elected candidate while the ‘value’ equal to the surplus is distributed to other candidates. Note that if the ballot pool making up the elected candidate’s tally includes any ballots transferred to that candidate from prior surplus transfers at earlier stages of counting, the fractional value of those specific ballots should be taken into account, such that ballots which have passed through two or more surplus transfers will have diminishing weightings equal to the product of all the surplus transfer weightings applied to them.

Finally, an alternative approach designed to avoid the need for ballot weighting is the Wright method, which is based on the concept of iteratively recounting the entire pool of ballots afresh after each elimination of a lowest-ranked candidate, rather than of progressively accumulated vote tallies through multiple’ stages’ of vote counting.

swingometer – see pendulum.

system of government – a system by which a nation (or other jurisdiction) provides for the exercise of the executive power of government, and establishes its relationship with the role of the national parliament. The traditional division of systems is into two broad classes, parliamentary and presidential. These can be subdivided into four more detailed categories, each of which features a different balance of power between an independent chief executive and control by a national assembly. In decreasing order of parliamentary control of the executive these four categories are the representative-parliamentary system, the plurality-parliamentary system, the premier-presidential system and the presidential system. Presidential systems vary greatly in the effective degree of democratic control that citizens have over their government. For completeness two categories of non-democratic system can also be defined: the party state system and the monarchical system. Many nations have nominal or ‘constitutional’ monarchies, but for practical purposes these may be classified with one of the parliamentary systems.

tactical voting – in plurality voting systems, the decision by a voter whose most preferred candidate is known to have insufficient electoral support to win the election to cast their vote not for that most preferred candidate, but for one of the two candidates thought best-placed to have a realistic chance of winning.

threshold MNTV – a majoritarian version of multiple non-transferable voting in which each voter may cast up to as many rate votes as there are candidates, and any candidates who achieve a threshold of 50% of the average number of votes cast per voter (that is, the total number of votes cast by all voters divided by the maximum number of votes each may cast) is allocated a seat, with any remaining seats filled by some alternative method (often through a second ballot). The result is that every elected member will have been ‘approved’ by at least half the voters. This system is used in many municipal elections in Switzerland.

threshold – a formal minimum requirement – in terms of votes won at an election – for parties to be eligible to win any seats. Formal thresholds (as distinct from implicit thresholds) are actual rules eliminating from the distribution of seats all parties which fall below the relevant vote target. In related cases, used in parallel voting systems such as in Germany and New Zealand, an alternative threshold rule may also deem a party eligible to be allocated party list seats if it wins a set minimum number of local district seats. Another form of threshold is that used in the two-round runoff system in France, where (if there is no majority winner in the first round) all candidates who achieve a 12.5% threshold (not merely the candidates with the two highest votes) may advance to the second round.

Ticino – an Italian-speaking canton of Switzerland. To address political unrest in the period 1870-1891 a form of representative voting was granted which constituted the earliest attempt to implement a system of party list seat allocation. The process was essentially that which became known as the free list system.

Tideman’s STV – see CPO-STV.

Tideman’s method – one of the pairwise comparison winner determination rules, designed to declare a winner where no Condorcet winner is present. Nicholas Tideman’s method orders the pairwise victories from strongest to weakest and ‘locks’ them in order, ignoring those that contradict stronger victories (ie: as the victories are examined in order, any later victory which contradicts an earlier one is ignored). Tideman’s method then picks as the winner the candidate who has locked victories over all other candidates. (The Dodgson and Simpson methods very rarely choose a Condorcet loser, but Copeland, Schulze and Tideman never do.) [xxxiii]

tiered systems – any form of electoral system which uses two or more levels of electoral division as the basis for electing members to, or allocating seats in, an assembly, but using the same type of electoral system at each level. Such systems should not be referred to as ‘parallel’, ‘mixed’ or ‘composite assembly’ systems, as these terms indicate the use of different electoral methods for different components of the assembly. The most common form of tiered system are those using seat allocation systems at both a regional and a national level, for example South Africa, Denmark and Sri Lanka. Some systems may be described as having ‘integrated tiers’, in which the counting procedure for lower levels directly links to that used at higher levels, a key example being the system used in Austria. Papua New Guinea elects its assembly in two groups (local district MPs and also regional governors who become ex officio MPs), with the same limited preferential voting method used for both groups.

turnout – see participation.

two-party preferred vote – (Australian usage); in a preferential ballot for a single position which is contested by candidates from each of a system’s two major political parties (and typically other candidates as well), the two-party-preferred result is the percentage support each major party achieves after the preferences of all other candidates are re-distributed between those of the two major parties. In the common circumstance where the two major party candidates are the two highest vote winners, distributing the preferences between these two rivals reveals the extent of the winner’s underlying preferential support in the electoral division against their main rival. Note that in many cases the identity of the winner may have been clear after the count of first preferences (ie: because one candidate won more than 50% of the votes), regardless of the ‘2PP’ result.

The Australian Electoral Commission has fully counted two-party preferred results for all House of Representatives elections since 1984. In cases where the full distribution of preferences results in one or more of the two final highest vote-winners not being a major-party candidate, the balance between the actual two finalists is termed a two-candidate preferred result, and a special distribution is separately conducted between the candidates of the two major parties for purposes of analysis only. Finally, the sum of two-party-preferred results for all divisions across the nation forms the aggregate two-party preferred national total, a statistic often used for political analysis and comment.

two-round runoff – a form of preferential voting for single positions, commonly used to elect presidents, and also occasionally used to elect members to an assembly in single member divisions. Two-round runoff is closely equivalent to the contingent vote preferential voting method, but is conducted over two rounds of voting on different dates. Ballot papers are marked in the first round as in plurality voting. If a candidate secures 50% of the formal votes in the first round, they are declared the winner. If no candidate achieves 50%, the two highest-placed candidates contest a second round of voting, at which the highest-placed will then be the winner.

Unlike the contingent vote by a single ballot, the pool of electors who participate and cast formal votes may be different in the second round from that which voted in the first round. Voter participation (turnout) is often lower, as supporters of excluded candidates do not all wish to express support for either of the remaining candidates. Voter participation fatigue may also be a factor, as may be the situation that the results of the assembly election as a whole are largely determined by first round results and publicly known, if that is the case. Conversely, the election of a national president on the same dates may act to keep turnout strong.

Variations of the threshold for entry into the second round are possible, such as with the elections to the French Assemblée Nationale, where the threshold is 12.5% of the registered voters in each electoral division, often allowing three and sometimes four candidates (hypothetically as many as seven) into the second round, which is then determined by plurality voting.

Second rounds are also used in the threshold variety of plurality voting, in which a minimum threshold (usually of 40% or 45% of the formal vote, or perhaps of a lead of 10% over the next highest candidate) is required to win at the first round of voting. If the lead candidate fails that requirement then a second round of voting is held between the two leading candidates. Argentina uses this threshold method to elect its presidents.

At present the use of the two-round runoff system is common for electing national presidents. The use of two-round runoff for parliamentary elections is mainly associated with France and with several former French colonial territories, although Byelorussia, Uzbekistan and Bhutan have also adopted it. It was used for elections to the New Zealand House of Representatives in 1908 and 1911. The system is also used in the US states of Louisiana and Georgia at present.

Überhang seats – a feature of MMP electoral systems; see overhang and compensation seats.

unrestricted domain – a criterion of social choice theory. In regard to ranked voting systems (or as social welfare theorists would express it, of ‘social welfare functions’) a requirement that all the votes (whether ranked as preferences or indicated by some rate voting method) allocated to all candidates by every voter is taken into account (in some way), so as to yield a ranking of the collective voter choice, which ranking will be both unique and ‘complete’. Unrestricted domain is one of the four criterion used in Arrow’s theorem.

V – standard shorthand for a reference to the number of formal votes in the statement of a formula, for example the determination of a STV quota such as “Q = (V/(N+1))+1.

Vinton method – see largest remainder method.

vote-eligible population – (VEP): an estimate of the population of a nation which is, using every applicable legal definitions and restrictions (ie: citizenship, age, residency, absence of criminal convictions etc) is potentially eligible to vote. Note that not all members of the VEP will be enrolled (or ‘registered’) to vote. See also voting age population.

vote wastage – the extent to which votes cast in an election fail to elect representatives. The term is primarily used in commenting on elections using plurality voting in single-member electoral divisions (first-past-the-post), where the extent of wastage is very high, although for comparative purposes vote wastage rates can also be calculated for electoral systems with much better representation outcomes. Vote wastage can be calculated as the sum of all votes cast for unsuccessful candidates (which therefore fail to elect a representative) together with votes cast for the sole successful candidate in excess of the number needed to win the seat (that is, votes in excess of the number of votes cast for the largest runner-up candidate). See also efficiency gap.

voting age population – (VAP) : an estimate of the population of a nation which is of voting age (usually 18 or over). VAP may not be an accurate measure of the potential electorate because of the application of other criteria for eligibility of vote such as citizenship, residency, absence of criminal convictions etc. The estimated vote-eligible population is more useful as a measure of the total size of the electorate, against which enrolment, participation and formal voting results may be statistically compared.

voting power index – any of several formulae (such as the Banzhof index) developed by social choice theorists to attempt to present estimates of the relative political strength of parties according to their numbers in a decision-making assembly. The key focus of such indices is the reality that effective influence is not the same thing as voting numbers. The indices are generally values between 0 and 1, whereby 0 represents a total lack of political power and 1 indicates total control. The indices are generally based on simple numerical counts of number of multi-party alliance scenarios in which a party may find itself in the majority group, without regard to factors such as the real political likelihood of specific combinations. The indices are of significant interest in nations with multi-party political systems. In majority-controlled parliamentary assemblies, the majority party would be defined by these indices as having voting power of 1, and all other parties as having 0.

Ware elimination – see sequential elimination.

Warren’s method – a specific method of managing the vote transfer calculations in STV elections, developed by CE Warren in response to Meek’s method.

wasted vote – the converse of effective vote: any vote which has no effect in determining the election of successful candidates, at least at the final stage of counting of an election determining whether one candidate is elected and another not. In plurality elections, any vote which is cast for neither the winning candidate or the runner-up is generally regarded as a wasted vote. The expression highlights the diminished influence of voters whose true preference is to vote for a minor party or independent in an electoral contest where major parties (normally just two) dominate the contest. Such voters are under pressure to cast tactical votes for one of the dominant parties/candidates.[xxxiv]

Webster’s method – (also the method of major fractions): an adjusted divisor form of quotient rounding apportionment method. This method determines an initial divisor by dividing the national total population by the intended total number of seats, rounding all the resulting quotients up or down according to standard rounding, and taking the resulting values as the seat allocation to each state. The method is named after its proponent, US Senator Daniel Webster, who in the 1830s proposed it for use in allocating the seats in the US House of Representatives to the several states of the Union. The method is mathematically equivalent to the Sainte-Laguë escalating divisors method in that the two approaches always yield the same results.

winning on preferences – in a preferential voting system, a victory for a candidate who did not achieve a first preference win, but wins after the distribution of preferences.

within quota – a criterion for evaluating quotient rounding methods of apportionment. An apportionment of seats to a state is said to be ‘within quota’ if the exact number of seats allocated to the state is either the integer above or the integer below the exact value of that state’s quotient.

Woodall free riding – a concept, or voting tactic, occurring in STV voting, in which in specific circumstances a voter can strategically maintain the full value of their vote in the later stages of the vote counting compared to other voters. A Woodall free rider is described by Markus Schulz as “a voter who gives his first preference to a candidate who is [expected] by this voter to be eliminated early in the count even with this voter’s first preference. With this strategy this voter assures that he does not waste his vote for a candidate who is elected already during the transfer of the initial surpluses.” Woodall free riding does not occur if the specific Meek, Warren, or Wright STV counting methods are used.

Wright method – a variant approach to vote counting in STV voting systems, described by Australian psephologist Antony Van der Kraats, named by him after earlier Australian psephologist Jack Wright. The system aims to replace the standard method in which the vote count is progressed through multiple stages, in each of which votes of varying fractional values accumulate in each candidate’s vote tally. Under the standard approach, ballots that have passed through two or more surplus transfers will have been repeatedly weighted and have small ‘values’ attributed to them. An academic debate between electoral system designers (Meek, Warren and others) over which pool of ballot papers to use, and which corresponding weights to allot them, led Van der Kraats to propose an approach which escaped the perceived difficulty altogether.

In the Wright method each of potentially several vote counts nominally have only a single stage, consisting of the identification of candidates with quotas, and only one round of surplus transfers from those quota-holding candidates (potentially generating additional quota-holding candidates). Where a vote count fails to produce a number of candidates with quotas equal to the number of seats available, a single candidate (that with the lowest vote tally) is eliminated, their votes transferred to the next preference on the ballots in the usual manner, and a fresh ‘single round’ count of first preferences and surplus transfers is conducted. The process continues until the number of un-eliminated candidates begins to approach the number of available seats, and a vote count round must eventually occur where the vote tallies for each candidate followed by a single round of surplus transfers is sufficient to produce the required number of seat-winning candidates. The winning candidates are then all ‘declared elected’ at that final count.

This description gives the appearance of considerable extra counting effort, but in fact the actual counting and tallying work is more or less identical to the standard method of counting, and really requires only a different presentation of vote tallying tables (and computerisation of the tallying data makes this a simple task). The Wright method has the additional benefit, compared to the standard counting method, that it can be used neatly with the optional preferential balloting rule, with the result that the quota would be recalculated automatically for each vote count to take into account exhausted ballots.

References

[i] Arrow 1950, Arrow 1951
[ii] Penrose, LS, The elementary statistics of majority voting, Journal of the Royal Statistical Society 109, 1029-1050 (1946)
[iii] Banzhaf, John F, Weighted voting doesn’t work: A mathematical analysis, Rutgers Law Review 19: 317–343 (1965)
[iv] Coleman, J, Control of Collectivities and the Power of a Collectivity to Act, in: B Lieberman (ed.), Social Choice, New York: Gordon and Breach, 269-300 (1971)
[v] McLean 2004
[vi] 2010 election results.
[vii] Pringle 2011- Is Compulsory Voting an Illusion
[viii] John and Hargreaves 2011, analysis of hypothetical results of recent South Australian House of Assembly elections using alternative forms of preferential vote counting.
[ix] Fairvote 2012 (2)
[x] Fairvote 2012 (2)
[xi] SeeWilliam Riker, Duverger’s Law Revisited, in Grofman and Ljiphardt 1986.
[xii] Adapted from Maurice Duverger, Political Parties: their Organisation and Activity in the Modern State (1965) using the terminology of the present treatiste.
[xiii]Effective number of parties: a measure with application to Western Europe, M Laakso and Rein Taagepera, Comparative Political Studies, vol 12 (1979), pp 3-27.
[xiv] OPPD 2011, p 10
[xv] Wikipedia – ‘Gallagher Index’, 19 January 2012
[xvi] Shiratori 1995, p.81
[xvii] See ID Hill, 2011
[xviii] See ID Hill, 2011
[xix] There is still a degree of rural weghting in the divisions for the Queensland Legislative Assembly and the Legislative Council of Western Australia.
[xx] The law relating to the Australian House of Representatives requires intra-state boundaries to be reviewed at least every 7 years, and boundaries are constrained by tight margins for variance (a permitted variance of 10% from mean projected enrolment figures of each division at the date of boundary determination, together with a targeted variance of just 3.5% at the point 3.5 years from that date.)
[xxi] Engstrom and Engstrom 2008, p,415
[xxii] McLean 1996, p.2
[xxiii] See also Green-Armitage 2011, p.9
[xxiv] McLean 1996, p.2
[xxv] Penrose 1946, p.53–57
[xxvi] Engstrom and Engstrom 2008, p.415
[xxvii]European Commission for Democracy Through Law: Code of Good Practice in Electoral Matters, Guidelines and Explanatory Reports adopted October 2002, Venice Commission, http://www.venice.coe.int.
[xxviii] http://www.idea.int
[xxix] Fairvote 2012 (2)
[xxx] Blais and Loewen 2009, p.356
[xxxi] Fairvote 2012 (2)
[xxxii]Aggregation of Preferences with Variable Electorates, JH Smith, Econometrica, vol 41(6), pp 1027-41, 1973; see: also see Green-Armytage, 2011.
[xxxiii] Fairvote 2012 (2)
[xxxiv] Anckar 1997; ID Hill 2011 p.2