On Elections

How people elect parliaments

Not all voters are equal after all

“One vote, one value” – the famous and moving slogan which emerged out of the American debate over congressional district apportionment in the late 1950s and early 1960s – sums up one of the most basic principles of liberal democracy: each and every individual voter is equal.

But in what way are we equal?

Is this merely a preliminary concept, a preamble-like recitation, sung at the commencement of an argument, of little further practical significance?

Is it, at least, a guarantee that the institutions of state will not pretend otherwise?

In electoral terms, one vote one value appears to be a commitment to ensuring that each individual voter has the same influence on the results of elections. Millions in number though we may be, so that our individual influence appears almost undetectable, we are assured that each atom of influence is at least the same in mass.

The principle manifests in the practice that every vote is weighted the same – each worth a score of 1, none more nor less – when ballots are counted. Not since the late 19th century has anyone seriously suggested that any formal rule for vote weighting be used.

Indeed, the franchise was once linked to property ownership. There were even specific practices in 19th century Britain and Australia that the asset rich might have a vote in every consistency where they owned property. Eliminating this multiple voting was considered one of the democratic reforms of that era.

But does the modern practice of equal, unitary weighting of ballots actually mean that votes really have equal influence?

In practice, in almost all the world’s electoral systems, it does not. The chosen voting system used in almost all direct democracies in fact damages equality of influence dramatically. In most electoral democracies, a small minority of the voters enjoy influence which is equivalent to giving them two or even more votes, while a majority are actually handed a ballot that has an influence of only a fraction of a vote.

The legal achievements of the one vote – one value campaign in America – which came to be accepted by courts and laws in many other nations – were thought to overcome this problem. They didn’t. America, Britain, Canada, Australia, India – all the large direct democracies of the world – still impose major differences in vote weighting on their citizens, in a manner that is apparently invisible to those who focus on the legal form of ballots, and yet is hidden in plain sight.

The mid-20th century equality campaign failed – certainly in America – because the underlying problems of the voting method itself were not confronted.

This result is far from necessary or inevitable. In a few countries, and in a few legislatures, equal voting exists today.

But before we examine the heart of the problem, we need a meaningful method to measure such influence.

Measuring inequality

Democracy is measured in arrays of numbers. Endless columns of numbers under column headings: populations, enrolled voters, votes cast, totals for each party.

If every number in a column is the same – for example, if the enrolment in every electoral division was identical – we would say there was an equality between the numbers in the column.

Of course the numbers never are the same. But by how much are they different? How do we measure inequality?

Mathematicians provide a fairly widely agreed answer. It’s called the coefficient of variation. It’s a statistical concept that works to identify the degree of inequality in a set of numbers at a single value. If the co-efficient of variation of your list of numbers is zero, then your numbers are all the same. The more they vary, the larger the coefficient gets.

The value for a CoV can be expressed as a percentage of the mean, such at 81.5%, or as a unit value, such as 0.815 (in this essay I’ll generally use the percentage format).

Here’s an example: the following table shows the numbers of voters recorded to be enrolled to vote in the nine circonscriptions (the French term for electoral districts) of the départment of Bas-Rhin (lower Rhine) – home to the city of Strasbourg – in the region of Alsace, France, at the Assemblée Nationale elections in May 2012. (The Assemblée is France’s national parliament. Strasbourg happens to be the seat of the European Parliament, although that’s not relevant here.)

Circonscription #1 63,978
Circonscription #2 71,161
Circonscription #3 68,329
Circonscription #4 89,933
Circonscription #5 98,427
Circonscription #6 91,200
Circonscription #7 84,670
Circonscription #8 92,744
Circonscription #9 88,326

So Bas-Rhin had a total of 748,768 voters, and the mean enrolment of the nine divisions was 83,196. Six of the divisions had enrolments above that mean, and three had enrolments that are somewhat more noticeably below the mean. They ranged from 63,978 to 98,427, so the largest was more than half again as large as the smallest. Overall, we can see that they are noticeably unequal.

The CoV measures that inequality, allowing it to be compared to any other list of numbers.

The method? Take a deep breath … the coefficient of variation is the ratio of the square root of the mean of the squares of the amounts by which each number in the list varies from the mean of the numbers to that mean.

Again, in slower English: first work out the mean of the 9 numbers (which is 83,196 voters in our example above), and then work out the 9 amounts by which each number varies from that mean. Square each of those values (ie: multiply them by themselves), and then work out the mean of those 9 squares. Then take the square root of that number. The ratio between this resulting square root and the original mean of the 9 numbers is the CoV.

It’s a mouthful, but the logic is sound. The squaring and square-rooting steps arise from the need to ensure that the answer goes back down not just to the nine electoral division populations, but to the level of the 748,768 individual voters. It is really counting how much variation each individual voter has from the average influence of all the individuals.

There are, by the way, other mathematical ways of measuring variation, designed for different purposes such as measuring how extreme the most far-out variations are. But CoV, which focuses simply on the average amounts of variation, is the most uncomplicated of the methods.

Anyway, for the départment of Bas-Rhin in 2012, this calculation comes out at a CoV of 13.9% (or 0.139 if you prefer).

Later on we’ll put this value in context by comparing it to other results for whole national elections; for the moment lets just say that this value is mid-range – a fair bit of inequality, but not as bad as some we will see.

Here is the table again, showing one more detail: the specific relative variance from the mean of each of the nine circonscriptions:

voters variation
Circonscription #1 63,978 1.30
Circonscription #3 68,329 1.22
Circonscription #2 71,161 1.17
Circonscription #7 84,670 0.98
Circonscription #9 88,326 0.94
Circonscription #4 89,933 0.92
Circonscription #6 91,200 0.91
Circonscription #8 92,744 0.90
Circonscription #5 98,427 0.84

Notice that the table is now sorted according to the values in the last column, and the values below 1.00 are shown in red. What does this signify?

This is where the meaning of inequality hits home for the individual voters. In the first three circonscriptions, voters had one form of above-average influence. They were enrolled in divisions with fewer voters, so each individual had more influence in the election of a deputy to the Assemblée than did the voters in the other six divisions.

Clearly it’s a bad thing to be enrolled in an electoral division with larger numbers of other voters; it dilutes your voting power. Better to live in a division with fewer people.

And that, of course, is inequality.

In reality, the voters in the 1st circonscription had ballots worth 1.30 votes, while those in the 5th circonscription had ballots worth 0.84 votes.

They may as well have marked the ballot papers with these weighted scores, and counted them accordingly.

We’ve only just begun – it gets more complicated. You may have noticed the little words “one form” a few sentences above. What was that about?

But lets pause to consolidate the basic maths of inequality as it applies to electoral enrolments, with a little history.

A brief history of equality reform

Even in the 19th century electoral system designers (who were mainly politicians and lawyers) in Britain, America, Australia and other places could see the obvious inequality of electoral divisions with different enrolment numbers. The advantage of living in a small-enrolment division was very clear.

It used to be a total joke. Before the British reforms of 1832, 44 House of Commons constituencies had fewer than 50 voters in them. The most extreme example was the constituency of Old Sarum, consisting of a hill in county Wiltshire, which famously had just 3 dwellings boasting a mere 7 voters. (Apart from the obvious mathematical unfairness, the votes from Old Sarum and other small constituencies were also utterly in the pocket of wealthy landowners – the voters were actually only tenants, who usually voted as they were told.)

As electoral system reform unfolded during the 19th century and into the 20th, equality of division enrolments came to be understood as a test of fairness. But while some improvements were made, little was done to secure formal equality; in all these democracies redrawing of divisional boundaries remained a matter for parliaments themselves to vote on, despite the obvious self-interest of the sitting members.

In Australia (and also in some other countries) the unequal notion that rural areas deserved some ‘weighting’ – that is, permission to have smaller enrolment numbers – actually persisted until the late 20th century. That’s now been almost entirely dealt with.

But it was in mid-20th century America that the matter came to a head. The boundaries of congressional districts were set by state legislatures (they still are), and across the South of the United States they were clearly being deliberately set in unequal ways to the disadvantage of African American voters.

(The district boundaries were also being gerrymandered, but that is another problem for another essay. Moreover, the legal suppression of African Americans from even getting registered and voting were in fact a more serious problem than the issues relating to district boundaries and enrolment equalities – but again, for another time.)

Following the American Civil War, America’s Constitution was amended to include protections of the right to vote regardless of race, and to include a crucial amendment guaranteeing to all citizens “the equal protection of the laws”. That protection has been used to achieve a myriad of legal outcomes every since.

In the late 1950s, American civil rights lawyers and their clients argued that it meant that populations living in the districts in which politicians were elected had to be equal.

The litigation took a few years, and several court cases, but in the end the US Supreme Court clearly agreed that yes, for the sake of the legal equality of all citizens when participating in elections, electoral districts had to have equal numbers of people. This was the great victory of the one vote one value campaign.

The cause propagated to other nations. In 1983 Australia adopted laws for regular and independent reviews of district boundaries with low tolerances for variation in enrolment numbers. Britain and Canada have reached a similar legal position in just the last decade or so.

As political reform, hard-won against entrenched opposition, it was a great success. Except that, for not one but two reasons, it didn’t actually work.

One vote one value, understood as equality of electoral enrolments, doesn’t actually create equality of voting influence, either in the legal sense the US Constitution was thought to guarantee, or as a matter of practical reality.

The more important reason why comes down to the choice of voting method.

But lets deal with the first, simpler reason, which is just about maths and population statistics.

Who should count in a democracy?

Who makes up the people of an electoral division?

Is it the whole population, including even children who don’t vote? Including even people who aren’t citizens, but are living there permanently?

Or is it only the eligible voters? Constitutions go to a lot of trouble to define who can and can’t vote; shouldn’t this be the relevant group of people?

Or is it only the list of people who are actually enrolled? This is at least something where we would have exact numbers, because the earlier two numbers can really only be estimates.

Does one of these answers seem more correct that the others?

What approach do democratic countries today actually use? Well, the United States has always used population data to decide how many congressional seats each state is allotted, and in the wake of the one vote one value decisions it seemed obvious to keep using population data to draw district boundaries ‘equally’.

Australia got into the practice of universal, indeed compulsory, voter registration after 1911, so it eventually seemed natural to use the very detailed data on enrolments as the basis of boundary changes. The thorough system put in place in the 1980s has continued to be based on enrolment numbers, not populations.

For a long time Britain had more or less always allowed subjects of the monarch to vote, meaning that residents visiting from other commonwealth nations could often do so. They have tightened up their rules only recently with a proper registration system.

Canada has recently developed into an approach similar to Australia – based on enrolment numbers – but without having Australia’s compulsory enrolment system to fully support it.

So there’s no single agreed world practice.

In fact, it’s hard to see that there is an axiomatic answer at all. It’s really about different visions of democracy.

Using population figures would seem to say that parliaments are meant to represent all people, even non-citizens and children.

Using eligible voters seems to link parliaments more closely with constitutional definitions of who can elect them. Using actual enrolment numbers is similar, with the advantage of detailed data being available.

In Australia the last two options seem hard to differentiate, because of comprehensive enrolment. But in other nations the latter two options can yield quite different numbers.

Now, it’s clearly a fact that the results of elections might turn on which choice is made. Elections are often close enough that a few seats won on differently drawn boundaries could change who forms governments. Politicians (at least in the United States) put great efforts into gerrymandering because such results are so easily achieved.

Indeed, this very question had drifted up to the US Supreme Court for resolution in the past 12 months. A conservative NGO claims that the boundaries in Texas – state with a demographic profile including many non-citizens, above-average rates of people under the age of 18 and above-average birth rates – should use the estimates of ‘vote-eligible population’ (“VEP”), not the estimates of total population that have always been used in the past.

The whole American jurisprudence on electoral districts has always been based on total population. The plaintiff’s argument, if upheld, would mean the current boundaries in 43 states would have to be redrawn. If they won such a court order in an election year, all the arrangements for the coming elections, including candidate primaries, would be thrown into confusion.

Election boundaries in the US are already a heavily partisan affair, with boundary drawing in more than half the states under the partisan control of one or other political party; hence the gerrymandering.

Every observer of the court action can also see the partisan implications of the argument. Given the ‘clumpy’ nature of American demographics and partisan geography, new VEP-based boundaries would almost certainly result in more seats going to Republican candidates over Democratic ones. This is therefore a very partisan legal dispute.

In late 2015 the Supreme Court agreed to hear the case, but the untimely death of conservative Justice Antonin Scalia in February 2016 has thrown it (and several other pending cases) into an awkward position. It would now seem most unlikely that five of the remaining eight judges will vote to reorganise the US congressional districts system this year. It’s already a dramatic enough election year, and the Court is already deeply entangled in that drama.

But lets emerge from such speculations, and simply note that there is no absolute answer to the question of which population data should be used in the cause of ‘equalising’ the size of electoral divisions.

Because there is a more important issue on which the deeper equality of real electoral influence for voters turns.

The truth is, equality in the population size – or even the specific enrolment numbers – of electoral divisions is just one input into the degree of equality of influence of each voter.

Most modern democracies now attempt to make these divisions at least roughly equal, using at least one of the population measures discussed above. These efforts minimise – no more – one potential cause of unequal voter influence. That’s a good thing as far as it goes.

But population and enrolments do not on their own determine how many people actually cast votes. In all the major democracies – other than in Australia – the rate of turnout has a significant impact on variations in voter equality. Turnout rates often differ markedly across large countries such as the United States, leaving a pattern of variations quite different from the underlying population or enrolment variations.

Categories-within-categories of population

Other than in Australia – which has had compulsory enrolment of voters since 1911 and compulsory voting since 1922 – voting is voluntary in the major democracies.

British and Canadian turnout rates fluctuate around percentages from the low 60s to the mid 70s.

Turnout rates in the United States are lower, at around 60% in presidential election years and a little below 40% in mid-term congressional elections. The 2014 election’s voter turnout of just 35.5% was the lowest since the enfranchisement of women other than the war-affected election of 1942.

Marked differences between the total population, vote-eligible population and registration rates across the 50 American states mean that the geographical variations on which seat allocations and district boundaries are based (using population estimates) are significantly different from the variations in actual votes cast.

The variations in the various categories of population or voting in the 436 US House districts (including the District of Columbia) in the last presidential election year – 2012 – were as follows:

Population set CoV 5 most advantaged states (and relative weighting)
Voting aged population (VAP) 5.0% Rhode Island (1.32), Wyoming (1.26), Nebraska (1.18), Utah (1.13), West Virginia (1.11)
Vote-eligible population (VEP) 7.9% Rhode Island (1.30), Wyoming (1.17), Nebraska (1.14), California (1.12), Texas (1.11)
Registered voters 12.6% Wyoming (1.31), Hawaii (1.28), Rhode Island (1.27), Utah (1.23), California (1.21)
Votes actually cast 20.8% 5 most advantaged districts:
Texas 29th (uncontested) (2.93), California 21st (2.41), Texas 33rd (2.39), California 40th (a 2-Democrat contest) (2.23), Arizona 7th (2.19)
5 most disadvantaged districts:
Iowa 1st (0.72), Wisconsin 2nd (0.72), Missouri 2nd (0.71), Colorado 2nd (0.66), Montana (0.58)

(Note that I don’t have data on the full population – in the first row I have used the voting-aged population, which is the population aged 18 or over, but including non-citizens. For this exercise I have used US Census Bureau estimates together with actual election result data for the final row.)

Notice how at the highest population set, the voting aged population (VAP), the degree of variation between districts starts out very low – just 5% (or 0.050). This indicates that across the whole of the 50 states, the effect of the national seat apportionment rule is working well.

Then when we take out the non-citizens and move to the vote-eligible population (VEP), the degree of inequality between districts starts to rise.

By the time we get to numbers of registered voters, the degree is up to 12.6%. It’s starting to get significant.

Finally, when we look at the election result numbers for votes actually cast (which is, incidentally, the first calculation based on a real known set of data, not a census bureau estimate), the variation jumps again, and each district now varies from the mean by around 21%. This is starting to get serious.

Just for illustration’s sake, I’ve included above details of the 5 most advantaged states (or for the final row, the most and least advantaged individual congressional districts). The point is really to show that the advantages do vary as the population set used in the calculations changes.

Around the world, these rates vary, and using actual statistical evidence (inequality-national-summary) we can turn to look at what constitutes low, medium and high rates of variation.

In regard to enrolment data, the best results at the moment are those in Australia. Apart from the compulsory enrolment regime – which will flatten regional variation in enrolment rates out as the whole nation averages a 93% rate of enrolment – since the reforms of 1984 introduced regular, independent boundary reviews Australia’s variation rate has been around 5% to 8% of mean enrolments. From 1901 to 1983 the rate varied in a band broadly between 15% and 25%, so the modern regime has clearly made a big difference to enrolment equality.

Some Australian state electoral administrations have done even better. The NSW Legislative Assembly election of 2015 was conducted on the basis of divisional enrolment variations of just 2.1%, down from 2.3% in 2011. Better still, at its 2010 election the Tasmanian House of Assembly was divided into 5 divisions with an enrolment variation of an astonishingly low 0.97%, or 0.0097.

In results from other major democracies with similar electoral systems, the past three UK House of Commons elections have shown 11-12% variation in constituency enrolments, France was at 17% in 2012, Canada at 22% in 2011 and 17% in 2015, India at 19% in 2009, and Pakistan’s last two elections were at 22% and 24%.

Malaysia’s last two elections had district variations of 39% and 42%, highlighting the major pro-rural seat malapportionment system that is still in force there. Malaysia remains one of the few nations to maintain institutionalised voter enrolment inequality.

The United States had district variation of just under 21% in 2012, as discussed above. Because US registration rates by district are not generally available, this calculation is based on US Census Bureau estimates. It is of doubtful use to make the effort to calculate such values other than for the first election after each census – such as 2012 – when the boundary reviews are still fresh. During the decade which follows each national boundary review, we can assume that inequalities will steadily increase.

There is a loose general correlation between the size of the assembly and the degree of variation. The UK House of Commons is the largest assembly at 650 seats, and shows a fairly low 11-12% rate of variation.

But clearly regular independent reviewing of boundaries carried out against tight criteria, as is practiced in Australia, is the factor with the strongest effect on keeping enrolment inequality down.

As with the American example shown above, in almost all these cases the votes cast in the elections generated slightly higher rates of variation between electoral divisions than those arising from enrolment figures.

The impact of plurality voting on real influence

Which brings us to the final – and mathematically the most significant – reason why votes have different values.

We saw a moment ago that, depending on the system of boundary setting, most of the democracies mentioned above were achieving enrolment inequality between electoral divisions with CoV results in a range between 5% and around 25%.

But look at what happens when we turn to data on the margins by which candidates win their seats.

The PDF document linked above shows dramatically higher rates of inequality in the impact voters have in actually electing people to parliaments and congresses. Here is one final row to add to the table above giving the US results from their 2012 elections:

Population set CoV 5 most advantaged states (and relative weighting)
Margins of victory
in individual seats
57.3% 5 most advantaged districts:
Minnesota 2nd (540), Michigan 8th (523), Florida 22nd (139), New Jersey 12th (128), Connecticut 2nd (65)
5 most disadvantaged districts:
(Not counting districts in which there was no contest at all) Texas 8th (0.39), Massachusetts 3rd (0.39), Colorado 5th (0.39), Florida 1st (0.37), Florida 14th (0.34)

The CoV ratings for all the same countries mentioned above range from around 57% up to around 80%.

Two recent Indian and Pakistani elections had ratings of over 100%, which is only possible when a significant number of electoral districts have extraordinarily high influence over election results, achieved at the expense of many districts having well below average influence.

In the United States data given just above, note the details of most advantaged congressional districts. The voters in the most marginal congressional result in 2012 – the 2nd district of Minnesota – had an influence 540 times the national average. This is what it actually means to be a voter in the most marginal single-member contest in an election.

These data provide a statistical measurement to the problem known universally as safe vs marginal districts. A few electoral areas see such close contests that they hold all the real influence in the election results. The rest – always the majority of districts – are so safe that their voters can be taken for granted. The candidates – and their party campaigns – typically do exactly that; to live and vote in a safe district is to be generally ignored.

This is, obviously, very far from equality of influence. And such inequality would still be entirely possible even if the populations or the enrolments in districts were set at exactly the same numbers.

Even the very well-distributed electoral divisions in Australia and its states manifest roughly the same levels of final inequality as the other nations cited above.

This is where the US courts went wrong in the 1960s. They believed that equality of enrolments was the same thing as equality of influence. They have laboured to define and insist on specific rules to implement equality of enrolment, all for little real gain in equality.

Enrolment equality is not a complete failure, lets be clear. If it were not in place, the malapportionment of former decades would make voter equality even greater. Malapportionment was also used for deliberate discriminatory purposes (as of course gerrymandering still is). It is a good thing that such distortions have been minimised. But they have not created equality of influence.

Specifically, the promise of the American constitutional amendment that voters, in exercising their vote, would have “the equal protection of the laws” has not been meaningfully achieved. If that proposition meant equality of legal impact, in reality and not merely in form, then it should not have stopped merely at equal population of districts. It needed to do something more.

The problem, of course, is the use of the plurality rule to declare winners in single-member districts. This system – in use in India, Britain, the United States and Canada, in Australia (with preferential voting instead of plurality), in several other nations touched by the history of British electoral practice, and also in France (with the two-round runoff method), continues to deny voters equal influence on elections.

This problem has long been known. The British were debating it and considering solutions from the 1850s. Americans know it too, though they had deliberately moved to single-member districts in the 1840s to get away from the even worse results caused by the ‘block vote’ in multi-member (or ‘at-large’) districts.

The answer – the only answer that keeps elections being direct votes for candidates – was the single transferable vote based on election by identical quotas of votes.

This ‘STV’ system was put forward in Britain – and independently in Denmark – in the 1850s, and it was known in Australian and American circles by the 1860s. By shortly after 1900 the system’s key techniques had been settled. It had been trailed at real elections in the state of Tasmania. In 1902 it almost became the voting system for the Senate of the new Australian nation (and some people even advocated for it to be used for the lower house of parliament).

Look again at the table in the attached document, specifically at the election results listed in the bottom third of the table. Leaving aside the Australian Senate – which has fundamental malapportionment – the other examples from Éire, Malta and certain Australian states have very low enrolment inequality results. Their margin inequalities for actually electing candidates aren’t even shown, because their margins are precisely identical by virtue of the quota rule itself.

Even these STV-based national electoral systems cannot technically create perfect equality, because some voters fail to vote, or spoil their ballots, or vote for minor candidates and thus end up unrepresented. But no electoral system can eliminate those outcomes. STV at the least offers every voter real equality as they go into the polling booth.

STV also achieves actual representation rates (another key test of a representative voting system) of 75-80% of all voters, as well as gifting equality of influence to all those who are represented.

The Australian Senate also meets this test of equality within each state, because each state forms a separate voting pool. It only fails the test of equality at a national level because the 6 Australian states are each allocated the same number of senate seats – 12 – despite their greatly different populations of voters.

The same inequality of influence applies to the US Senate – although the US Senate has no equality of voting within each state, because senators are again elected by single-member plurality rule.

Results for the US Senate are shown on the table, but in fact any estimate for the US Senate at any point in time is really the aggregate of three different elections, with different combinations of states and different turnout rates. Aggregated across the three most recent electoral cycles, the inequality of voter influence of the US Senate during the 2015-16 term can be estimated as a CoV of 87% – slightly higher than the Australian upper house.

The inequality of the margins of victory in the 97 contests won by current US senators (setting aside 3 senators from states so safe that no-one ran against them at their last elections) is a staggering 131%.

The US Senate is, therefore, the most unequally elected legislative chamber in the democratic world.

But the US Senate, dramatically illustrative though it is of the problem of inequality, is not a parliament in which government is formed (some US Senators may of course think otherwise). Perhaps more important is the fact that in Britain and France, Canada and Australia, India and Pakistan and several other nations parliaments are being elected, and governments created, on the basis of dramatic levels of inequality of voter influence.

Many of the constitutions of these countries, either expressly or by implication, hold out the promise, or at least the claim, that all voters are equal.

Nowhere – outside of Éire and Malta – are such constitutional promises being kept.

Malcolm Baalman
February 2016

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