Last week we looked at some of the numbers associated with the previous week’s elections in Britain and in France. So how do they compare, and what do they tell us about representation in general?
Consider a country with three competitive parties; call them (unimaginatively) A, B and C. Assume that A has roughly 40% support and the other two about 30% each, and that there’s nothing out of the ordinary in the way that support is distributed – none of them is heavily regionalised. Assume also that while none of the three are allies, B and C are more sympathetic to each other than either is to A.
Now consider three different voting systems (not the only ones possible, but by far the most common). First, proportional representation: either the whole country is a single electorate, or it has large multi-member electorates that approximate the same effect. Party A will win about 40% of the seats, and the other two about 30% each. This is how the majority of democracies work.
Second, single-member districts with first-past-the-post, as in Britain and most of its former possessions. Unless something unusual happens (we’ll come back to that), Party A will win a majority, probably a fairly substantial one: maybe something like 54% of the seats, with 23% for each of the other two.
Third, single-member districts with preferential or two-round voting, as in France and Australia. Being closer to each other, parties B and C will mostly exchange preferences, and as a result will win a majority between them. How big it is will depend on (among other things) how tight their preference flow is, but if it’s very tight (as it is, for example, between Labor and the Greens in Australia) it could well be something like 70% of the seats (35% each), with 30% for party A.
Note a couple of things about these cases. In an obvious sense, the first is the one that gives the voters what they actually voted for. The second, at least in this instance (things may be different if voting strength is unevenly distributed), gets the parties in the right order, but exaggerates the support of the one in front. The third may get the parties in the wrong order, but it reproduces (again, in this instance but not always) the critical fact about them: that none of the three has the support of a majority.
Political scientists have a standard measure of unfairness or disproportionality, known as a Gallagher index (Wikipedia will show you how to calculate it). Higher numbers mean greater disproportion, from zero, the perfect case where shares of votes and of seats match exactly, to 100, the theoretical maximum where one party wins all the votes but another party wins all the seats.
Proportional systems routinely get scores under five; last year’s New Zealand election, for example, was 4.25. The 2022 Australian election, by contrast, came out at 16.2, which is unusually poor, driven mostly by the over-representation of the ALP.* France’s election last week I calculate as 7.45, which for a non-proportional system is very good. But the British election was a truly awful 23.7. (Although things can get worse: a Queensland election in 2012 recorded an extraordinary 31.2.)
Since the French election resolved itself into something like a two-party contest, you can also look at the result in terms of the Cube law – the reasonably well-established principle that in a two-party system with single-member districts, the parties will win seats roughly in proportion to the cubes of the strength of their vote. On that basis, National Rally would only have won about a hundred seats rather than its actual 143; the difference between the two is the measure of the extent to which the preference flow among its opponents was less than perfect.
In Britain, if you just ignore the other parties, the Cube law predicts a ratio of Labour to Conservative of 33.7^3: 23.7^3, which is 38,254:13,320, or a bit less than 3:1. But with 411 seats to 121 Labour did better than that, with a ratio of 3.4 to 1. That could be because the Cube law is just an approximation, but I think it’s probably also due to the point that I said we’d come back to: how a first-past-the-post system can sometimes behave a little bit like a preferential one.
Parties that are not quite allies but have a certain amount of common sympathy – as, for example, Labour, Liberal Democrats and Greens did in Britain – will to some extent pool their votes. In seats where only one of them is seen to have a realistic chance of beating their common enemy, votes from the other(s) will gravitate towards them; the parties will concentrate their resources on their more winnable seats, and at least some of their voters will make tactical decisions to cast their vote where it will have most effect.
At the extreme, this sort of tactical voting can go a long way to reproduce the effect of a preferential system. The British election was not a case like that, but it does look as if Labour and the Lib Dems both benefited from votes that would not have represented the voter’s first choice, but where that was subordinated to the priority of beating the Conservatives. Their tally of seats reflects that.
It’s also an important reminder that electoral systems are not just passive conduits for voter behavior; they also help to shape that behavior. Changing the system will change the way people vote, and therefore in practice may deliver results that are quite different from what’s expected.
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* I notice that at the time I calculated it as 15.5; the difference is probably something to do with the treatment of independents.
The World is Not Enough 