A vote worth a square root

An assembly of countries is to decide by weighted vote. The countries differ in size by a factor of a hundred. How many votes should each have?

There are two obvious answers, and each has a respectable argument. One vote per country treats states as equals, as the United Nations General Assembly does. Votes in proportion to population treat citizens as equals, or seem to. The second looks like the arithmetic of fairness, and it is what most people would accept without further discussion.

In 1946 the British geneticist Lionel Penrose published a short paper in the Journal of the Royal Statistical Society pointing out that the second answer is wrong, and is wrong in a precise way. What should be equal is a citizen’s chance of affecting the outcome. That chance has two factors: the chance that the citizen’s vote decides how their country votes, and the chance that their country’s vote decides how the assembly votes. The first factor shrinks as the country grows, and it shrinks by less than proportion. The same rule that lets an average settle while its wobble does not sets the rate: the fluctuation of a total of $N$ votes grows as $\sqrt N$, so the chance that those votes come out exactly level falls as $1/\sqrt N$. Then the second factor must grow as $\sqrt N$ to compensate, and the country’s voting power should be proportional to the square root of its population.

One voter in a crowd #

The calculation is small enough to do exactly. Suppose a district has an odd number $N$ of voters, each voting independently and either way with equal chance. A given voter decides the district’s vote exactly when the other $N - 1$ split evenly, and the chance of that is a central binomial coefficient:

$$P(\text{tie}) = \binom{N - 1}{(N - 1)/2}\frac{1}{2^{N-1}} \approx \sqrt{\frac{2}{\pi N}}.$$

That approximation is Stirling’s, and it is the same quantity that gave the fifty-match box its remainder and that gives the bell curve its height at the middle. It says something that is not obvious at first glance about the value of a vote: it does not fall as one over the number of voters.

At 101 voters the chance is 7.96%. At 10,001 it is 0.80%, and at a million and one it is 0.08%. A hundredfold electorate leaves a voter with a tenth of the say. Intuition usually wants a hundredth. The reason there is more than the intuition expects is that a voter in a big crowd is far likelier to be decisive than a single-voter share would suggest, because most of the crowd cancels itself, and the outcome turns on the balance of what is left. Only the fluctuation counts, and the fluctuation is of size $\sqrt N$.

Now the two-tier vote. The chance that a voter in district $i$ decides the overall outcome, assuming independence, is the product of the chance of a tie in the district and the chance that the district’s votes turn a losing coalition of districts into a winning one. That second chance is exactly the Banzhaf count from a share of the votes is not a share of the power: the number of coalitions of other districts for which the district’s vote is decisive. Equal say for every citizen demands that this product be the same in every district.

The opening figure computes that product exactly for seven districts of sizes 9 to 225. With one vote per district, a voter in the smallest district has about 2.3 times the average say, and a voter in the largest has 0.45, a ratio of about 5. With votes in proportion to population, the order is roughly reversed: 0.50 in the smallest and 2.05 in the largest. Its line is jagged because in a small assembly the weights make some districts powerless in odd ways, as Luxembourg’s single vote was in 1958. And with votes in proportion to the square root of the size, the relative say stays between 0.994 and 1.021. In a seven-member assembly, that level of agreement is more than the argument had any right to promise.

Twenty-seven states #

The real test of a rule like this is a real assembly, and the Council of the European Union has been the standard example since 2004. The twenty-seven members run in population from about 83 million to about half a million. That is a span over two orders of magnitude, the shape of real inequality of size.

The argument needs Banzhaf counts for twenty-seven weighted players, and that means considering $2^{26}$ coalitions, for each player. A dynamic programme over total weight reduces it to something very cheap: build the counting polynomial $\prod (1 + x^{w_i})$ once, divide out one factor to remove a player, and add up the coalitions whose total lands in the window where that player’s weight swings the result. It takes a fraction of a second, and it is exact.

One vote per state makes a citizen of Malta 12.9 times as influential as a citizen of Germany. Votes in proportion to population, with a simple majority, make it about 14.9 times the other way. Square-root weights, at the right quota, hold the ratio between any two states’ citizens to 1.006. The figure shows those three curves, and the third is flat almost to the width of a line. Neither of the obvious answers is even close, and the two err in opposite directions by about the same factor. On logarithmic scales they lie nearly symmetric about the flat line, which is what they must do, since the correct exponent lies at the half-way point between 0 and 1.

Voting power is not voting weight #

There is a step in that argument which hides real work. Square-root weights give equal say to citizens only if each state’s power, its chance of swinging the assembly, is proportional to the square root of its population. Weight is what appears in the treaty. Power is what happens. The whole of the earlier essays on weighted voting was about how those differ, sometimes drastically. It looks as if the rule might require a table of power indices to be negotiated, and might have to be renegotiated whenever a state joins.

The surprise is that it need not. For a fixed list of weights, power depends on the quota, the proportion of total weight needed to pass a motion, and there is a single quota at which power is almost exactly proportional to weight.

The misfit between power and weight is 0.0062 at a quota of 50% and falls to a sharp minimum of 0.0003 at 61.5%, twenty times smaller. Beyond that it rises steeply: it is 0.013 at 70% and 0.022 at 74%. Wojciech Słomczyński and Karol Życzkowski noticed this in 2004 and gave the estimate

$$q \approx \frac12\left(1 + \frac{\sqrt{\sum w_i^2}}{\sum w_i}\right).$$

For square-root weights $\sum w_i^2$ is just the total population, and the estimate here comes to 61.6%, within a tenth of a point of the exact optimum. The formula is not an accident of the data. The total weight of a random coalition is approximately normally distributed, by the central limit theorem: mean half the total, variance $\tfrac14 \sum w_i^2$. The power of a player of weight $w$ is the probability that a random coalition lands in a window of width $w$ just below the quota, and for a small player that window’s probability is $w$ times the density of the bell at the quota. So small players have power proportional to weight at any quota. The correction for large players’ weights is what fixes the quota, and $\sqrt{\sum w_i^2}$ is the spread of the bell. The right quota is one spread above the middle, in those units, where the bell has its point of inflection and is locally straight. This is the same central limit theorem at both tiers: once inside each district, and once for the coalition of districts.

At the optimal quota the ratio of power to weight runs from 0.998 to 1.002 across all twenty-seven states. At 50% it runs from 0.977 to 1.046. Those differences do not look important until one recalls that this is the ratio the entire argument is about. The rule that came out of this is known as the Jagiellonian compromise, named after the Kraków university where it was developed. Its appeal is that it has a single line: votes as the square root of population, quota about 61.5%. A new member would adjust the quota slightly. No negotiation over tables is needed, and that is an extraordinary property for a system whose power indices are generally as unruly as the earlier examples of weighted assemblies showed.

The proposal was discussed seriously when the Treaty of Lisbon was negotiated. Poland’s government argued for it in 2007, under the slogan “the square root or death”. It was not adopted. The Council’s rule since 2014 is a double majority of 55% of states representing 65% of the population, which is simple to state and has been analysed in detail, and which has by construction some of the proportional rule’s overweighting of the largest states.

A square root rests on an assumption #

Everything so far rested on a single modelling assumption: voters vote independently, with equal chance either way. This is an assumption that a sceptic should attack first. Real voters are not independent coins. They are influenced by common events, national economies and shared media, and the fractional support for a side moves as a whole. A model with this feature is not hard to write down and it changes the conclusion.

Suppose each election has a common support level $p$, drawn from a bell around one half with some spread $s$, and voters are independent once $p$ is fixed. Then the district’s vote is close to $pN$ with a fluctuation $\sqrt N/2$, and what matters is whether the common swing lands close enough to one half for the fluctuation to bridge the gap. For small $N$, with $\sqrt N/2$ large against $sN$, the swing makes no difference and the slope is minus one half. For large $N$ the chance of a tie is the chance that $p$ lands in a window of width about $1/N$ near one half, so it falls as $1/N$. The change happens where $N$ is of the order of $1/s^2$. With a swing of spread 1% that is some thousands of voters, and with 5% it is only a few hundred. So the numbers in the figure at the largest electorates have slope close to minus one, for both spreads.

So in large electorates with appreciable common swings, the proportional rule becomes the right one. If the chance of a tie in a district falls as $1/N$, power must grow as $N$ for each citizen to have an equal say. Andrew Gelman, Jonathan Katz and Joseph Bafumi argued in a 2004 paper that this is the case for the actual elections they examined: measured from real US data, the probability of decisive votes falls more like $1/N$ across districts than like $1/\sqrt N$. It is an empirical conclusion for which actual elections are the evidence, and the assumption of independent, evenly balanced voters that gives Penrose’s law its exact form is not a fact about any electorate.

That does not make the square root wrong. It makes it conditional, and it says clearly on what. In a council where national governments cast the votes after a public debate, the chance that the governments’ positions are independent is itself debatable, and there is a respectable argument for treating a rule like this as a principle of constitutional neutrality, ignorant by design of how anybody will vote, rather than a prediction. Penrose’s own justification was of this kind. A constitution is designed behind a veil. Nobody is asking how a specific vote will go; the question is what a voter is entitled to if nothing is known about the future at all. On that reading the independence of coin-tosses is no more a claim about voters than a uniform prior is a claim about a coin.

A count that had already been done #

This line of thought had been anticipated. The Banzhaf count that gives a voter’s share of power in the second tier, and the Shapley value that gives it from random orders of arrival, were both invented later than Penrose’s paper, and his arithmetic is the same as Banzhaf’s. John Banzhaf’s work of 1965 reinvented the idea for a court case about a New York county board, and Douglas Rae, Lionel Penrose and James Coleman all had related formulations before and after. The measure was invented at least three times because almost nobody reads statistics journals for constitutional arguments. Dan Felsenthal and Moshé Machover’s book of 1998 is what finally consolidated the history.

There is also a real difference, though not a large one, between the two indices in this context. The Shapley–Shubik index, which a block in an ocean of small holders treated as a limit, measures how a fixed pot of power is shared out, and it agrees closely with the Banzhaf count for small members of large assemblies. For this problem the Banzhaf count is the one that multiplies correctly with the probability of a tie, because a voter’s chance of deciding an election in a model of independent voters is exactly the product. An index that shares out a fixed total cannot be multiplied in that way. So the choice of index here is determined by the question.

What the exact numbers cover #

The Banzhaf counts for all twenty-seven members are exact integers, computed with integer weights scaled and rounded from the square roots. A coarser or finer rounding moves the third decimal place of the ratios and not the shape of any curve. The population figures are approximations for one year, rounded to the nearest hundred thousand, and they are a fixed input rather than a claim: rerunning them for another year would shift the optimal quota slightly. The quota was scanned in steps of half a per cent, and 61.5% is the best on that grid.

The figures assume an electorate with independent fair voters at both levels, except in the one figure with a common swing, where a single normal law with a fixed spread stands in for a whole political system. Nothing computed here tests any actual election. The historical and political statements about the Lisbon negotiation and the current rule are quoted, and neither the empirical analysis of Gelman, Katz and Bafumi nor the wider literature on what real voting data show was reproduced.

Still open: what is the right model of a voter? #

Penrose’s rule answers exactly the question it asks. The question that remains is whether it is the right question. For a real electorate, what law connects the size of a district to the chance that one of its votes is decisive? The two models here give exponents of one half and one, and the data appear to give values in between, varying with how contested the district is. Estimating a very small probability from elections in which it hardly ever occurs is intrinsically hard. Nobody has observed a tied national election, so the rate has to be extrapolated from close ones, and the extrapolation depends on the same model it is trying to test. A rule for setting weights that is robust across the plausible range would need a measure of how far between the two exponents a real council sits, and there is no agreement on one.

There is a sharper mathematical question as well. The single-quota phenomenon was found by computation on real data and then explained by a central limit theorem whose accuracy for a specific short list of very unequal weights is not obvious. Precise statements about how close power can be made to weight, in terms of the weights, and for which lists of weights no quota works at all, have only partial answers.

A proportional rule’s mistake #

A vote does not lose its say in proportion to the crowd: it loses it in proportion to the square root. So citizens have equal say when their country’s voting power goes as the square root of its population. Computed for twenty-seven real populations, the obvious rules give ratios of 12.9 and 14.9 between citizens of the largest and smallest states, in opposite directions, while square-root weights at a quota of 61.5% give 1.006. The central limit theorem appears twice, in each district’s tie and in each coalition’s weight, and it is the second appearance that lets a single quota do what would otherwise need a table.

And the square root is exactly as firm as the coin-tossing behind it. Let voters move together with a common swing, and for electorates large enough the exponent bends to one, which brings back the proportional rule that the argument seemed to have refuted.