How often the rules find the majority favourite

Five rules and five winners counted one set of twenty-seven ballots five reasonable ways and got five different winners, and four conditions, and no rule that has all of them showed that no way of counting can be right in every case. Both are statements about what can happen. They leave open how often it does. A rule that goes wrong on a contrived profile and almost never on a real one is a different thing from a rule that goes wrong every few elections, and the impossibility theorems cannot tell the two apart.

This essay measures one kind of going wrong. Call a candidate the Condorcet winner if it beats every rival in a head-to-head majority — the candidate the majority prefers to each alternative, one at a time. When there is such a candidate, the case for electing it is strong; the majority that goes in a circle showed that sometimes there is none, and how often the majority goes in a circle measured how often. Here the question is the next one: when a Condorcet winner exists, how often does each standard rule actually elect it? The share is the rule’s Condorcet efficiency, and it can be computed exactly.

Counting every election #

The model is the one the cycle essay used, the impartial culture: every voter, independently, ranks the three candidates $A$, $B$ and $C$ in one of the six possible orders, each with chance one in six. It is not a model of any real electorate, and that is its use — it is the case in which no structure of opinion helps or hinders any rule, so a rule’s success there is a measure of the rule rather than of the voters.

Under the model an election with $n$ voters is described completely by six numbers, how many voters chose each ranking, and a split $(c_1, \ldots, c_6)$ of $n$ has probability $n!/(c_1!\cdots c_6!\,6^n)$. So the efficiencies can be found exactly by running through every split. For each one the figure decides the three head-to-head contests, keeps the split if one candidate wins both of its contests, and then runs four rules:

• plurality: the most first places;
• the runoff: drop the candidate with the fewest first places and let a majority choose between the other two, which for three candidates is the same as instant runoff;
• Borda: two points for a first place, one for a second, nought for a last;
• anti-plurality: the fewest last places, as though each voter vetoed one candidate.

A tie inside a rule is split evenly, so a three-way tie for first places credits the Condorcet winner with a third. The electorates are odd so that no head-to-head contest is tied. At 51 voters there are 3,819,816 splits, and every odd electorate from 3 to 51 is enumerated in full.

The hero shows the four efficiencies settling as the electorate grows. At 51 voters the runoff elects the Condorcet winner 96.24% of the time, Borda 90.46%, plurality 78.04% and anti-plurality 73.39%, and all four have nearly reached the values at the right, which are the limits for a very large electorate.

Three, five and seven voters, every ballot #

The smallest electorates are worth seeing whole, because they show where the differences begin. With three voters there are $6^3 = 216$ ordered profiles, and 204 have a Condorcet winner; the other twelve are the cycles. Of the 204, plurality elects the winner in 180 once ties are split, the runoff in 192, Borda in 195, and anti-plurality in only 132.

Plurality’s losses at three voters have a simple source. A Condorcet winner needs two of the three voters to rank it above each rival, but it can be first on only one ballot — the profile $ABC$, $BAC$, $CAB$ makes $A$ the winner, beating $B$ with the first and third voters and $C$ with the first and second, while each candidate has exactly one first place. Plurality sees a three-way tie, credits $A$ with a third, and in the long run elects someone else two times in three in profiles of that shape. The runoff does better on the same profile because, whichever candidate it drops, $A$ survives with probability two in three and then wins its head-to-head contest.

By seven voters, with 279,936 ordered profiles and 258,936 Condorcet winners among them, plurality’s record has fallen to 82.81% and the runoff’s has risen to 95.78%. The rules are already in the order they keep for ever.

The fewest voters each rule needs to miss #

A rule can be judged by how small an electorate can trip it, and the search runs through every split of 3, 5, 7 voters in turn until each rule has failed outright, with no tie to soften it. Anti-plurality fails with three: two voters ranking $ACB$ and one ranking $CBA$ make $A$ the Condorcet winner, but $A$ has one last place and $C$ none, so the veto count elects $C$. Plurality and the runoff both fail with five, on the same profile — one $ACB$, two $BAC$, two $CAB$ — where $A$ wins both contests three to two and has the fewest first places, so that plurality cannot pick it and the runoff eliminates it at once. Borda fails with five too: three voters ranking $ACB$ and two ranking $CBA$ give $A$ six points and $C$ seven, though $A$ beats $C$ three to two.

The last case is the clearest picture of Borda’s weakness. Both groups of voters agree that $B$ is no good to them, and the two voters who prefer $C$ prefer it to $A$ by two places while the three who prefer $A$ prefer it to $C$ by only one. Borda counts intensity — how far apart the ballot puts the candidates — and the head-to-head majority does not. Neither is wrong; they are answers to different questions, which is what the minimal profiles make plain.

The minimal electorates say little about frequency, though. Plurality and Borda can both be tripped by five voters, and in large electorates one fails a quarter of the time and the other a tenth. How easily a rule can go wrong and how often it does are separate measurements, which is the whole reason for counting.

The large-electorate limit #

As the electorate grows, the six counts behave like $n/6$ plus fluctuations of size $\sqrt n$, and every decision any of these rules makes depends only on the fluctuations: a head-to-head contest asks whether a sum of three counts exceeds half the electorate, and the halves cancel. So in the limit an election is a draw of six correlated normal variables — the multivariate version of the bell curve from coin flips — and the efficiencies are probabilities of regions of a five-dimensional space, bounded by flat walls through the origin.

The figure samples that space a million times. A Condorcet winner exists 91.1% of the time, against the exact limit $\tfrac34 + \tfrac{3}{2\pi}\arcsin\tfrac13 \approx 91.23\%$ that the cycle essay derived, a gap of two standard errors. Given a winner, the runoff elects it 96.3% of the time, Borda 90.1%, plurality 75.7% and anti-plurality 75.6%. William Gehrlein and Peter Fishburn computed limits of this kind analytically in the 1970s, as orthant probabilities of normal distributions, and plurality’s 0.7572 is their value to four places.

Two features of the limit are worth noticing. The rules converge from different sides — plurality falls towards its limit and anti-plurality rises to meet it — so the small electorates are not a uniformly worse or better version of the large ones. And the convergence is slow and wobbles with parity: at 51 voters plurality is still more than two points above its limit, because the share of near-ties, where small numbers matter, falls only like $1/\sqrt n$.

Why the runoff is so hard to beat #

The runoff’s lead deserves an explanation, because it is a rule that throws away almost as much information as plurality does. It looks at first places, discards one candidate, and asks a single head-to-head question. The reason it does so well with three candidates is that the head-to-head question is exactly the one a Condorcet winner always answers correctly. If the winner survives the first round, it meets one rival in the second, and it beats every rival by definition. So the runoff fails precisely when the Condorcet winner has the fewest first places and is eliminated before it can be asked about.

That is a much rarer event than plurality’s failure, which happens whenever the Condorcet winner fails to have the most first places. A candidate preferred by a majority to each rival tends to be ranked first by a fair number of voters, and coming bottom of three on first places while beating both rivals head to head needs a particular shape: the winner must be most voters’ second choice. In the limit that shape occurs in about 3.7% of elections with a Condorcet winner, which is the runoff’s whole shortfall, against plurality’s 24.3%. The difference between the two rules is the difference between “not last” and “first” on one statistic, and for a candidate whose strength is breadth rather than depth that difference is large.

The same reasoning predicts where the runoff weakens. With more candidates, instant runoff eliminates one candidate at a time, and the Condorcet winner must avoid being bottom on first places in every round, against a field that grows thinner as transferred ballots pile up. A compromise candidate who is most voters’ second choice can be eliminated in an early round long before its broad support is counted, and the candidates figure below shows instant runoff sliding from 0.96 to 0.84 as the field grows from three to seven. Borda reads the second choices directly and slides less. The elimination rules and the scoring rules trade places, and which is better depends on how many candidates there are — a fact that the minimal profiles, all of which use three candidates, could never show.

More candidates, and plurality falls behind #

Three candidates is the gentlest case. With more, the impartial culture spreads first places more thinly, and the figure samples elections of 201 voters with three to seven candidates — exact enumeration is out of reach, since seven candidates have 5,040 rankings — and counts how often each rule finds the Condorcet winner. The runoff becomes instant runoff, eliminating the candidate with the fewest first places one at a time.

The share of elections with a Condorcet winner falls, from about nine in ten with three candidates to about six in ten with seven. Among those that have one, plurality’s record falls fastest, to 0.47 at seven candidates, and anti-plurality’s to 0.42, while Borda holds at 0.85 and instant runoff at 0.84. The reason is the same as at three voters, scaled up. A Condorcet winner needs only to be preferred to each rival by a majority, which a candidate can manage while being almost nobody’s first choice — a compromise everyone ranks second or third. Plurality looks only at first places and so misses exactly those compromise winners, and the more candidates there are, the more of the field’s support is first-place support for somebody else. Borda reads every position on every ballot and is much harder to fool.

Why Borda misses when it does #

Borda’s failures have a precise description, because Borda’s count is the head-to-head contests in disguise. A candidate gets two points from each ballot that ranks it first and one from each that ranks it second, which is the same as one point for every rival it is ranked above. So a candidate’s Borda score is the total number of head-to-head votes it receives, summed over its contests with every rival — an identity the figure checks in each of its 3,650 elections. The Condorcet winner wins every contest; Borda elects whoever collects the most head-to-head votes in total. They disagree exactly when a rival loses narrowly to the winner and beats the third candidate by much more than the winner does.

The scatter plots that picture. Each dot is an election with a Condorcet winner, placed by the winner’s smallest margin and by its Borda lead over the best rival. The dots form a fan rising to the right: a winner with large margins everywhere has a large Borda lead. The 262 elections below the line, where Borda goes wrong, sit almost entirely at the left, where the winner’s closest contest is won by one or three votes. Borda misses the Condorcet winner only when that winner is barely one, and in those cases it elects a candidate who would have been the Condorcet winner with a vote or two changed — which is part of why many writers on voting, from Borda himself to Donald Saari, have argued that its failures are not failures at all.

The most disordered electorate, not a real one #

The three-candidate efficiencies for 3 to 51 voters are exact — every split enumerated, weights computed from factorials in floating point, ties split by an explicit rule — and the tables for three, five and seven voters count every ordered profile. The minimal profiles are exact searches, and each is the first failure found in a scan that started at three voters, so no smaller electorate trips the rule; there may be other profiles of the same size. The limits are Monte Carlo estimates from a million samples of the normal limit, with standard errors near three hundredths of a percentage point, and the candidates figure samples three thousand elections a point, so its values are good to about a percentage point.

None of this says how often any rule finds the Condorcet winner in a real election. The impartial culture is the most disordered electorate there is, with no shared sense of which candidate is moderate, and real preferences are far from it; a party that loses every issue can win found the same limitation for issue voting. Under a culture of single-peaked preferences, where every voter ranks candidates by distance along one left–right line, a Condorcet winner always exists — the median voter’s favourite — and the rules’ efficiencies change, plurality’s typically for the worse when the median candidate is squeezed between two neighbours.

And efficiency is one criterion. A rule that always elects the Condorcet winner exists — elect it when there is one, and fall back on something else when there is not — and such Condorcet methods, of which the fewest swaps to a winner described Dodgson’s, score one by construction. What the counts measure is how far the familiar rules, which were not designed around this criterion, happen to satisfy it anyway.

Still open: efficiency with structured voters #

For the impartial culture the main questions are settled, with limits known analytically for small numbers of candidates and computable for more. The open questions concern every other model. How efficient is plurality when preferences are spatial in two dimensions, with voters and candidates scattered in a plane and rankings by distance? The answers are known by simulation for particular distributions and depend strongly on them, and no analytic theory covers the general case. How do the rules compare when voters vote strategically, misreporting rankings in the way a lie that pays described, so that the profile counted is not the profile held? The efficiencies above assume sincere ballots throughout, and with strategic ballots the ordering of rules can change.

There is also a question about how to read the limit itself. The convergence in the hero is slow and parity-dependent, and the exact finite-$n$ efficiencies have been worked out as closed forms only for very small electorates. Whether there is a useful expansion of the efficiency in powers of $1/\sqrt n$, with computable coefficients, that would let a few small electorates predict the limit without sampling, is a question the exact counts here raise and do not answer.

A majority favourite is easy to miss #

When every ranking is equally likely and one of three candidates beats both rivals head to head, plurality elects it about three times in four, Borda nine times in ten and the runoff between the top two ninety-six times in a hundred — exactly for every electorate up to 51 voters and in the limit by sampling. Anti-plurality and plurality can be tripped by three and five voters, Borda and the runoff by five. Borda’s score is the candidate’s total of head-to-head votes, so it misses only when the Condorcet winner’s closest contest is narrow; plurality misses compromise winners that few voters rank first, and with seven candidates it finds the Condorcet winner less than half the time.