Paradoxes and Limits

Every method in How Voting Systems Work 2 to 5 had a weak spot: vote splitting for plurality, the centre squeeze for runoffs, cycles for pairwise contests, easy point-shading for Borda. It is natural to ask whether a cleverer rule could have none of them. The mathematical answer, for ranked ballots, is a discouraging “not quite”. This chapter looks at the two best-known results, Arrow’s theorem (about fairness) and the Gibbard–Satterthwaite theorem (about honesty), and then does something more useful than quoting them: it writes small programs that test each method against named fairness criteria and count how often it fails. The maths stays at counting and comparing.

The Theorems in Plain Words

Arrow’s theorem, published by the economist Kenneth Arrow in his 1951 book Social Choice and Individual Values, concerns rules that turn everyone’s full rankings of three or more options into one ranking for the group. In the usual statement, no such rule satisfies all of these at once: it works for every possible set of ballots (unrestricted domain); if everyone prefers X to Y then the group does (unanimity, or Pareto efficiency); the group’s order of X and Y depends only on how voters ordered X and Y (independence of irrelevant alternatives, IIA); and no single voter always decides (non-dictatorship). The sources I found agree on the result and the 1951 date; the exact wording of the four conditions varies between textbooks. They also note the theorem can be avoided by restricting the ballots allowed, for example to preferences that follow a single left-right line.

The Gibbard–Satterthwaite theorem (credited to Allan Gibbard in 1973 and Mark Satterthwaite in 1975) is about honesty. Roughly: with three or more possible winners, any rule that can elect any of them and is not a dictatorship will sometimes give some voter a reason to put down a ballot that is not their true ranking. Neither theorem says every method is equally bad, or that failures are common. They say a perfect method cannot exist. How often and how badly each one fails is a question for counting, which is what the rest of this chapter does.

Independence: Test It by Code

A criterion becomes testable once you state it as a check on two profiles. For IIA: take a profile, change only where one third candidate (Cy) appears on some ballots, so that no voter’s Ana-versus-Bo order moves, and see whether the method’s verdict on Ana versus Bo flips. The code below does that for the Borda count with five invented voters. It tries every way of moving Cy on the two ballot types and prints the first one that flips the verdict.

Before, Bo wins the Borda count 7 to 6 over Ana. Then Cy is moved to the top of both kinds of ballot, and the order of Ana and Bo in each ballot is unchanged (three voters still prefer Ana to Bo, two still prefer Bo). Yet Ana now beats Bo 3 to 2 on points. The voters said nothing new about Ana or Bo; the points changed because Cy’s position changed. That is an IIA failure. The Borda count is not alone: one search result said plurality and ranked runoffs fail IIA as well, which How Voting Systems Work 2 showed in the vote-splitting form (check).

Strategic Voting

The oldest tactic is the compromise vote under plurality. Imagine 100 invented voters: 40 prefer Ana, 35 prefer Bo, 25 prefer Cy and would rank Bo second. If everyone is honest Ana wins. If the 25 move to Bo, Bo wins.

The numbers are zero-based places: the honest winner, Ana, was last on the 25 Cy voters’ true ballots (place 2), and Bo, their second choice, is place 1. By lying, the 25 voters swapped an outcome they liked least for one they liked more. Unless they are sure of the other voters’ plans this is a gamble, and a poll can persuade many people to make the same move. Another tactic, burying, is to rank a dangerous rival last on purpose; the exercises below try it under the Borda count.

Counting Failures

Now the whole test bench. The code draws 3,000 random elections of nine voters and three candidates, each ballot chosen uniformly from the six possible orders. When a Condorcet winner exists, it checks whether plurality, Borda and instant-runoff elected that candidate; when some candidate has a majority of first choices it checks the same. Ties are broken alphabetically, which is an assumption of this chapter.

A Condorcet winner existed in 2,775 of 3,000 elections. Plurality missed it in 514 of those (about 18.5%), Borda in 227 (about 8.2%) and instant-runoff in 109 (about 3.9%). A first-choice majority winner existed in 1,298 elections; plurality and instant-runoff always chose it, and Borda missed it in 50 (about 3.9%). That matches the textbook picture: plurality and runoffs meet the majority criterion, Borda does not, and none of the three always meets the Condorcet criterion. But the percentages belong to this random-ballot model, in which voters have no shared political shape at all. Real electorates are more structured, and the rates would change.

One Voter Lying

The Gibbard–Satterthwaite theorem says opportunities to gain by lying exist. This simulation measures how often a single voter could use one. For each of 1,000 random elections of seven voters and three candidates, it tries every possible ballot for every voter, keeping the others fixed, and asks whether any lie leads to a winner that voter truly ranks higher.

Plurality had a profitable lie in 25.3% of elections, Borda in 24.3%, and instant-runoff in 11.2%. Instant-runoff did better here, in line with a source I found saying it resists some tactics such as burying, but note what the test counts: only a lone voter, only three candidates, only seven voters. Coordinated groups, more candidates and strategies that need guessing others’ votes are outside it. The result is a measurement of one model, not a verdict on any system.

Monotonicity: a stranger failure

Instant-runoff has a quirk that none of the others shows here: it can fail monotonicity, meaning that a candidate can lose because some voters ranked them higher. Sources I found agree on the failure. The advocacy group FairVote argues it matters little in practice and that other properties matter more; that is one side of a real dispute, and I did not check how often the failure arises in real elections (check). The code searches random 21-voter elections for one, skipping any profile where a tie for last place would need a tie-break, and then lifts the winner one place on more and more of the ballots that do not rank it first.

Before, Ana is eliminated first (6 first choices against 7 and 8). Ana’s five “Ana, Cy, Bo” ballots move to Cy, so Cy wins with 13. After two ballots lift Cy a place, so that Cy has 10 first choices, Bo (5) is eliminated instead, Bo’s remaining five voters all rank Ana next, and Ana wins 11 to 10. Cy gained support and lost. The cause is that the candidate who is removed first changes with the first-choice counts, so extra first choices can move who is eliminated and where those votes then go.