The flaw in every voting system

Source: The flaw in every voting system, Polylog, 18:25, uploaded 2023-08-10, Watch Later position 1137.

Polylog opens on an island where monkeys argue about fruit. Their disagreement gives the video a small setting in which the basic problem of voting becomes visible: the same preferences can produce different winners when the voting rule changes, and voters can sometimes improve the result for themselves by pretending to prefer a different candidate.

Three fruits and two election rules

The monkeys fall into three groups. One group ranks avocado first, banana second and coconut last. A second group prefers banana, then coconut, then avocado. The third group puts coconut first, avocado second and banana last.

In a one-round election, every monkey votes for one favourite. Avocado receives four votes, banana two and coconut three, so avocado wins. A two-round election sends avocado and coconut into a second round. The banana supporters then choose coconut, which wins with five votes. The preferences have stayed fixed while the winner has changed because the rule has changed.

The monkeys run the two-round election again. This time the first round produces six votes for banana, three for coconut and none for avocado. Banana wins the runoff. The four avocado supporters have voted for their second choice because they know that avocado cannot beat coconut in the second round. Their sincere vote would leave them with their least preferred candidate. The election has turned their ballot into a strategic decision.

The useful question follows from this small example. Could a voting system always reward a voter who reports their preferences honestly? Polylog answers by proving a simplified version of the Gibbard–Satterthwaite theorem, which says that every voting system with a modest notion of reasonableness sometimes gives a voter an incentive to vote strategically.

What counts as a voting system

The video defines each voter by a ranking of the candidates. All those rankings go into a function, and the function returns the winner. The two-round system fits this definition because the function can first select the two candidates with the most first-place votes and then compare the rankings between them. A tie-breaking rule, such as preferring avocado to banana to coconut, completes the definition for every possible input.

Strategic voting needs a distinction between a voter’s real preference and the ranking written on the ballot. Suppose the other voters have already voted and the voter’s honest ballot would make coconut win. If the voter swaps avocado and banana on the ballot, the two-round system can instead elect avocado. Since the voter prefers avocado to coconut, the lie improves the outcome. The term refers to the change in the voter’s action that becomes useful in a particular arrangement of the other ballots.

The word “reasonable” excludes a system that always elects the top choice of voter number three. That dictatorship is a voting system in the formal sense, yet nobody else can influence its result. Polylog uses a simpler condition: whenever a candidate is the first choice of a majority of voters, that candidate must win. Both plurality voting and the two-round system meet this condition. The condition is chosen because it is intuitive and makes the proof accessible, although the companion blog post says that it is not the least restrictive definition used in the full theorem.

The Condorcet cycle

The three groups produce a cycle in pairwise comparisons. Seven monkeys prefer avocado to banana, six prefer banana to coconut and five prefer coconut to avocado. Every comparison has a majority, yet the majorities point in a circle. Whichever fruit wins, another fruit can defeat it in a head-to-head vote. This is the Condorcet paradox, and the circular pattern is a Condorcet cycle.

The cycle explains why a majority preference cannot settle the whole election. With two candidates, the majority relation has a clear direction. Three or more candidates allow the pairwise comparisons to return to their starting point, leaving the voting system to decide which part of the cycle should count most.

A single voter at the point of change

Take any voting system that satisfies Polylog’s majority condition and apply it to this cycle. The system has to elect one fruit. Assume that it elects coconut. The monkeys who rank coconut last have the strongest reason to change the result. If this whole group swaps avocado and banana on its ballots, banana becomes the first choice of a majority, so the system has to elect banana. Coordination has moved the result from the group’s last choice to its second choice.

The theorem needs a single voter who can benefit after the group argument. Start with the original cycle and make a sequence of intermediate scenarios in which one voter at a time swaps avocado and banana. The first scenario elects coconut and the final scenario elects banana, so the winner must change somewhere along the sequence. At the first change, only one ballot differs between the two adjacent scenarios.

If the voter’s ballot in the earlier scenario records their honest ranking, the system elects coconut, their worst option. The voter can submit the adjacent ballot with avocado and banana swapped, which changes the winner to something they prefer to coconut. That ballot is strategic, and the existence of the first change completes the simplified proof.

The source calls this a “baby version” of Gibbard–Satterthwaite. The full theorem uses a broader definition of a voting system and a weaker condition than the majority rule used here. The companion blog post explains that the general result covers systems in which voters choose from a set of actions, including approval voting, and that the video leaves the formal proof at this simpler level.

Arrow’s theorem and irrelevant alternatives

Arrow’s theorem concerns a more ambitious system that ranks every candidate from best to worst instead of selecting only a winner. One desirable property is independence of irrelevant alternatives. If the final ranking places avocado above banana, changing the voters’ opinions about coconut should leave that relative order intact. Coconut is irrelevant to the comparison between avocado and banana.

The video presents Arrow’s theorem as the related impossibility result: no reasonable ranking system can preserve this independence property. A change in the position of a third candidate can alter which of two candidates appears above the other. The Condorcet cycle supplies the underlying pressure, since the system has to break a circle of pairwise majorities somewhere.

Polylog’s larger point concerns what these results reveal. Strategic voting and the failure of independence are mathematical consequences of the formal setup, even when political culture also shapes how people vote. The claim still needs its conditions. Arrow’s theorem applies to ranking systems built from candidate rankings, and the simplified argument uses the video’s own definition of reasonableness.

Approval voting changes the input

Approval voting does not ask each person to provide a complete ranking. A voter can approve as many candidates as they like, after which the candidates are ordered by their approval totals. The number of likes on a set of videos gives a familiar approximation, although approval voting has no requirement that a voter approve only a favourite.

This difference matters because a ranking does not reveal where a voter’s approval threshold lies. Someone might approve only their first choice, their first two choices or none of the candidates. The video’s proofs therefore do not apply directly to approval voting. Arrow’s theorem also fails to apply in this setting because approval totals satisfy independence of irrelevant alternatives: if one video has 35,000 likes and another has 5,000, the first remains above the second regardless of the other videos’ totals.

Approval voting still gives voters strategic reasons to shape their ballot. Someone who wants one video to receive the highest score should approve it and disapprove every other video, even when they like some of those other videos. The companion blog post supplies the general formulation that extends Gibbard’s result to systems with arbitrary voter actions. Strategic behaviour therefore returns once the ballot is allowed to express more than a ranking.

Worst cases and practical choices

The theorems describe worst-case possibilities and leave ordinary performance open. When thousands of entries compete in a mathematics exposition contest, the voting system’s ability to handle the number of candidates may matter more than strategic voting. Kenneth Arrow’s own summary, as quoted by the video, is that a system can work badly at times without working badly all the time.

Polylog says that a small group of voting theorists once ranked systems for political elections and placed approval voting at the top. The vote itself used approval voting, which gives the result a built-in joke and a clear limit. The video supplies no study or method for this ranking, so it remains an illustration rather than evidence that approval voting is the best system in general.

The same informal ranking places alternative vote, also called instant-runoff voting, among the popular systems, the two-round system in a middling group and plurality voting at the bottom. Plurality is used in elections in several countries despite its familiar strategic problems, including the spoiler effect that can split support between similar candidates. The source treats these labels as a practical comparison, not as a theorem that orders every voting system for every electorate.

The final proposal changes the rules again. Each monkey writes only its favourite fruit on a ballot, and the winner comes from a random draw of those ballots. Strategic voting cannot improve a monkey’s chance by naming a second choice, while a candidate supported by more monkeys becomes more likely to win. The monkeys reject the result because randomness feels excessive when it selects banana, and the narrator leaves the island while they continue arguing about which system deserves their trust.

Further reading / references

• Václav Rozhoň’s companion post on the video, which gives the standard statements of Arrow’s and Gibbard–Satterthwaite’s theorems, discusses approval voting and the problem of defining a “reasonable” system, and lists further sources.
• Polylog’s voting-systems repository, which contains the code and project materials for the video.
• Comparison of electoral systems, the comparison page linked in the video’s description. It collects criteria, models and studies for evaluating voting methods, including resistance to tactical voting.