Why Democracy Is Mathematically Impossible

Source: Why Democracy Is Mathematically Impossible, Veritasium, 23:34, uploaded 2024-08-27, playlist index 645.

Veritasium opens with a claim that sounds like a judgement about democracy itself: the system might be mathematically impossible. The claim is narrower. The problem lies in the ways modern democracies aggregate individual preferences into one social decision. The video follows the mathematical history of that problem, from plurality elections through ranked-choice voting and the work of Kenneth Arrow, then looks for voting systems that avoid the worst failures.

First-past-the-post and the spoiler effect

The simplest election asks each voter to mark one favourite. The candidate with the most votes wins, even when that candidate receives less than half of all votes. This is first-past-the-post, a system with roots in antiquity that has elected members of the English House of Commons since the fourteenth century. The video says that 44 countries still use it to elect their leaders, with 30 of those countries formerly belonging to the British Empire. In the United States, most states use it to choose representatives to the Electoral College.

Plurality voting can give a party control of a parliament after most voters chose another party. Veritasium reports that, over the previous hundred years, a single party won a majority of seats in the British Parliament 21 times whilst receiving a majority of the votes only twice. The system turns a minority of votes into a majority of seats, which gives one party control of the government.

The same rule creates a second problem when similar candidates compete. The 2000 United States presidential election offers the example. George W. Bush won Florida by fewer than 600 votes, whilst Ralph Nader received almost 100,000 votes in the state. Nader stood to the left of both Bush and Al Gore, and most of his voters preferred Gore to Bush. Since the ballot let them choose one candidate only, their vote for Nader helped Bush win Florida. The result is the spoiler effect: a candidate can lose whilst changing which of the remaining candidates wins.

The system therefore rewards strategic voting. A voter who prefers a small party has to consider whether that party can win, since a vote for it may help the least preferred major candidate. As voters and candidates respond to that pressure, larger parties collect support and smaller parties disappear. The tendency towards two dominant parties is known as Duverger’s Law.

Ranked preferences and the Minneapolis debate

One response asks voters to rank candidates from first to last. If nobody wins a majority of first choices, the candidate with the fewest votes is removed and those ballots move to their second choices. The process repeats until a candidate has a majority. This is instant runoff, also called preferential or ranked-choice voting. Mathematically, it gives one election the effect of several rounds whilst avoiding the cost of holding them separately.

Ranked choice changes the candidates’ incentives as well as the voters’. In the 2013 Minneapolis mayoral election, 35 people ran after the incumbent mayor stepped down. The candidates expected to need second- and third-choice support from one another’s voters, so they stayed unusually polite. At the final debate they gathered and sang “Kumbaya”. The scene is comic, although the reason matters: a voting rule can change the kind of campaign that makes sense.

Instant runoff carries its own failure. Imagine three candidates, Einstein, Curie and Bohr. Einstein receives 25 percent of the first choices, Curie 30 percent and Bohr 45 percent. Since nobody has a majority, Einstein is eliminated. Einstein’s voters prefer Curie as their second choice, so Curie wins.

Now Bohr gives a terrible speech or proposes a policy so unpopular that some of his voters move to Einstein. Curie then becomes the lowest-ranked candidate and leaves the race. Half of Curie’s voters move to Einstein and the other half to Bohr, which lets Bohr win. Bohr’s weaker first-round result has helped him. A candidate can improve the final outcome by losing support, which makes the method hard to regard as a stable expression of preference.

Condorcet’s pairwise method

The French mathematician Nicolas de Condorcet began applying mathematics and logic to voting during the French Revolution, when the question of how to determine the will of the people had immediate political force. In 1784, his contemporary Jean-Charles de Borda proposed a ranked points system. With five candidates, a first-place ranking earns four points, second place earns three and last place earns zero.

Condorcet objected that the Borda count makes the points depend on the number of candidates. Adding candidates who have no chance of winning can change the score of every other candidate and therefore change the winner. He called those extra candidates irrelevant factors in the judgement.

In 1785, Condorcet proposed a different rule. The winner should beat every other candidate in a head-to-head contest. Voters can still rank the candidates once, after which the count compares each pair and records how many voters prefer one candidate to the other. The video notes that Ramon Llull had described the same method around 450 years earlier whilst studying the election of church leaders. Llull’s book, Ars eleccionis, was lost and rediscovered only in 2001, so the method carries Condorcet’s name.

The pairwise rule seems fair until three options produce a cycle. Suppose three friends choose between burgers, pizza and sushi. One ranks the options as burgers, pizza, sushi. The second ranks them pizza, sushi, burgers. The third ranks them sushi, burgers, pizza. Two people prefer sushi to burgers, so sushi should beat burgers. Two prefer pizza to sushi, so pizza should beat sushi. Two prefer burgers to pizza, so burgers should beat pizza. The group prefers burgers to pizza to sushi to burgers. This is Condorcet’s paradox.

Condorcet died before resolving the problem. He helped draft a French constitution and criticised the constitution introduced when La Montagne took power during the Reign of Terror. The regime labelled him a traitor. He was arrested in 1794 and died in jail.

The conditions of a rational vote

For the next 150 years, mathematicians proposed variations on Condorcet’s and Borda’s systems. Charles Dodgson, better known as Lewis Carroll, worked on fair election methods when he was away from Alice in Wonderland. The same failures kept returning. A method could produce Condorcet cycles, or candidates with no chance of winning could affect the result.

Kenneth Arrow gathered these problems into a theorem. In his 1951 PhD thesis, he described five conditions that a rational voting system ought to satisfy when it combines individual rankings into one social ranking.

1. Unanimity. When every voter prefers one option to another, the social ranking should preserve that preference. If everyone prefers sushi to pizza, the group should prefer sushi to pizza.
2. Non-dictatorship. One person’s vote cannot override the preferences of everyone else. A result controlled by one decisive voter is a dictatorship rather than a democratic aggregation.
3. Unrestricted domain. The system must accept every possible set of ballots and produce one repeatable social ranking from it. It cannot avoid difficult preferences by ignoring the ballots or choosing at random.
4. Transitivity. If the group prefers burgers to pizza and pizza to sushi, it must prefer burgers to sushi.
5. Independence of irrelevant alternatives. Adding a third option should not change the group’s existing preference between two options. If the group prefers sushi to pizza, introducing burgers can alter the place of burgers in the ranking, although it should leave sushi’s position relative to pizza alone.

Arrow proved that a ranked voting system with three or more candidates cannot satisfy all five conditions at once. This is Arrow’s impossibility theorem, which earned him the 1972 Nobel Prize in Economics. The theorem does not show that every election produces absurd results. It shows that every ranked aggregation rule must abandon at least one condition.

A version of Arrow’s proof

The video presents a version of the proof based on a formulation by John Geanakoplos. Three candidates, Aristotle, Bohr and Curie, become A, B and C. The voters stand in a row from voter 1 through voter N, and each voter can rank the candidates in any order, including ties.

The proof first establishes that if every voter places one candidate at the top or bottom of their ranking, society must also place that candidate at the top or bottom. Choose B. Half the voters put B first and half put B last. Assume the social ranking puts B in the middle, with A above B and B above C. Now let every voter move C above A. Unanimity requires the social ranking to put C above A. The relative positions of A to B and C to B have stayed fixed, however, so independence of irrelevant alternatives preserves A above B and B above C. Transitivity then requires A above C. The same social ranking must place C above and below A, which is a contradiction.

Next, every voter puts B at the bottom, whilst the relative order of A and C remains open. Unanimity puts B at the bottom of the social ranking. This is Profile 0. Create Profile 1 by moving B from the bottom to the top for voter 1, then Profile 2 by doing the same for voter 2, and continue until every voter places B at the top. At some point one voter’s change first moves B from the bottom of the social ranking to the top. That voter is pivotal. Profile p is the profile after the pivotal change and Profile o is the one before it.

Now make Profile q identical to p, except that the pivotal voter moves A above B. Independence of irrelevant alternatives says that the social ranking must also place A above B. Because the other voters preserve the old relation between A and B, and the move to Profile p has put B above C, transitivity forces A above C. The other voters can rearrange A and C in any way they like without changing those two pairwise relations to B. The pivotal voter therefore controls society’s preference between A and C.

The same construction with C at the bottom shows that the pivotal voter controls society’s preference between A and B. It is the same pivotal voter, so that voter determines the social preference between every pair. The five reasonable conditions have produced a dictator.

The single axis and approval voting

Arrow’s result leaves room for more hopeful cases. Duncan Black showed that when voters and candidates sit along one natural dimension, such as a liberal-to-conservative political scale, the median voter’s preference can reflect the majority decision. The median voter’s choice often determines the result, which avoids the cycles and inconsistencies that appear when preferences move across several dimensions.

Arrow’s theorem also concerns ordinal systems, where voters rank candidates relative to one another. Rated systems ask a different question. In approval voting, voters tick every candidate they approve of, and the candidate with the highest approval percentage wins. Other rated systems let voters express intensity on a scale such as -10 for strong disapproval through +10 for strong approval.

The video reports research claiming that approval voting increases turnout, reduces negative campaigning and avoids the spoiler effect. A voter can approve of a small-party candidate without sacrificing an expression of approval for a larger candidate. The count remains simple: add the approvals and compare the percentages.

Arrow initially doubted rated systems, although the video says he came to regard them late in life as probably the best method. Approval voting has historical precedents. Vatican priests used it to elect the Pope between 1294 and 1621, and the United Nations uses it to elect its Secretary-General. Large public elections have tested it far less often, so the video leaves its practical performance open.

The limit in the title

Democracy is mathematically impossible when democracy means a ranked-choice method that must satisfy all of Arrow’s conditions for three or more candidates. Most countries use ranked methods to elect their leaders, and some of those methods aggregate preferences more coherently than others. Veritasium treats first-past-the-post as especially hard to defend once its minority victories, spoiler effects and two-party pressure are visible.

The result does not settle the political question. People can remain interested in public life, care about issues and take part in elections even when every voting rule carries a cost. The video closes with Churchill’s line that democracy is “the worst form of government except for all the other forms that have been tried.” Its final judgement is modest: democracy is imperfect, and it remains the best available system in the source’s view.

Limits

The historical episodes, British election figures, Minneapolis account, reported research on approval voting and the proof reconstruction in this note follow Veritasium’s English captions and description. The description credits Eric Maskin for help with the script, Chris Dong for inspiring the video, Latif Nasser for appearing in it, and Gabriel Strong for additional research. I have not independently checked the underlying papers, election records or the reported approval-voting findings. The source’s conclusion about democracy applies to ranked aggregation under Arrow’s conditions, not to every possible democratic institution or voting procedure.

Further reading / references

• Yu, “A one-shot proof of Arrow’s impossibility theorem”
• Geanakoplos, “Three brief proofs of Arrow’s impossibility theorem”
• Arrow, Social Choice and Individual Values
• Arrow, “A Difficulty in the Concept of Social Welfare”
• Black, “On the rationale of group decision-making”
• Black, “On Arrow’s impossibility theorem”
• Maskin and Sen, The Arrow Impossibility Theorem
• Gehrlein and Valognes, “Condorcet efficiency: A preference for indifference”
• Dardanoni, “A pedagogical proof of Arrow’s Impossibility Theorem”
• McCune and Wilson, “Ranked-choice voting and the spoiler effect”
• Santucci, “Variants of ranked-choice voting from a strategic perspective”
• Kaminski, “Spoiler effects in proportional representation systems”
• Brams and Fishburn, “Approval voting”
• Radiolab, “Tweak the Vote”, linked in the video description.