The Condorcet Paradox

Title page of an essay The experience of applying analysis to the problem of probability of decisions by a majority vote , in which Condorcet formulated his paradox.

The Condorcet paradox is a paradox of the theory of public choice, first described by the Marquis of Condorcet in 1785 .

It consists in the fact that if there are more than two alternatives and more than two voters, the collective ranking of alternatives can be cyclical (not transitive ), even if the ratings of all voters are not cyclical (transitive). Thus, the will of different groups of voters, each of which represents a majority, can come into paradoxical conflict with each other.

Generalized by the theorem of "impossibility" Arrow in 1951.

In practice, the idea of the need for ranking candidates was implemented by voting according to the Schulze method .

Content

Condorcet Principle

Condorcet defined a rule according to which a comparison of the chosen alternatives (candidates) is made taking into account the complete ordinal information on voters' preferences.

According to the Condorcet principle, to determine the true will of the majority, it is imperative that each voter rank all candidates in the order of their preference. After that, for each pair of candidates it is determined how many voters prefer one candidate to another - a complete matrix of pairwise preferences of voters is formed.

Based on this matrix, using the transitivity of the preference relation, we can try to build a collective ranking of candidates.

Example of application of the principle

Here is a numerical example from Condorcet.

For brevity, we introduce the notation: $A\succ B\succ C$ ${\ displaystyle A \ succ B \ succ C}$ $A\succ B\succ C$ will mean that the voter prefers candidate A to candidate B and candidate B to candidate C.

Let 60 voters give the following preferences:

23 people: $A\succ C\succ B$ ${\ displaystyle A \ succ C \ succ B}$ $A\succ C\succ B$
19 people: $B\succ C\succ A$ ${\ displaystyle B \ succ C \ succ A}$ $B\succ C\succ A$
16 people: $C\succ B\succ A$ ${\ displaystyle C \ succ B \ succ A}$ $C\succ B\succ A$
2 people: $C\succ A\succ B$ ${\ displaystyle C \ succ A \ succ B}$

When comparing A with B, we have: 23 + 2 = 25 people for the fact that $A\succ B$ ${\ displaystyle A \ succ B}$ , and 19 + 16 = 35 people for $B\succ A$ ${\ displaystyle B \ succ A}$ .

According to the Condorcet principle, the majority opinion is that B is better than A.

Comparing A and C, we will have: 23 people per $A\succ C$ ${\ displaystyle A \ succ C}$ and 37 people for $C\succ A$ ${\ displaystyle C \ succ A}$ . From here, according to Condorcet, we conclude that the majority prefers candidate C to candidate A. Similarly (19 people for $B\succ C$ ${\ displaystyle B \ succ C}$ , 41 people for $C\succ B$ ${\ displaystyle C \ succ B}$ ) C is more preferable than B.

Thus, according to Condorcet, the will of the majority is expressed in the form of three judgments: $C\succ B$ ${\ displaystyle C \ succ B}$ ; $B\succ A$ ${\ displaystyle B \ succ A}$ ; $C\succ A$ ${\ displaystyle C \ succ A}$ which can be combined into one relationship of preference $C\succ B\succ A$ ${\ displaystyle C \ succ B \ succ A}$ and if it is necessary to choose one of the candidates, then, according to the Condorcet principle, candidate C.

Contradiction to majority voting system

Let us compare this conclusion with the possible outcome of voting by the majority system of the relative or absolute majority.

For the above example, voting on an absolute majority system will give the following results: for A - 23 people, for B - 19 people, for C - 18 people. Thus, in this case, candidate A. wins.
When voting on a two-round relative majority system, candidates A and B will enter the second round, where candidate A will receive 25 votes and candidate B will receive 35 votes and win.

We get that the rules of the game will determine the winner, and these winners will be different under different voting rules. According to the second procedure widely used in the world, a candidate can win, who would lose to a candidate who was eliminated in the first round by a pairwise vote with a ratio of up to 1 to 1.99 ... The paradox of this situation in real elections is sometimes confused with the Condorcet paradox itself. ^[1] The Condorcet principle eliminates such errors associated with incomplete consideration of voters' preferences in the first round, but can lead to an insoluble contradiction.

The Condorcet Paradox

In another example reviewed by Condorce:

1 person: $A\succ B\succ C$ ${\ displaystyle A \ succ B \ succ C}$
1 person: $C\succ A\succ B$ ${\ displaystyle C \ succ A \ succ B}$
1 person: $B\succ C\succ A$ ${\ displaystyle B \ succ C \ succ A}$

based on the results of voting, two-thirds of the votes receive three statements: $B\succ C$ ${\ displaystyle B \ succ C}$ , $C\succ A$ ${\ displaystyle C \ succ A}$ , $A\succ B$ ${\ displaystyle A \ succ B}$ . But together, these statements are contradictory. This is the Condorcet paradox or the paradox of collective choice. It turns out to be impossible to determine the will of the majority and make some kind of coordinated decision. If in order to assess the consistency of the preferences of these voters, we apply the Spearman rank correlation coefficient developed later, then the correlation coefficients between the preferences of any two voters from this three are negative and equal to −0.5 ^[2] .

By virtue of symmetry in this form, the paradox is insoluble by any tricks. But if we replace individual voters in this example with three groups with close, but not the same number of voters, for example, 9, 10 and 11, then the Schulze method allows us to formally determine the winner. Although the paradoxical cyclicality of collective ranking persists.

The paradox of composite voting

In another form, the Condorcet paradox arises when the adoption of a certain decree or law by article, when each of the articles of the law is adopted by a majority of votes, and the law put to a vote is generally rejected (sometimes even by a one hundred percent majority). Or vice versa, it is entirely possible that decisions will be collectively taken that were not supported at the individual level by any of the voters.

An example . Let us have three people voting on three issues. The first one votes “yes” on the first question, “yes” on the second and “no” on the third (“yes” / “yes” / “no”), the second - “yes” / “no” / “yes”, the third is “no” / “yes” / “yes”. The total voting result is calculated as the ratio of the sum of the yes and no votes for each of the issues. In the considered case, the total voting result will be “yes” / “yes” / “yes”. This result does not reflect the views of any of the voters and, of course, does not satisfy anyone.

Alternative Voting

In practice, Condorcet’s idea of the need for ranking candidates was implemented in an alternative vote . This method is used in elections to various authorities of Australia , New Zealand , Papua New Guinea , Fiji , Ireland , the USA , as well as in a number of political parties, non-governmental organizations, etc.

Anti-ratings

Condorcet’s paradox echoes the idea of an “anti-rating” policy. When determining the anti-ratings of potential voters, they are asked to name not only the most, but also the least supported candidates, that is, actually rank all the candidates by degree of preference.

Sources

Literature

Arrow KJ Social Choice and Individual Values, London, 1951
Granger G. G. La mathématique sociale du Marquis de Condorcet, Paris, 1956
Sen AK Collective Choice and Social Welfare, London, 1970

Notes

[1] Illustration du paradoxe de Condorcet dans les sondages de l'élection présidentielle française 2007

[2] Poddyakov A.N. Intransitivity (non-transitivity) of relations of superiority and decision-making // Psychology. Journal of the Higher School of Economics. 2006. No. 3. S. 88-111.

The Condorcet Paradox

Content

Condorcet Principle

Example of application of the principle

Contradiction to majority voting system

The Condorcet Paradox

The paradox of composite voting

Alternative Voting

Anti-ratings

Sources

Literature

Notes

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