Condorcet paradox

In social choice theory, a Condorcet paradox (or voting paradox) is a situation where majority rule behaves in a way that is self-contradictory. In such a situation, every possible choice is rejected by the electorate in favor of another, because there is always some other outcome that a majority of voters consider to be better.

History
Condorcet's paradox was first discovered by Spanish philosopher and theologian Ramon Llull in the 13th century, during his investigations into church governance, but his work was lost until the 21st century. The mathematician and political philosopher Marquis de Condorcet rediscovered the paradox in the late 18th century.

Condorcet's discovery means he arguably identified the key result of Arrow's impossibility theorem, albeit under stronger conditions than required by Arrow: Condorcet cycles create situations where any ranked voting system that respects majorities must have a spoiler effect.

Example
Suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows:

If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus the society's preferences show cycling: A is preferred over B which is preferred over C which is preferred over A.

As a result, any attempt to appeal to majority rule will cause self-contradictory behavior. Regardless of which alternative we select, we can find another alternative that would be preferred by most voters.

Likelihood of the paradox
It is possible to estimate the probability of the paradox by extrapolating from real election data, or using mathematical models of voter behavior, though the results depend strongly on which model is used.

Impartial culture model
We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates. (This is the "impartial culture" model, which is known to be a "worst-case scenario"  —most models show substantially lower probabilities of Condorcet cycles.)

For $$ n $$ voters providing a preference list of three candidates A, B, C, we write $$ X_n $$ (resp. $$ Y_n $$, $$ Z_n $$) the random variable equal to the number of voters who placed A in front of B (respectively B in front of C, C in front of A). The sought probability is $$ p_n = 2P (X_n> n / 2, Y_n> n / 2, Z_n> n / 2) $$ (we double because there is also the symmetric case A> C> B> A). We show that, for odd $$ n $$, $$ p_n = 3q_n-1/2 $$ where $$ q_n = P (X_n> n / 2, Y_n> n / 2) $$ which makes one need to know only the joint distribution of $$ X_n $$ and $$ Y_n $$.

If we put $$ p_{n, i, j} = P (X_n = i, Y_n = j) $$, we show the relation which makes it possible to compute this distribution by recurrence: $$ p_ { n + 1, i, j} = {1 \over 6} p_ {n, i, j} + {1 \over 3} p_ {n, i-1, j} + {1 \over 3} p_ {n, i, j-1} + {1 \over 6} p_ {n, i-1, j-1} $$.

The following results are then obtained: The sequence seems to be tending towards a finite limit.

Using the central limit theorem, we show that $$ q_n $$ tends to $$q = \frac{1}{4} P\left(|T| > \frac{\sqrt{2}}{4}\right),$$ where $$ T $$ is a variable following a Cauchy distribution, which gives $$q=\dfrac{1}{2\pi }\int_{\sqrt{2}/4}^{+\infty }\frac{dt}{1+t^{2}}=\dfrac{ \arctan 2\sqrt{2}}{2\pi }=\dfrac{\arccos \frac{1}{3}}{2\pi }$$ (constant quoted in the OEIS).

The asymptotic probability of encountering the Condorcet paradox is therefore $${{3\arccos{1\over3}}\over{2\pi}}-{1\over2}={\arcsin{\sqrt 6\over 9}\over \pi}$$ which gives the value 8.77%.

Some results for the case of more than three candidates have been calculated and simulated. The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates: The likelihood of a Condorcet cycle for related models approach these values for three-candidate elections with large electorates:
 * Impartial anonymous culture (IAC): 6.25%
 * Uniform culture (UC): 6.25%
 * Maximal culture condition (MC): 9.17%

All of these models are unrealistic, and are investigated to establish an upper bound on the likelihood of a cycle.

Group coherence models
When modeled with more realistic voter preferences, Condorcet paradoxes in elections with a small number of candidates and a large number of voters become very rare.

Spatial model
A study of three-candidate elections analyzed 12 different models of voter behavior, and found the spatial model of voting to be the most accurate to real-world ranked-ballot election data. Analyzing this spatial model, they found the likelihood of a cycle to decrease to zero as the number of voters increases, with likelihoods of 5% for 100 voters, 0.5% for 1000 voters, and 0.06% for 10,000 voters.

Another spatial model found likelihoods of 2% or less in all simulations of 201 voters and 5 candidates, whether two or four-dimensional, with or without correlation between dimensions, and with two different dispersions of candidates.

Empirical studies
Many attempts have been made at finding empirical examples of the paradox. Empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.

While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.

A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4% (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).

An analysis of 883 three-candidate elections extracted from 84 real-world ranked-ballot elections of the Electoral Reform Society found a Condorcet cycle likelihood of 0.7%. These derived elections had between 350 and 1,957 voters. A similar analysis of data from the 1970–2004 American National Election Studies thermometer scale surveys found a Condorcet cycle likelihood of 0.4%. These derived elections had between 759 and 2,521 "voters".

A database of 189 ranked United States elections from 2004 to 2022 contained only one Condorcet cycle: the 2021 Minneapolis Ward 2 city council election. While this indicates a very low rate of Condorcet cycles (0.5%), it's possible that some of the effect is due to general two-party domination.

Andrew Myers, who operates the Condorcet Internet Voting Service, analyzed 10,354 nonpolitical CIVS elections and found cycles in 17% of elections with at least 10 votes, with the figure dropping to 2.1% for elections with at least 100 votes, and 1.2% for ≥300 votes.

Implications
When a Condorcet method is used to determine an election, the voting paradox of cyclical societal preferences implies that the election has no Condorcet winner: no candidate who can win a one-on-one election against each other candidate. There will still be a smallest group of candidates, known as the Smith set, such that each candidate in the group can win a one-on-one election against each of the candidates outside the group. The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner. The Condorcet methods which always elect someone from the Smith set when there is no Condorcet winner are known as Smith-efficient. Note that using only rankings, there is no fair and deterministic resolution to the trivial example given earlier because each candidate is in an exactly symmetrical situation.

Situations having the voting paradox can cause voting mechanisms to violate the axiom of independence of irrelevant alternatives—the choice of winner by a voting mechanism could be influenced by whether or not a losing candidate is available to be voted for.

Two-stage voting processes
One important implication of the possible existence of the voting paradox in a practical situation is that in a two-stage voting process, the eventual winner may depend on the way the two stages are structured. For example, suppose the winner of A versus B in the open primary contest for one party's leadership will then face the second party's leader, C, in the general election. In the earlier example, A would defeat B for the first party's nomination, and then would lose to C in the general election. But if B were in the second party instead of the first, B would defeat C for that party's nomination, and then would lose to A in the general election. Thus the structure of the two stages makes a difference for whether A or C is the ultimate winner.

Likewise, the structure of a sequence of votes in a legislature can be manipulated by the person arranging the votes, to ensure a preferred outcome.

Spoiler effects
Condorcet paradoxes imply majoritarian methods fail independence of irrelevant alternatives. Label the three candidates in a race Rock, Paper, and Scissors. In a one-on-one race, Rock loses to Paper, Paper to Scissors, etc.

Without loss of generality, say that Rock wins the election with a certain method. Then, Scissors is a spoiler candidate for Paper: if Scissors were to drop out, Paper would win the only one-on-one race (Paper defeats Rock). The same reasoning applies regardless of the winner.

This example also shows why Condorcet elections are rarely (if ever) spoiled: spoilers can only happen when there is no Condorcet winner. Condorcet cycles are rare in large elections, and the median voter theorem shows cycles are impossible whenever candidates are arrayed on a left-right spectrum.