Condorcet paradox
Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory.
In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters.
Thus, any attempt to ground social decision-making in majoritarianism must accept such self-contradictions (commonly called spoiler effects).
Systems that attempt to do so, while minimizing the rate of such self-contradictions, are called Condorcet methods.
Condorcet's paradox is a special case of Arrow's paradox, which shows that any kind of social decision-making process is either self-contradictory, a dictatorship, or incorporates information about the strength of different voters' preferences (e.g. cardinal utility or rated voting).
Condorcet's paradox was first discovered by Catalan 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.
[1][2][3] 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.
As a result, any attempt to appeal to the principle of majority rule will lead to logical self-contradiction.
The voters in Cactus County prefer the incumbent county executive Alex of the Farmers' Party over rival Beatrice of the Solar Panel Party by about a 2-to-1 margin.
Charlie is a wealthy and outspoken businessman, of whom the voters hold polarized views.
A majority of voters are either Beatrice-lovers or Charlie-haters, so prefer Beatrice to Charlie (B > C).
And a majority of voters are either Charlie-lovers or Alex-haters, so prefer Charlie to Alex (C > A).
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.
We can calculate the probability of seeing the paradox for the special case where voter preferences are uniformly distributed among the candidates.
[11] The simulated likelihood for an impartial culture model with 25 voters increases with the number of candidates:[11]: 28 The likelihood of a Condorcet cycle for related models approach these values for three-candidate elections with large electorates:[9] All of these models are unrealistic, but can be investigated to establish an upper bound on the likelihood of a cycle.
[5]: 78 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.
[11]: 31 Many attempts have been made at finding empirical examples of the paradox.
[13] Empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available.
[14] 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%[6]: 325 (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without).
[12] A similar analysis of data from the 1970–2004 American National Election Studies thermometer scale surveys found a Condorcet cycle likelihood of 0.4%.
[15] A database of 189 ranked United States elections from 2004 to 2022 contained only one Condorcet cycle: the 2021 Minneapolis City Council election in Ward 2, with a narrow circular tie between candidates of the Green Party (Cam Gordon), the Minnesota Democratic–Farmer–Labor Party, (Yusra Arab) and an independent democratic socialist (Robin Wonsley).
[16] Voters' preferences were non-transitive: Arab was preferred over Gordon, Gordon over Wonsley, and Wonsley over Arab, creating a cyclical pattern with no clear winner.
The several variants of the Condorcet method differ on how they resolve such ambiguities when they arise to determine a winner.
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.
This logical inconsistency is the origin of the poison pill amendment, which deliberately engineers a false Condorcet cycle to kill a bill.
Likewise, the order of votes in a legislature can be manipulated by the person arranging them to ensure their preferred outcome wins.
Despite frequent objections by social choice theorists about the logically incoherent results of such procedures, and the existence of better alternatives for choosing between multiple versions of a bill, the procedure of pairwise majority-rule is widely-used and is codified into the by-laws or parliamentary procedures of almost every kind of deliberative assembly.
Condorcet paradoxes imply that majoritarian methods fail independence of irrelevant alternatives.
Condorcet cycles are rare in large elections,[18][19] and the median voter theorem shows cycles are impossible whenever candidates are arrayed on a left-right spectrum.
