{ "id": "eppq57jz6d", "slug": "condorcet-paradox", "title": "Condorcet paradox", "updated": "2026-10-25", "topic_path": [ { "id": "society", "label": "Society and history", "api_url": "https://www.edgepedia.org/api/v1/topics/society" }, { "id": "society.politics", "label": "Politics and government", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics" }, { "id": "society.politics.elections", "label": "Elections and representation", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics.elections" }, { "id": "society.politics.elections.principles", "label": "Electoral systems and principles", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics.elections.principles" }, { "id": "society.politics.elections.principles.theory", "label": "Electoral theory and criteria", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics.elections.principles.theory" }, { "id": "society.politics.elections.principles.theory_paradoxes", "label": "Voting paradoxes and impossibility results", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics.elections.principles.theory_paradoxes" }, { "id": "society.politics.elections.principles.theory_paradoxes.condorcet_cycle", "label": "Condorcet cycle paradox", "api_url": "https://www.edgepedia.org/api/v1/topics/society.politics.elections.principles.theory_paradoxes.condorcet_cycle" } ], "geo": [ { "id": "geo.weu.t1000.society.politics", "label": "Western Europe · 1000 to 1799: Politics and government", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1000.society.politics", "path": [ { "id": "geo.weu", "label": "Western Europe", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu" }, { "id": "geo.weu.t1000", "label": "Western Europe · 1000 to 1799", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1000" }, { "id": "geo.weu.t1000.society", "label": "Society and history", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1000.society" }, { "id": "geo.weu.t1000.society.politics", "label": "Politics and government", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.weu.t1000.society.politics" } ] } ], "excerpt": "The Condorcet paradox is a voting situation where majorities cycle: A beats B, B beats C, and C beats A, described by Condorcet in 1785.", "snippet": "The Condorcet paradox is a voting situation where majorities cycle: A beats B, B beats C, and C beats A, described by Condorcet in 1785.", "node": "society.politics.elections.principles.theory_paradoxes.condorcet_cycle", "markdown": "# Condorcet paradox\n\nThe Condorcet paradox is the situation in which pairwise majority comparisons produce an intransitive group preference: alternative A beats B, B beats C, and C beats A, even though every individual voter ranks the three alternatives transitively. The [Marquis de Condorcet](https://www.edgepedia.org/marquis-de-condorcet) described the problem in 1785, and it became central to social choice theory after Duncan Black and [Kenneth Arrow](https://www.edgepedia.org/kenneth-arrow) rediscovered its importance for economic and political theory in the twentieth century[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\n| Key fact | Detail |\n|---|---|\n| Definition | A cycle in pairwise majorities: A defeats B, B defeats C, C defeats A[2](https://encyclopediaofmath.org/wiki/Condorcet_paradox) |\n| Condorcet winner | An alternative that beats every other alternative in pairwise comparisons; one may not exist even when all individual rankings are transitive[2](https://encyclopediaofmath.org/wiki/Condorcet_paradox) |\n| Three-candidate large electorates | About 6.25% under impartial anonymous culture, 9.17% under maximal culture[8](https://sites.socsci.uci.edu/~bgrofman/129%20Tsetlin%20et%20al.%202003.%20The%20Impartial%20Culture%20Maximizes%20the%20Probability...Soc.%20Choice.pdf) |\n| Real elections | A 2025 study of 253 elections in 59 countries found cycles in 5 of 8,099 three-candidate triplets (0.06%)[1](https://link.springer.com/article/10.1007/s11127-025-01353-7) |\n| Escape condition | Single-peaked preferences guarantee a Condorcet winner on one policy dimension[1](https://link.springer.com/article/10.1007/s11127-025-01353-7) |\n\n## What the paradox says\n\nCandidate A defeats candidate B if a majority of voters prefer A to B. With two candidates, one of them defeats the other (barring ties). With three or more candidates, cyclic situations can occur in which A defeats B, B defeats C, and C in turn defeats A[2](https://encyclopediaofmath.org/wiki/Condorcet_paradox). A Condorcet cycle occurs when there is a violation of transitivity in the social preference ordering[3](https://ocw.mit.edu/courses/14-75-political-economy-and-economic-development-fall-2012/a9fd8e5ab75a325016094e6bbe625b2a_MIT14_75F12_Lec12.pdf).\n\nA Condorcet winner is an alternative that gains a majority of votes when paired against each of the other alternatives[3](https://ocw.mit.edu/courses/14-75-political-economy-and-economic-development-fall-2012/a9fd8e5ab75a325016094e6bbe625b2a_MIT14_75F12_Lec12.pdf). If such a candidate exists it wins every pairwise comparison, but such a winner may not exist even when all individual preferences are transitive[2](https://encyclopediaofmath.org/wiki/Condorcet_paradox).\n\n## Why it happens\n\nPairwise majority comparison aggregates preferences locally, one pair at a time. [Majority rule](https://www.edgepedia.org/majority-rule) between two options is well behaved; the paradox arises when pairwise results over three or more options are combined into a single social ordering, and the pieces fail to fit together transitively[2](https://encyclopediaofmath.org/wiki/Condorcet_paradox).\n\nThe consequence is that the outcome can depend on the order of comparison. Agenda setting therefore acquires real power over the result, which is why the problem of Condorcet cycles and agenda setting is regarded as a deep, fundamental problem in social choice[3](https://ocw.mit.edu/courses/14-75-political-economy-and-economic-development-fall-2012/a9fd8e5ab75a325016094e6bbe625b2a_MIT14_75F12_Lec12.pdf).\n\n## By the numbers\n\nProbability models ask how often cycles arise under stated assumptions about how voters' preferences are drawn.\n\n**Impartial culture**, where each voter's ranking is drawn independently with every ranking equally likely, is the classic pessimistic benchmark. In one model, top-cycles occur about 9% of the time with 3 candidates but about 68% of the time with 20 candidates, and in the limit of many candidates the probability that a Condorcet winner exists tends to zero[4](https://rangevoting.org/CondorcetCycles).\n\n**More realistic culture models** give lower figures for three-candidate elections with large electorates: about 6.25% under impartial anonymous culture (where only the counts of each ranking are random) and 9.17% under the maximal culture condition[8](https://sites.socsci.uci.edu/~bgrofman/129%20Tsetlin%20et%20al.%202003.%20The%20Impartial%20Culture%20Maximizes%20the%20Probability...Soc.%20Choice.pdf).\n\n**Spatial voting models**, where voters have ideal points in a policy space, are more forgiving still.\n\nMethod matters as well as model. Formal solutions have a clear advantage for estimating the limiting values of cycle probability as the electorate or number of alternatives goes to infinity, while computer simulation models are far more tractable for finite values and extensions[5](https://journals.sagepub.com/doi/10.1177/089443939801600109).\n\n## How it compares with related results\n\nCondorcet exposed a limitation of majority-based pairwise comparison by showing that, for specific preference profiles over three alternatives, it leads to a contradiction, and [Arrow's impossibility theorem](https://www.edgepedia.org/arrows-impossibility-theorem) is often introduced as a generalization of this finding[6](https://link.springer.com/article/10.1007/s00355-024-01557-8). The connection is precise: a 2024 proof shows that for any non-dictatorial social choice rule satisfying independence of irrelevant alternatives, which can be represented as a set of pairwise comparison maps, a contradiction-generating preference profile always exists and can be identified by a straightforward procedure[6](https://link.springer.com/article/10.1007/s00355-024-01557-8).\n\nThe MIT formulation states the trade-off directly: if you want a social ordering with universal domain, Pareto optimality, independence of irrelevant alternatives, and no dictatorship, you cannot also have transitivity; you will get cycles[3](https://ocw.mit.edu/courses/14-75-political-economy-and-economic-development-fall-2012/a9fd8e5ab75a325016094e6bbe625b2a_MIT14_75F12_Lec12.pdf).\n\n## Escaping the cycle: domain restrictions\n\nIn spatial voting models, a Condorcet winner exists if voters' ideal points are value-restricted, such as single-peaked, on the underlying dimension; this is Black's 1948 condition[1](https://link.springer.com/article/10.1007/s11127-025-01353-7). When single-peakedness is a sufficiently good approximation of voter preferences, a Condorcet winner exists[7](https://maskin.scholars.harvard.edu/sites/g/files/omnuum10606/files/condorcet_voting_e.maskin_e.foley_march_2025_0.pdf).\n\nThe restriction is fragile in higher dimensions. With multidimensional policy spaces, Plott's 1967 radial symmetry condition is required, and without it cycles can extend across the whole option set, the result associated with McKelvey (1979)[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\n## Empirical evidence and controversies\n\nDocumented cases exist but are scarce. Before recent work, the literature identified only two likely real-election cases of the paradox: the 2016 US election (Potthoff and Munger 2021) and the 1994 Danish election (Kurrild-Klitgaard 2008)[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\n**The frequency dispute.** A 2025 study points much lower. Analyzing 253 electoral polls across 59 countries using CSES data, it found no robust evidence of cyclical majorities in any of the 253 elections and concluded the paradox has virtually no empirical relevance[1](https://link.springer.com/article/10.1007/s11127-025-01353-7). Across the 8,099 three-candidate triplets analyzed, cyclical majorities appeared in only five cases (0.06%)[1](https://link.springer.com/article/10.1007/s11127-025-01353-7). No instance was found among 212 parliamentary elections; among 41 presidential elections one case was identified, the 2011 Peruvian presidential election[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\nSmaller datasets fall between these poles. Small electorates cycle more often than large ones, matching the spatial-model prediction.\n\n**Cycles and Condorcet winners can coexist.** Cyclical majorities can appear below a Condorcet winner, meaning a candidate who beats everyone pairwise while lower-ranked alternatives form a cycle among themselves. The 2025 study observed such cases four times, including the 2005 Finnish election, the only instance where no Condorcet loser was present[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\nUncertainty in the underlying data also matters. In 198 of 212 parliamentary elections, not a single one of the 10,000 bootstrap replications exhibited a cyclical majority; the 2011 Peruvian parliamentary election showed cycles in about 18% of replications and the 2003 Icelandic election about 11%, dropping to 2% after resolving indifferences[1](https://link.springer.com/article/10.1007/s11127-025-01353-7).\n\n## References\n\n1. On the prevalence of Condorcet's paradox (Public Choice, 2025). https://link.springer.com/article/10.1007/s11127-025-01353-7\n2. Condorcet paradox, Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Condorcet_paradox\n3. MIT OCW Lecture 12: Condorcet's Paradox and Arrow's Impossibility Theorem. https://ocw.mit.edu/courses/14-75-political-economy-and-economic-development-fall-2012/a9fd8e5ab75a325016094e6bbe625b2a_MIT14_75F12_Lec12.pdf\n4. Condorcet cycles, RangeVoting.org. https://rangevoting.org/CondorcetCycles\n5. Simulation: Analytic and Algorithmic Analyses of Condorcet's Paradox (Social Science Computer Review, 1998). https://journals.sagepub.com/doi/10.1177/089443939801600109\n6. From Condorcet's paradox to Arrow: yet another simple proof of the impossibility theorem (Social Choice and Welfare, 2024). https://link.springer.com/article/10.1007/s00355-024-01557-8\n7. Condorcet voting (Maskin & Foley, March 2025). https://maskin.scholars.harvard.edu/sites/g/files/omnuum10606/files/condorcet_voting_e.maskin_e.foley_march_2025_0.pdf\n8. [The impartial culture maximizes the probability of majority cycles, sites.socsci.uci.edu](https://sites.socsci.uci.edu/~bgrofman/129%20Tsetlin%20et%20al.%202003.%20The%20Impartial%20Culture%20Maximizes%20the%20Probability...Soc.%20Choice.pdf)\n\n---\n*Topic: Encyclopedia › Society and history › Politics and government › Elections and representation › Electoral systems and principles › Electoral theory and criteria › Voting paradoxes and impossibility results › Condorcet cycle paradox*\n\n*Initially written Sep 17, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 25, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://sites.socsci.uci.edu/~bgrofman/129%20Tsetlin%20et%20al.%202003.%20The%20Impartial%20Culture%20Maximizes%20the%20Probability...Soc.%20Choice.pdf" ], "url": "https://www.edgepedia.org/condorcet-paradox", "markdown_url": "https://www.edgepedia.org/condorcet-paradox.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. 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