Voting paradox

A voting paradox is a counterintuitive or inconsistent outcome produced by collective choice, even when individual preferences or the voting rule seem reasonable.

Condorcet paradox: A cycle in pairwise majority comparisons in which each of three options defeats one other option. It is the clearest case of majority rule producing no stable winner.
Social choice theory: The mathematical study of how individual preferences are combined into collective decisions. Voting paradoxes are central problems in this broader study.
Condorcet criterion: A criterion requiring a voting rule to elect a candidate who defeats every rival in pairwise majority contests, when one exists. It defines a desirable property that some paradox-prone rules fail to satisfy.
Gibbard–Satterthwaite theorem: A theorem stating that every non-dictatorial deterministic voting rule with at least three outcomes is manipulable under broad conditions. It connects counterintuitive outcomes to incentives for misrepresenting preferences.
Arrow's impossibility theorem: A theorem showing that no ranked-choice rule for three or more options satisfies a specified set of fairness conditions in every case. It formalizes why plausible social-choice requirements can conflict.
Preference relation: A formal representation of how an agent ranks or compares available options. Paradoxical collective rankings arise from aggregating such individual comparisons.
Condorcet winner: A candidate who defeats every other candidate in separate pairwise majority comparisons. A clear winner can exist even when a voting rule selects someone else.
Agenda setting: The selection and ordering of issues or alternatives considered in a decision process. With cyclic majority preferences, changing the pairwise voting order can change the final winner.
Borda count: A ranked voting method that assigns points according to each voter's ordering of the candidates. Its winner can differ from the pairwise-majority winner, exposing dependence on the aggregation rule.
Majority rule: A decision rule that selects the option supported by more than half of the relevant votes. Its pairwise use can generate cycles even when every voter's ranking is consistent.

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