Social choice theory

Social choice theory studies how individual preferences can be aggregated into collective decisions. It analyzes voting rules, welfare criteria, and the properties and limits of these methods.

Arrow's impossibility theorem: A theorem showing that no ranked-preference voting rule satisfies several appealing fairness conditions for three or more options. It establishes a fundamental limit on what preference aggregation can guarantee.
Preference relation: A formal relation representing how an agent ranks or compares alternatives. Individual rankings are the inputs that aggregation rules combine.
Nicolas de Condorcet: An eighteenth-century French mathematician and political thinker who analyzed majority voting and electoral procedures. His work exposed cycles in majority preferences that remain central to the field.
Electoral system: A set of rules governing how votes are cast, counted, and converted into elected offices. Social choice results help compare the consequences of electoral rules.
Gibbard–Satterthwaite theorem: A theorem stating that every nondictatorial voting rule with at least three alternatives can be manipulated under broad conditions. It shows that truthful voting cannot be guaranteed by a wide class of rules.
Condorcet method: A voting approach that selects an option defeating every rival in pairwise majority comparisons, if one exists. It defines a winner through pairwise majority comparisons rather than first-place votes.
Social welfare function: A rule that maps profiles of individual preferences to a collective ranking of alternatives. It formalizes the central task of turning many rankings into one.
Jean-Charles de Borda: An eighteenth-century French mathematician who proposed a points-based method for ranking candidates. His voting method offered an early alternative to simple majority choice.
Proportional representation: An electoral principle in which parties or groups receive seats roughly in proportion to their share of votes. It applies aggregation to representation across multiple seats rather than a single winner.
Strategic voting: Voting in a way that differs from sincere preference expression to influence the election's outcome. It tests how voters adapt when outcomes depend on the voting rule.

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