Social choice theory studies how individual preferences determine a collective decision. A social choice function selects one alternative from a preference profile; strategyproofness asks whether agents can benefit by misrepresenting their preferences.
A Condorcet winner defeats every other alternative by a strict majority in pairwise voting. It is unique when it exists. For an odd electorate with single-peaked preferences, the median peak is a Condorcet winner: alternatives on either side lose to the majority whose peaks lie at or beyond the median in the other direction. With an even electorate ties may prevent a strict winner.
A weak Condorcet winner is never defeated by a strict pairwise majority, allowing ties. For even electorates with single-peaked preferences, all alternatives between the two middle peaks have this property. A single-valued selection requires a tie convention; arbitrary tie selection is not automatically strategyproof.
A strict preference order is single-peaked on an axis when moving away from its most preferred alternative along either one side of the axis makes alternatives less preferred. Comparisons between opposite sides need not follow symmetric physical distance. A profile is single-peaked when all voters' orders share such an axis. This restricted domain admits the median voter rule, unlike the unrestricted-domain conclusion of the Gibbard-Satterthwaite theorem.
On a fixed common axis, select the median of reported peaks. For odd this is the unique Condorcet winner. For even , a fixed lower-median or upper-median convention selects a weak Condorcet winner. Holding other peaks fixed, the attainable outcomes form an interval between two order statistics. Truthful reporting obtains the true peak if it is inside that interval, or its nearest interval endpoint otherwise, so no allowed report improves the outcome under single-peaked preferences.
A deterministic social choice function maps profiles of preference orders to single alternatives. Unrestricted domain permits every profile in ; onto means every alternative lies in the range. A unanimous rule selects an alternative whenever every agent ranks it first.
With an odd number of agents and two alternatives, select the alternative ranked first by a strict majority. The rule is onto, strategyproof, and for at least three agents has no dictator in social choice. A pivotal agent already gets its preferred alternative by reporting truthfully; a nonpivotal agent cannot change the outcome alone.
A dictatorship always selects the top alternative of one fixed agent . Other agents cannot alter the selection against that agent's preference. This is the social-choice meaning of dictatorship, rather than a political classification.
A dictator is the fixed agent whose top-ranked alternative is always selected by a dictatorship in social choice.
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Social choice theory is a theoretical framework that explores how individuals' preferences can be aggregated to make collective decisions. It encompasses a variety of methods and principles for assessing and determining the best course of action in situations where multiple individuals have differing preferences, needs, or choices. Key aspects of social choice theory include: 1. **Voting Systems**: The study of various electoral systems and how they influence the outcomes of elections. This includes methods such as plurality voting, ranked-choice voting, and others.