Social Choice

Game Theory · Axiom Academy

Aggregating individual preferences into collective decisions, voting systems, preference orderings, and social welfare functions In social choice theory, we begin with individual preferences . Each person has a ranking or ordering over a set of alternatives (candidates, policies, outcomes). A preference ordering is a complete, transitive ranking. If there are three candidates A, B, and C, a voter might prefer: A > B > C (meaning they prefer A to B, and B to C). A voting system (or voting rule) is a method for aggregating individual preferences into a collective choice. Different systems can produce different outcomes from the same preferences! Common voting systems include: Plurality (most first-place votes), Borda Count (points for rankings), Runoff , and Condorcet methods. Plurality: Each candidate gets 1 first-place vote (tie!) Borda Count: A gets 4 points, B gets 4 points, C gets 4 points (tie!) A social welfare function (SWF) aggregates individual preference orderings into a complete social ordering of all alternatives. Unlike voting rules that just pick a winner, SWFs produce a full ranking. Formally, an SWF takes as input a preference profile (the collection of all individual orderings) and outputs a social preference ordering. Social welfare functions must satisfy certain desirable properties to be considered "fair." These properties include unanimity (Pareto efficiency), independence of irrelevant alternatives, and non-dictatorship. 4 Arrow's Impossibility Theorem

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