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| Smith set | |
|---|---|
| Name | Smith set |
| Field | Social choice theory |
| Introduced | 1970s |
| Known for | Condorcet-consistent majority cycle resolution |
Smith set The Smith set is a construct in social choice theory and voting theory identifying a minimal nonempty set of alternatives that each pairwise defeats every alternative outside the set by majority preference. Originating in analyses of Condorcet paradox and pairwise comparison methods, it informs designs of electoral systems, voting methods, and decision theory frameworks seeking consistency with majority judgments.
The Smith set is defined as the smallest subset of alternatives such that every alternative in the set beats every alternative outside the set in head-to-head majorities. The notion refines concepts from Condorcet method, Condorcet winner, top cycle, and Schwartz set by imposing minimality and pairwise dominance conditions. In practice it interacts with profiles characterized in works by Kenneth Arrow, Marian Rejewski (note: contributor to voting literature), Donald Saari, Amartya Sen, and researchers at institutions like Institute for Advanced Study and RAND Corporation.
Mathematical properties include closure under pairwise majority relations, minimality, and nonemptiness for finite electorates. The Smith set always contains any Condorcet winner and is a subset of the top cycle and the Smith–Minimax relations studied in theoretical treatments by Nobel Prize in Economic Sciences laureates who addressed aggregation problems. It satisfies criteria such as inclusion of pairwise dominant alternatives and is invariant under permutation of voter labels and monotone reinforcement for many voting methods. The size can range from one (a Condorcet winner) to the full set of alternatives in instances of universal cycles like those analyzed by Lewis Carroll and Condorcet.
Computing the Smith set reduces to finding the minimal dominating set in a tournament graph constructed from pairwise majority comparisons. Standard graph algorithms such as depth-first search and strongly connected component decomposition by Tarjan algorithm, or reachability via Floyd–Warshall algorithm and Kosaraju's algorithm, are commonly applied. Computational complexity is polynomial in the number of alternatives using algorithms inspired by Edmonds' algorithm for directed graphs and implementations in software from GNU projects and research groups at MIT, Stanford University, and University of Oxford. Practical systems integrate Smith-set identification into procedures like Copeland method filters, Ranked Pairs preprocessing, and Condorcet-compliant incarnations of IRV adaptations.
Simple examples illustrate behavior: with three alternatives forming a majority cycle as in classic analyses by Marquis de Condorcet and narratives in Bayesian treatments, the Smith set equals the full set of three alternatives. In elections studied by Pew Research Center and case studies at Harvard Kennedy School, when a Condorcet winner such as a broadly acceptable candidate emerges, the Smith set is a singleton containing that candidate. Multi-candidate primaries analyzed by Electoral Reform Society and simulations by Cambridge University researchers often show Smith sets of size two or three, influencing strategic nominations in contexts like United Kingdom general elections and Australian preferential ballots.
The Smith set relates to the Condorcet criterion, Schwartz set, and top cycle; it refines and is refined by these constructs depending on tie-breaking conventions. It interacts with methods such as Plurality voting, Instant-runoff voting, Borda count, Approval voting, Score voting, Minimax method, Copeland's method, Dodgson's method, Kemeny–Young method, Ranked Pairs, and Single Transferable Vote. The set figures in normative comparisons in literature by scholars at Princeton University, Yale University, and London School of Economics, informing debates on criteria like participation, monotonicity, and consistency examined by Kenneth Arrow and Martha Nussbaum-adjacent ethicists.
Knowledge of the Smith set affects strategic behavior in nomination, coalition formation, and ballot design. Campaign strategies by parties such as Democratic Party and Conservative Party may aim to place preferred candidates inside the Smith set to survive head-to-head elimination. Tactical voting literature from Game theory studies at University of Chicago and Centre for Economic Policy Research shows incentives for strategic entry and withdrawal, vote splitting, and agenda control by actors like legislatures and electoral commissions. The Smith set underpins criteria for resistance to manipulation in mechanisms proposed by Alan Gibbard and Mark Satterthwaite and informs complexity-theoretic defenses against strategic control studied by researchers at Carnegie Mellon University and Microsoft Research.
Category:Voting theory