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| Schwartz set | |
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| Name | Schwartz set |
Schwartz set is a concept in social choice theory that identifies a particular set of alternatives in a pairwise majority tournament. It selects alternatives that are unbeaten within a minimal unbeaten subset, offering a refinement of Condorcet-consistent ideas. The notion is discussed alongside concepts developed by scholars in voting theory and has practical implications for electoral systems, committee selection, and multiagent decision-making.
The Schwartz set is defined for a finite set of alternatives under a binary majority relation induced by individual preferences. For a given profile of preferences studied by researchers such as Kenneth Arrow, Amartya Sen, Herbert Simon, Donald Saari, and John H. Smith (economist), construct the directed graph where vertices represent alternatives and an edge from x to y indicates a majority prefers x over y. The Schwartz set consists of all alternatives that belong to some nonempty subset S such that no alternative outside S is majority-preferred to any alternative in S, and S is minimal with this unbeaten property. This definition relates to foundational work by Maurice Allais, Nobel Prize in Economic Sciences laureates like Kenneth Arrow and Amartya Sen, and subsequent elaborations in journals where scholars such as Peter C. Fishburn, William Vickrey, and Donald G. Saari contributed to tournament solution concepts.
The Schwartz set has several key mathematical and normative properties explored by authors including Merrill Flood, Thomas Schelling, and Kenneth J. Arrow: - Nonemptiness: For any finite electorate with strict preferences, the Schwartz set is nonempty by construction, a property examined in the context of fixed-point results by John Nash and equilibrium concepts by Lloyd Shapley. - Superset relations: It contains the set of Condorcet winners when they exist, linking to concepts associated with Marquis de Condorcet and treatments in texts by Condorcet scholars and Duncan Black. - Minimal unbeaten sets: The Schwartz set is formed from minimal externally unbeaten components, a notion related to coalition and closure properties studied by Robert Aumann and Thomas R. Palfrey. - Stability and monotonicity: The set satisfies certain stability criteria similar to those in the study of solution concepts like the top cycle and uncovered set investigated by Peter C. Fishburn and Howard Raiffa. - Inclusion relationships: It is contained in the Smith set and contains the uncovered set under specific conditions, connecting to literature by John H. Smith (economist), Mark Satterthwaite, and Iain McLean.
Computing the Schwartz set reduces to graph-theoretic procedures familiar to computer scientists and mathematicians such as Edsger W. Dijkstra, Donald Knuth, and Leslie Lamport: - Tournament graph construction: Build the directed tournament using pairwise majority comparisons; algorithms for majority graph construction echo methods in work by Michael Rabin and Noam Nisan on preference aggregation. - Strongly connected components: The Schwartz set can be obtained by computing strongly connected components and identifying those with no incoming edges from other components, employing linear-time algorithms by Robert Tarjan or Andrew V. Goldberg. - Transitive reductions and closures: Computing minimal unbeaten subsets involves analysis related to transitive closure algorithms associated with Stephen A. Cook and graph reachability techniques in the tradition of Alfred Aho. - Complexity: For n alternatives, baseline algorithms run in O(n^2) to build pairwise comparisons and O(n^2) for component decomposition; complexity discussions parallel topics in computational social choice covered by Tuomas Sandholm and Craig Boutilier. - Implementation: Practical implementations appear in software libraries used in research by groups at institutions like MIT, Stanford University, and University of Oxford.
Illustrative examples draw on canonical profiles and historical voting analyses cited by scholars such as Kenneth Arrow, Amartya Sen, Duncan Black, and Condorcet: - Three-alternative cycle: In the classic Condorcet cycle where A beats B, B beats C, and C beats A, the entire set {A, B, C} forms a single minimal unbeaten component, so the Schwartz set equals the full set of alternatives. This example is prominent in discussions by Condorcet and later expositions by Maurice Allais. - Presence of Condorcet winner: If an alternative D beats all others pairwise, then the Schwartz set is {D}, reflecting properties noted by John H. Smith (economist) and in textbooks by Amartya Sen. - Multiple components: In larger electorates with clusters of alternatives, Schwartz components correspond to subsets discussed in applied work by Kenneth Arrow and Donald Saari in which strategic behavior and agenda effects are analyzed.
The Schwartz set relates to several major solution concepts and paradoxes examined by prominent theorists: - Smith set: It is always a subset of the Smith set introduced in literature by John H. Smith (economist) and compared in surveys by Maurice J. S.. - Condorcet criterion: It refines the Condorcet criterion associated with Marquis de Condorcet and Condorcet methods. - Uncovered set, top cycle, Banks set: Inclusion and overlap properties connect to concepts studied by John H. Smith (economist), Thomas R. Palfrey, and Donald Saari. - Arrow’s impossibility and strategic manipulation: Discussions of manipulability and incentive compatibility reference foundational work by Kenneth Arrow and Amartya Sen, and mechanism-design implications drawn by Tibor Scitovsky.
The Schwartz set is applied in electoral engineering, committee decision rules, and multiagent systems researched at institutions such as Harvard University, Princeton University, Yale University, and London School of Economics: - Voting system design: It informs choice of aggregation rules in settings explored by William Riker and Amartya Sen. - Automated decision-making: Multiagent preference aggregation using the Schwartz set appears in work by Tuomas Sandholm and groups at Carnegie Mellon University. - Robustness to strategic voting: Comparative analyses of strategic vulnerability reference studies by Mark Satterthwaite and Allan Gibbard. - Empirical studies: Political scientists at University of California, Berkeley and Columbia University examine real-world ballots for occurrence of cycles and Schwartz components.
Category:Voting theory