The Computational Complexity of Choice Sets

Social choice rules are often evaluated and compared by inquiring whether they satisfy certain desirable criteria such as the Condorcet criterion, which states that an alternative should always be chosen when more than half of the voters prefer it over any other alternative. Many of these criteria can be formulated in terms of choice sets that single out reasonable alternatives based on the preferences of the voters. In this paper, we consider choice sets whose definition merely relies on the pairwise majority relation. These sets include the Copeland set, the Smith set, the Schwartz set, von Neumann-Morgenstern stable sets, the Banks set, and the Slater set. We investigate the relationships between these sets and completely characterize their computational complexity, which allows us to obtain hardness results for entire classes of social choice rules (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

[1]  Jeffrey S. Banks,et al.  Voting games, indifference, and consistent sequential choice rules , 1988 .

[2]  A. Sen,et al.  Social Choice Theory , 2005 .

[3]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[4]  Till Tantau.  A Note on the Complexity of the Reachability Problem for Tournaments , 2001, Electron. Colloquium Comput. Complex..

[5]  P. Fishburn.  The Theory Of Social Choice , 1973 .

[6]  Gerhard J. Woeginger,et al.  Banks winners in tournaments are difficult to recognize , 2003, Soc. Choice Welf..

[7]  Bhaskar Dutta.  Covering sets and a new condorcet choice correspondence , 1988 .

[8]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[9]  M. Richardson.  Extension Theorems for Solutions of Irreflexive Relations. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[10]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[11]  Nicholas R. Miller,et al.  Graph- Theoretical Approaches to the Theory of Voting* , 1977 .

[12]  I. Good.  A note on condorcet sets , 1971 .

[13]  P. Slater.  Inconsistencies in a schedule of paired comparisons , 1961 .

[14]  H. Landau.  On dominance relations and the structure of animal societies: III The condition for a score structure , 1953 .

[15]  T. Schwartz.  Rationality and the Myth of the Maximum , 1972 .

[16]  A. Sen,et al.  Collective Choice and Social Welfare , 2017 .

[17]  Noga Alon,et al.  Ranking Tournaments , 2006, SIAM J. Discret. Math..

[18]  William F. Lucas,et al.  Von Neumann-Morgenstern stable sets , 1992 .

[19]  J. H. Smith.  AGGREGATION OF PREFERENCES WITH VARIABLE ELECTORATE , 1973 .

[20]  Felix A. Fischer,et al.  A computational analysis of the tournament equilibrium set , 2008, Soc. Choice Welf..

[21]  Moshe Tennenholtz,et al.  On the Axiomatic Foundations of Ranking Systems , 2005, IJCAI.

[22]  P. Fishburn.  Condorcet Social Choice Functions , 1977 .

[23]  H. Moulin.  Choosing from a tournament , 1986 .

[24]  Rajat Deb.  On Schwartz's rule☆ , 1977 .

[25]  Ad M. A. van Deemen,et al.  A note on generalized stable sets , 1991 .

[26]  Felix A. Fischer,et al.  Computing the minimal covering set , 2008, Math. Soc. Sci..

[27]  T. Tideman,et al.  Collective Decisions and Voting: The Potential for Public Choice , 2006 .

[28]  Robin Farquharson,et al.  Theory of voting , 1969 .

[29]  Jean-François Laslier,et al.  Condorcet choice correspondences: A set-theoretical comparison , 1995 .

[30]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[31]  David C. Mcgarvey.  A THEOREMI ON THE CONSTRUCTION OF VOTING PARADOXES , 1953 .

[32]  Olivier Hudry.  A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger , 2004, Soc. Choice Welf..

[33]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[34]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[35]  Howard Raiffa,et al.  Games and Decisions: Introduction and Critical Survey. , 1958 .

[36]  M. Richardson,et al.  Solutions of Irreflexive Relations , 1953 .

[37]  M. Richardson,et al.  Relativization and extension of solutions of irreflexive relations. , 1955 .

[38]  Edith Hemaspaandra,et al.  The complexity of Kemeny elections , 2005, Theor. Comput. Sci..

[39]  Somdeb Lahiri,et al.  Stable Sets of Weak Tournaments , 2003 .

[40]  Jean-François Laslier,et al.  Tournament Solutions And Majority Voting , 1997 .

[41]  Kenneth O. May,et al.  A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision , 1952 .

[42]  J. Banks.  Sophisticated voting outcomes and agenda control , 1984 .