On the structure of stable tournament solutions

A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (1) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (2) we completely characterize which simple choice functions give rise to stable choice functions, and (3) we prove a strong relationship between stability and a new property of tournament solutions called local reversal symmetry. Based on these findings, we provide the first concrete tournament—consisting of 24 alternatives—in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.

[1]  Alex D. Scott,et al.  The minimal covering set in large tournaments , 2012, Soc. Choice Welf..

[2]  Jean-François Laslier,et al.  The Copeland Measure of Condorcet Choice Functions , 1994, Discret. Appl. Math..

[3]  H. Landau.  On dominance relations and the structure of animal societies: I. Effect of inherent characteristics , 1951 .

[4]  A. Sen,et al.  Choice Functions and Revealed Preference , 1971 .

[5]  Felix Brandt,et al.  Minimal extending sets in tournaments , 2014, Math. Soc. Sci..

[6]  Mark Fey,et al.  Choosing from a large tournament , 2008, Soc. Choice Welf..

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

[8]  Ayzerman,et al.  Theory of choice , 1995 .

[9]  Paul D. Seymour,et al.  A counterexample to a conjecture of Schwartz , 2013, Soc. Choice Welf..

[10]  Yongjie Yang,et al.  A Further Step Towards an Understanding of the Tournament Equilibrium Set , 2016, ArXiv.

[11]  P. Fishburn.  Probabilistic Social Choice Based on Simple Voting Comparisons , 1984 .

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

[13]  Felix Brandt,et al.  Tournament Solutions , 2016, Handbook of Computational Social Choice.

[14]  Scott Moser.  Majority rule and tournament solutions , 2015 .

[15]  Dan S. Felsenthal,et al.  After two centuries, should condorcet's voting procedure be implemented? , 1992 .

[16]  Felix Brandt,et al.  Set-rationalizable choice and self-stability , 2009, J. Econ. Theory.

[17]  Thomas Schwartz.  Cyclic tournaments and cooperative majority voting: A solution , 1990 .

[18]  Stefano Allesina,et al.  A competitive network theory of species diversity , 2011, Proceedings of the National Academy of Sciences.

[19]  Felix Brandt,et al.  Bounds on the disparity and separation of tournament solutions , 2015, Discret. Appl. Math..

[20]  Vincent Conitzer,et al.  Handbook of Computational Social Choice , 2016 .

[21]  H. Chernoff.  Rational Selection of Decision Functions , 1954 .

[22]  Felix Brandt,et al.  On the Discriminative Power of Tournament Solutions , 2014, OR.

[23]  M. Breton,et al.  The Bipartisan Set of a Tournament Game , 1993 .

[24]  Felix Brandt,et al.  Set-monotonicity implies Kelly-strategyproofness , 2010, Soc. Choice Welf..

[25]  Matthias Mnich,et al.  When Does Schwartz Conjecture Hold? , 2015, IJCAI.

[26]  Bernard Monjardet.  Statement of precedence and a comment on IIA terminology , 2008, Games Econ. Behav..

[27]  Jeffrey Richelson.  Some Further Results on Consistency, Rationality and Collective Choice , 1978 .

[28]  David C. Fisher,et al.  Optimal strategies for random tournament games , 1995 .

[29]  Erkut Y. Ozbay,et al.  Revealed Attention , 2009 .

[30]  Nicolas Houy,et al.  Still more on the Tournament Equilibrium Set , 2009, Soc. Choice Welf..

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

[32]  Jennifer Ryan,et al.  Tournament games and positive tournaments , 1995, J. Graph Theory.

[33]  Jean-François Laslier,et al.  More on the tournament equilibrium set , 1993 .

[34]  Donald G. Saari,et al.  Consequences of reversing preferences , 2003 .

[35]  T. Schjelderup-Ebbe,et al.  Beiträge zur Sozialpsychologie des Haushuhns. , 1922 .

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

[37]  Felix Brandt,et al.  Consistent Probabilistic Social Choice , 2015, ArXiv.

[38]  Felix A. Fischer,et al.  Minimal retentive sets in tournaments , 2010, AAMAS.

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

[40]  Felix Brandt,et al.  Minimal stable sets in tournaments , 2008, J. Econ. Theory.

[41]  Olivier Hudry,et al.  A survey on the complexity of tournament solutions , 2009, Math. Soc. Sci..