Characterization of dominance relations in finite coalitional games

McGarvey (Econometrica, 21(4), 608–610, 1953) has shown that any irreflexive and anti-symmetric relation can be obtained as a relation induced by majority rule. We address the analogous issue for dominance relations of finite cooperative games with non-transferable utility (coalitional NTU games). We find any irreflexive relation over a finite set can be obtained as the dominance relation of some finite coalitional NTU game. We also show that any such dominance relation is induced by a non-cooperative game through β-effectivity. Dominance relations obtainable through α-effectivity, however, have to comply with a more restrictive condition, which we refer to as the edge-mapping property.

[1]  E. Kalai,et al.  Monotonic Solutions to General Cooperative Games , 1985 .

[2]  Somdeb Lahiri,et al.  A Weak Bargaining Set for Contract Choice Problems , 2006 .

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

[4]  W. Wallis.  The Theory of Voting , 2012 .

[5]  R. Aumann,et al.  VON NEUMANN-MORGENSTERN SOLUTIONS TO COOPERATIVE GAMES WITHOUT SIDE PAYMENTS , 1960, Classics in Game Theory.

[6]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[7]  S. Hart,et al.  Bargaining and Value , 1996 .

[8]  Murat R. Sertel,et al.  Bargaining over a finite set of alternatives , 2007, Soc. Choice Welf..

[9]  John Duggan,et al.  An Implementation-Theoretic Approach to Non-cooperative Foundations , 1999 .

[10]  J. Nash.  Two-Person Cooperative Games , 1953 .

[11]  R. Aumann.  The core of a cooperative game without side payments , 1961 .

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

[13]  H.C.M. de Swart,et al.  Logic, Game Theory and Social Choice , 1999 .

[14]  William S. Zwicker,et al.  Simple games - desirability relations, trading, pseudoweightings , 1999 .

[15]  A. Rubinstein.  Perfect Equilibrium in a Bargaining Model , 1982 .

[16]  J. Nash.  THE BARGAINING PROBLEM , 1950, Classics in Game Theory.

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

[18]  Roberto Serrano,et al.  A comment on the Nash program and the theory of implementation , 1997 .

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

[20]  H. Keiding,et al.  Effectivity functions in social choice , 1991 .

[21]  S. Vajda,et al.  GAMES AND DECISIONS; INTRODUCTION AND CRITICAL SURVEY. , 1958 .

[22]  Bhaskar Dutta,et al.  Comparison functions and choice correspondences , 1999 .