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Bargaining Sets of Majority Voting Games

Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Institute of Mathematics and Center for the Study of Rationality, The Hebrew University of Jerusalem, Feldman Building, Givat Ram, 91904 Jerusalem, Israel
Department of Business and Economics, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
holzman{at}techunix.technion.ac.il
pelegba{at}math.huji.ac.il
psu{at}sam.sdu.dk

Let A be a finite set of m alternatives, let N be a finite set of n players, and let R^N be a profile of linear orders on A of the players. Let u^N be a profile of utility functions for R^N. We define the nontransferable utility (NTU) game V_u^N that corresponds to simple majority voting, and investigate its Aumann-Davis-Maschler and Mas-Colell bargaining sets. The first bargaining set is nonempty for m 3, and it may be empty for m 4. However, in a simple probabilistic model, for fixed m, the probability that the Aumann-Davis-Maschler bargaining set is nonempty tends to one if n tends to infinity. The Mas-Colell bargaining set is nonempty for m 5, and it may be empty for m 6. Furthermore, it may be empty even if we insist that n be odd, provided that m is sufficiently large. Nevertheless, we show that the Mas-Colell bargaining set of any simple majority voting game derived from the k-fold replication of R^N is nonempty, provided that k n+2.

Key Words: NTU game; voting game; majority rule; bargaining set
History: Received: November 17, 2005; revision received: August 6, 2006;