Bargaining Sets of Majority Voting Games

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.