Bargaining Sets of Voting Games

Let A be a finite set of m ³ 3 alternatives, let N be a finite set of n ³ 3 players and let R n be a profile of linear preference orderings on A of the players. Throughout most of the paper the considered voting system is the majority rule. Let u N be a profile of utility functions for R N . Using a -effectiveness we define the NTU game V u N 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 ³ 4. Moreover, 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 ³ 6. Moreover, we prove the following startling result: The Mas-Colell bargaining set of any simple majority voting game derived from the k-th replication of R N is nonempty, provided that k ³ n + 2. We also compute the NTU games which are derived from choice by plurality voting and approval voting, and we analyze some interesting examples.