# Bargaining Sets of Voting Games

Let A be a finite set of m ≥ 3 alternatives, let N be a finite set of n ≥ 3 players and let Rn 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 uN be a profile of utility functions for RN. Using α-effectiveness we define the NTU game VuN 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. 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 ≥ 6. Moreover, we prove the following startling result: The Mas-Colell bargaining set of anysimple majority voting game derived from the k-th replication of RN is nonempty, provided that k ≥ 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.