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.
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- Mas-Colell, Andreu, 1989. "An equivalence theorem for a bargaining set," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 129-139, April.
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- Gaertner,Wulf, 2001. "Domain Conditions in Social Choice Theory," Cambridge Books, Cambridge University Press, number 9780521791021, November.
- Peleg, Bezalel & Sudholter, Peter, 2005. "On the non-emptiness of the Mas-Colell bargaining set," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1060-1068, December.
- Bezalel Peleg & Peter Sudholter, 2004. "On the Non-Emptiness of the Mas-Colell Bargaining Set," Discussion Paper Series dp360, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.