Finite Horizon Bargaining and the Consistent Field

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show that for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent field converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value. Journal of Economic Literature Classification Numbers: C71, C72. This research was started in 1991, as part of the first author’s Ph.D. thesis at Harvard University. We gratefully acknowledge financial support from: CAPES, Brazil (Gomes); the U.S.-Israel Binational Science Foundation (Hart and Mas-Colell); the Israeli Academy of Sciences and Humanities (Hart); the Spanish Ministry of Education (Mas-Colell); and the Generalitat de Catalunya (Mas-Colell). †Department of Finance, The Wharton School, University of Pennsylvania, Philadelphia. e-mail: [email protected]. ‡Department of Economics; Department of Mathematics; and Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem. e-mail: [email protected]. §Department of Economics, Universitat Pompeu Fabra, Barcelona. e-mail: [email protected].