N-PERSON NASH BNARGAINING WITH VARIABEL THREATS

Abstract

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash’s variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large. An important constituent of bargaining is the specification of the payoff vector when players fail to achieve an agreement. This payoff vector is called a disagreement point. With the major exception of Nash (1953), a disagreement point is assumed to be exogenously given. I) In some economic examples, exogenous disagreement points such as an endowment point in an exchange economy are naturally determined. In general game situations and in economic situations with externalities, however, we may not find a natural definition of an exogenously given disagreement point. This fact requires us to consider a bargaining model with an endogenous determination of a disagreement point. Nash (1953) gave such a model with two players. The purpose of this paper is to investigate the behaviour of this model with n players. We compare, from both axiomatic and noncooperative game theoretic viewpoints, the n-player extension of Nash’s model with an alternative model where disagreement points are also endogenously determined. Nash’s (1953) model - the first model in this paper - is as follows. An underlying environment is described by a noncooperative strategic game G. The players are allowed to cooperate for obtaining higher payoffs and bargain over possible cooperative payoffs, subject to the possibility of failing to achieve an agreement. If the players fail, they return to the original noncooperative game. For this possibility, each player has to choose a strategy in the case of disagreement, prior to bargaining.