An Equilibrium-Point Interpretation of Stable Sets and a Proposed Alternative Definition

Abstract

The paper argues that the von Neumann-Morgenstern definition of stable sets is unsatisfactory because it neglects the destabilizing effect of indirect dominance relations. This argument is supported both by heuristic considerations and by construction of a bargaining game B(G), formalizing the bargaining process by which the players agree on their payoffs from an n-person cooperative game G. (G itself is assumed to be given in characteristic-function form allowing side payments.) The strategies ρi, the players use in this bargaining game will determine which payoff vectors x will be stationary, i.e., will have the property that, should such a payoff vector x be proposed to the players, all further bargaining will stop and x will be accepted as the outcome of the game. It will be suggested that a stable set should be defined as the set V of all stationary payoff vectors x, on the assumption that the players' bargaining strategies p, will form a canonical equilibrium point ρ in the bargaining game B(G). In a certain class of games, to be called absolutely stable games, indirect dominance relations turn out to be irrelevant, and in these games the suggested definition for stable sets is equivalent to the von Neumann-Morgenstern definition. But in general this is not the case. However, if we assume that the players will adopt bargaining strategies deliberately discouraging any use of indirect dominance relations, then, in every game, the stable sets our model yields will always be von Neumann-Morgenstern stable sets. At the end of the paper, we briefly discuss a possible modification in the suggested definition for stable sets, based on a modified bargaining game B0(G), in which the players are permitted to accept cuts in their provisional payoffs if they think that this move will increase their final payoffs.