A characterization of stable sets in assignment games

Abstract

We consider von Neumann-Morgenstern stable sets in assignment games. In the symmetric case Shapley (1959) proved some necessary conditions of vNM stability. In this paper we generalize this result for any assignment game. We show that a V set of imputation is stable if and only if (i) is internally stable, (ii) is connected, (iii) contains an imputation with 0 payoff to all buyers and an imputation with 0 payoff to all sellers, (iv) contains the core of the semi-imputations in the rectangular set spanned by any two points of V. With this characterization we give a new proof to the existence of stable sets. Moreover using these reult if the core is not stable we can construct infinite many stable set.