Core, value and equilibria for market games: On a problem of Aumann and Shapley

In this note a partial solution of Open Problem 41C of Aumann and Shapley is presented. A sufficient condition for the Aumann-Shapley value of a market game to exist, to be contained in its core, and to be the competitive payoff distribution of a transferable utility competitive equilibrium is given. In this context, balancedness and σ-balancedness criteria for large classes of cooperative games are proven.