On Scarf's theorem for Generalized Cooperative Games

Abstract

In this paper, we study a generalization of cooperative games with non-transferable utility. In our model, coalitions are replaced by firms: each firm is assigned a resource vector, while the set of utility vectors of a coalition is replaced by a general comprehensive set of feasible payoff vectors of this firm in a common payoff space. We introduce the notions of core and fractional core for such games and relate their existence to homotopy invariants of covers associated with the game. The main result shows that the fractional core is nonempty precisely when the associated cover is homotopicaly nontrivial. As a consequence, we obtain a Scarf-type theorem for generalized cooperative games.