Values of mixed games

Aumann andShapley [1973] have investigated values of games in which all players are individually insignificant, i.e. form a non-atomic continuum, or “ocean”. In this paper we treat games in which, in addition to such an ocean, there are also some “atoms”, i.e. players who are individually significant. We define spaces of such games that are analogous to those investigated byAumann andShapley, and prove the existence of values on some of them. Unlike in the non-atomic case, we find that in general there are infinitely many values, corresponding to various ways in which the atoms can be imbedded in the ocean. The results generalize those ofMilnor andShapley [1961]. Precise statements will be found in Section 2.