Star-shapedness of Richter-Aumann integral on a measure space with atoms: theory and economic applications (Q1764657)

This paper is an important addition to the theory of large economies and games, in their most general version where there is a countable set of atoms in a mass of non-atomic agents. Such theory of `mixed' economies and games lacks appropriate existence theorems, and the paper fills the gap both for large economies, for which it provides the existence of a Walrasian equilibrium, and for non-cooperative games, for which it gives an existence theorem of a Nash equilibrium. The unifying mathematical result which enables the author to obtain these non-trivial results gives sufficient conditions under which the Aumann integral, not being convex, is nonetheless star-shaped if atoms are not `too large'. The condition concerns the span of the two parts, the atomic and the non-atomic ones, of the integral. Intuitively, it requires that optimizing the behavior of the atomless agents in the economy or game be at least as flexible as that of the atoms.