On the Approximation of Upper Semi-Continuous Correspondences and the Equilibriums of Generalized Games

In Section 1 we explain some of the definitions and terminology that we use. In Section 2 we prove several theorems concerning the approximation of upper semi-continuous correspondences having for range a locally convex space. Theorems 1 and 2 (and Corollary 1) generalize certain approximation theorems by G. Haddad [16, pp. 1352213541, G. Haddad and J. M. Lasry [ 17, pp. 299-3001, and J. P. Aubin and A. Cellina [2, pp. 86891 (see also F. S. De Blasi [lo] and J. M. Lasry and R. Robert [21]). Although some of the details of the proofs of Theorems 1 and 2 are new, the basic ideas are taken from the above mentioned papers. Theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular (see Section 1). This theorem (see the remark following its proof) contains a classical result of M. Hukuhara [ 18, pp. 56573. Theorem 3 is proved using Theorem 2 and Propositions 1 and 4. Variants of Propositions 2, 3, and 4 were given in [ 193. In Section 3 we give, among others, Theorem 5, which concerns the existence of equilibriums of generalized games ( = abstract economies). The main purpose of this result is to replace the continuity hypothesis in the W. Shafer-H. Sonnenschein equilibrium theorem for generalized games by an upper semi-continuity one. The proof of Theorem 5 is based on the results on the approximation of upper semi-continuous correspondences obtained in Section 2. Theorem 5 us used in Section 4. In Section 4 we establish Theorems 6 and 7. These theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of K. Fan type. Theorems 1, 2, 3, 5, 6, and 7 are the main results of this paper. 267 0022-247X/88 $3.00