Equilibrium in Abstract Economies with a Non-compact Infinite Dimensional Strategy Space, an Infinite Number of Agents and without Ordered Preferences

The abstract economy defined by Debreu (1952) generalizes the Nash non-cooperative game in that a player’s strategy set depends on the strategy choices of all the other players. Debreu (1952) proved the existence of equilibrium in abstract economies with finitely many agents, finite dimensional strategy space, and quasi-concave utility functions. Since then, the Debreu’s result has been extended in several directions. For the finite number of agents and the finite dimensional strategy space case, Shafer and Sonnenschein (1975) and Borglin and Keiding (1976) extended the Debreu’s result to abstract economies without ordered preferences. For the infinite dimensional space and finite or infinitely many agents case, the existence results were given by Yannelis and Prabhakar (1983) Khan and Vohra (1984), Toussaint (1984) Kim, Prikry, and Yannelis (1985) Khan (1986) and Yannelis (1987) among others. However, all existence theorems mentioned above are proved upon the compactness of choice sets. In the finite-dimensional setting, the usual closed boundedness assumptions imply compactness of the feasible sets. However, in the infinite-dimensional setting, the usual closed boundedness assumptions do not imply compactness of the feasible sets and thus the feasible sets will not generally be (weakly) compact, a typical situation in infinite dimensional linear space. ’ To avoid this difficulty in the literature, people explicitly assume that the feasible set is (weakly) compact in some topology and ignore the case of non-compact feasible sets. Recently, Tian (1988) considered the existence of equilibrium for abstract economies with a non-compact infinite dimensional choice space and a countably infinite number of agents by the quasi-variational inequality approach. However, the quasi-variational inequality approach requires that preferences be representable by a concave utility function.