The Existence of Equilibrium in Discontinuous Economic Games, I: Theory

Abstract

The article presents information about the existence of equilibrium in discontinuous economic games. In this paper and its sequel presents study the existence of Nash equilibrium in games where agents' payoff functions are discontinuous. The enquiry is motivated by a number of recent studies that have uncovered serious existence problems in seemingly innocuous economic games. In the sequel to this paper, authors have explained the utility function in the economic games referred to earlier are neither continuous nor quasi-concave. However, they demonstrate that the payoff functions in mildly modified versions of these constructs exhibit two weaker forms of continuity which, together with the requirement of quasi-concavity, suffice for the existence of an equilibrium. From this one may conclude that, at least in the modified versions of these models, discontinuities in the payoff functions are not the real source of the problem. Rather, it is the failure of the payoff functions to the quasi-concave which is responsible for the non-existence of equilibrium. These observations bear on the existence of Nash equilibrium in pure strategies.