Games with Discontinuous Payoffs

Abstract

We prove an equilibrium existence result for a class of games with an infinite number of strategies. Our theorem generalises an earlier result by Dasgupta and Maskin. We also identify conditions under which the limit of pure-strategy equilibria of a sequence of finite games is an equilibrium for the limit game. We apply this result to obtain new existence results for the multi-firm, i-dimensional version of Hotellings's location game. The techniques used suggest a technique for computing such equilibria. This paper studies games with discontinuous payoffs. Any game with a compact set of strategies can be approximated by a sequence of finite games. We study the correspondence mapping each approximation to the limit game to its mixed-strategy equilibria and identify conditions under which the limit of such equilibria is an equilibrium for the limit game. When payoffs are discontinuous, this property will not hold in general. An equilibrium for the limit game may not exist or, if it does exist, the equilibrium correspondence may not be upper hemi-continuous. The topic is of concern to economists for two, rather different, reasons. The existence question is clearly important in its own right. In many games that interest economists, payoff functions fail to be continuous or even semi-continuous. Standard existence theorems (Debreu (1952), Glicksberg (1950), (1952), etc.), therefore, are inapplicable. Two classical examples are Bertrand's (1883) model of price competition and Hotelling's (1929) model of spatial competition. Recent work on auction theory (see Milgrom-Weber (1982)) and dynamic oligopoly models (surveyed in Fudenberg-Tirole (1983)) has generated many more instances. The second reason for interest in the equilibrium correspondence is pragmatic. Finite games are generally much more tractable than infinite games. When the equilibrium correspondence is upper hemi-continuous, the easiest way to find an equilibrium for an infinite game may be to calculate the equilibria of a sequence of finite games and take limits.1 The seminal paper on games with discontinuous payoffs is Dasgupta-Maskin (DM) (1986a). This paper proves an existence theorem under assumptions that are strictly weaker than DM's. The key difference between our approach and DM's is as follows. DM identify conditions under which the limit of equilibria of any sequence of approximating finite games is an equilibrium for the limit game. These conditions are quite restrictive. They may be violated even though dominant strategy equilibria exist for the limit game. Our approach to the existence problem is different. We consider a given sequence of approximating games and ask when the limit of equilibria for this particular sequence will be an equilibrium for the limit game.2