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, l-dimensional version of Hotellings's location game. The techniques used suggest a technique for computing such equilibria.

In this paper, we extend to a general equilibrium context Bertrand's classic critique of Cournot. We present a game-theoretic model of a pure exchange, monetary economy, in which buyers as well as sellers announce both quantities and prices. When buyers act strategically, the "Edgeworth nonexistence problem" is circumvented: under weak conditions, a pure strategy Nash equilibrium exists for this game. We make precise the Bertrand idea that when agents in a finite economy are permitted to compete-by-price the resulting allocations will be competitive. Specifically, the Nash equilibria for our game yield allocations that are "competitive" allocations for the underlying exchange economy, provided that there are at least two buyers and two sellers actively trading in every market. Under this characterization of strategic behaviour, then, "two is enough for competition."

The authors present three distinct approaches to perfect and proper equilibria for infinite normal form games. In the first two approaches, players 'tremble' in the infinite game playing full support approximate best responses to others' strategies. In the strong approach, a tremble assigns high probability to the set of pure best responses; in the weak approach, it assigns high probability to a neighborhood of this set. The third, limit-of-finite approach applies traditional refinements to sequences of successively larger finite games. Overall, the strong approach to equilibrium refinement most fully respects the structure of infinite games.

A new framework for games in continuous time is proposed. The continuous-time model conforms as closely as possible to the conventional discrete-time framework. Indeed, continuous time is viewed as "discrete time, but with a grid that is infinitely fine." The paper presents several examples illustrating the difficulties that arise in continuous-time game theory. Theorems relate the equilibria of continuous time games to the equilibria of approximating discrete time games. A variety of industrial organization applications are studied, yielding sharp predictions. Applications include continuously repeated games, preemption models, and patent races.

This paper proposes a new approach to the study of economic problems that have hitherto been modeled as games with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies in the underlying problem. The authors' point of departure from the conventional approach is to view the sharing rules that resolve these indeterminacies as part of the solution rather than as part of the description of the model. A solution to the authors' model is a sharing rule, together with a profile of (mixed) strategies that satisfies the usual (Nash) best response criterion. Their main result is that such a solution always exists.

This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual best-response criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence. Copyright The Econometric Society 2002.