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Extensive Form Games in Continuous Time: Pure Strategies

Leo K. Simon
Maxwell B. Stinchcombe

Abstract

We propose a new framework for games in continuous time that conforms as closely as possible to the conventional discrete-time framework. In this paper, we take the view that continuous time can be viewed as "discrete time, but with a grid that is infinitely fine." Specifically, we define a class of continuous-time strategies with the following property: when restricted to an arbitrary, increasingly fine sequence of discrete-time grids, any profile of strategies drawn from this class generates a convergent sequence of outcomes, whose limit is independent of the sequence of grids. We then define the continuous-time outcome to be this limit. Because our continuous-time model conforms so closely to the conventional discrete-time model, we can readily compare the predictions of the two frameworks. Specifically, we ask two questions. First, is discrete-time with a very find grid a good proxy for continuous time? Second, does every subgame perfect equilibrium in our model have a discrete-time analog? Our answer to the first question is the following "upper hemi-continuity" result: Suppose a sequence of discrete-time @?-subgame-perfect equilibria increasingly closely approximates (in a special sense) a given continuous-time profile, with @? converging to zero along as the period length shrinks. Then the continuous-time profile will be an exact equilibrium for the corresponding continuous-time game.Our second answer is a lower hemi-continuity result that holds under weak conditions. Fix a perfect equilibrium for a continuous-time game and a positive @?. Then for any sufficiently fine grid, there will exist an @?-subgame perfect equilibrium for the corresponding game played on that grid which "is within @? of" the continuous-time equilibrium. Our model yields sharp predictions in a variety of industrial organization applications. We first consider several variants of a familiar preemption model. Next, we analyze a stylized model of a patent race. Finally, we obtain a striking uniqueness result for a class of "repeated" games.