Extensive Form Games in Continuous Time: Pure Strategies

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 infinitely fine. Specifically, we define a class continuous-time strategies with the following property: when restricted to an arbitrary, increasingly fine sequence discrete-time grids, any profile strategies drawn from this class generates a convergent sequence outcomes, whose limit independent the sequence 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 the two frameworks. Specifically, we ask two questions. First, 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 the following upper result: Suppose a sequence discrete-time e-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 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 e-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 industrial organization applications. We first consider several variants a familiar preemption model. Next, we analyze a stylized model a patent race. Finally, we obtain a striking uniqueness result for a class repeated games.