SFB 303 Discussion Paper No. A - 092

Author: Hellwig, Martin, and Wolfgang Leininger
Title: A Note on the Relation between Discrete and Continuous Games of Perfect Information
Abstract: Most of the literature on extensive-form noncooperative games assumes that the players have finite strategy sets (see, e.g., Selten (1975) or Kreps and Wilson (1982)). In applications however the economic context often suggests a natural specification of strategies as continuous variables. Even if one feels that the relevant strategy sets are discrete, there may be no particular discretization that imposes itself as being the most natural. In these cases, the application of available concepts and results may require that the analysis be restricted to a fixed discretized version of the game in which one is interested. However, the results should then be reasonably insensitive to the particular discretization that has been chosen. For normal-form games with compact metric strategy spaces and continuous payoff functions, the preceding desideratum is unproblematic. In this case any sequence of Nash equilibria of successively finer discretizations of the game has a convergent subsequence; moreover if the discretizations approximate the continuous game, then any limit point of a sequence of Nash equilibria of the discrete games will be a Nash equilibrium of the continuous game. For extensive-form games the difficulty arises that the strategies are functions which indicate how each player's behaviour depends on the information he has about the history of the game. In general then the convergence behaviour of a given sequence of equilibria, i.e. a given sequence of constellations of such functions, is unclear. To be sure, the corresponding sequence of equilibrium paths will have a convergent subsequence. However, it is not in general clear that the limit path is an equilibrium path of the continuous extensive-form game. Even if the limit path is an equilibrium path of the continuous game, it is not clear what is the relation between the equilibrium strategies in the continuous game and the equilibrium strategies in the finite approximations of the continuous game. In the following we resolve these issues for the special case of games of perfect information.
Creation-Date: December 1986
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