| The purpose of this thesis is to extend fictitious play to games with a continuum of strategies. Two classes of games are particularly considered: zero-sum games and potential games. We extend the Brown-Robinson and Monderer-Shapley fictitious play theorems for two-person zero-sum games and potential games to games with continuous strategy spaces. In the zero-sum case, the payoff function is not required to be the same at every stage: it needs only to be better known as the number of stages increases. Moreover, at each stage players do not have to make the best reply against the average of the past choices of the other players, but need only choose a strategy that approximates the true best response as time goes to infinity. Our result implies some recent results obtained by Vrieze and Tijs (1982) and by Van Der Genugten (1998), as special cases. Our result for potential games significantly relaxes the requirement on step sizes in the iteration process when the potential itself is concave. The only requirement is that the (infinite) sum of all the step sizes should diverge (this could further be generalized to allow different players to choose different step sizes at each iteration). Turning to more general games (which need not be two-person zero-sum games or potential games), we have the following results. First assume that all strategy sets are closed intervals in the real line. Fictitious play converges to Nash equilibrium when the best response is single valued and is a linear function. In the two person case, linearity can be relaxed to monotonicity.; A dynamic process related to fictitious play is the sequential best-response process. This is studied for potential games. We show that if the number of players is 2, or if the potential is strictly concave in each player's strategy (varied unilaterally), the sequential best-response converges to Nash equilibrium. Unfortunately, the same conclusion does not apply to simultaneous best response.; To apply these results to economics models, we explicitly construct a strictly concave potential so that the Cournot oligopoly is best response equivalent to such a potential game. Thus all our convergence results hold in this model, generalizing and improving earlier work on this topic done by Thorlund-Pertersen (1990), Fisher (1960) and Leleno (1994). A similar construction also applies to other economic models, which include technological competition, pollution games and model of group interaction such as rent-seeking and public goods and etc. |
