Abstract

this paper, we present a simple algorithm for solving this problem, and give a simple analysis of the algorithm. The bounds we obtain are not asymptotic and hold for any finite number of rounds. The algorithm and its analysis are based directly on the "on-line prediction" methods of Littlestone and Warmuth [24]. The analysis of this algorithm yields a new (as far as we know) and simple proof of von Neumann's minmax theorem, as well as a provable method of approximately solving a game. We also give more refined variants of the algorithm for this purpose, and we show that one of these is optimal in a very strong sense. The paper is organized as follows. In Section 2 we define the mathematical setup and notation. In Section 3 we introduce the basic multiplicative weights algorithm whose average performance is guaranteed to be almost as good as that of the best fixed mixed strategy. In Section 4 we outline the relationship between our work and some of the extensive existing work on the use of multiplicative weights algorithms for on-line prediction. In Section 5 we show how the algorithm can be used to give a simple proof of Von-Neumann's min-max theorem. In Section 6 we give a version of the algorithm whose distributions are guaranteed to converge to an optimal mixed strategy. We note the possible application of this algorithm to solving linear programming problems and reference other work that have used multiplicative weights to this end. Finally, in Section 7 we show that the convergence rate of the second version of the algorithm is asymptotically optimal. 2 Playing repeated games

Citations