The Entropy Method in Decision and Game Theory

In recent years connections between information theory and repeated games have been established. Three such connections will be exemplified.
The first example will be a one-player game (decision problem). A fixed game (maximisation of an objection function) is repeated n times. The player (decision maker) has to device an m(n)-space algorithm with the objective of maximising the average of the objectives of the n problems. The asymptotic optimal solution is an analogue of Shannon's channel capacity.

The second example will be a two-player repeated game in which each player is subject to a different memory (recall) constraint. The asymptotic min-max value of this game is characterised in terms of the value of a one-stage game in which the player with smaller memory is restricted to play strategies with high entropy.

The third example is will be a three-player repeated game with memory constraints. Here, the existence of an asymptotic min-max value is an intriguing open problem. If time permits, a few recent developments towards this problem will be mentioned.