Bayesian models and repeated games
A game is a theoretical model of a social situation where the people involved have individually only partial control over the outcomes. Game theory is then the method used to analyse these models. As a player's outcome from a game depends upon the actions of his opponents, there is some uncertainty in these models. This uncertainty is described probabilistically, in terms of a player's subjective beliefs about the future play of his opponent. Any additional information that is acquired by the player can be incorporated into the analysis and these subjective beliefs are revised. Hence, the approach taken is `Bayesian'. Each outcome from the game has a value to each of the players, and the measure of merit from an outcome is referred to as a player's utility. This concept of utility is combined with a player's subjective probabilities to form an expected utility, and it is assumed that each player is trying to maximise his expected utility. Bayesian models for games are constructed in order to determine strategies for the players that are expected utility maximising. These models are guided by the belief that the other players are also trying to maximise their own expected utilities. It is shown that a player's beliefs about the other players form an infinite regress. This regress can be truncated to a finite number of levels of beliefs, under some assumptions about the utility functions and beliefs of the other players. It is shown how the dichotomy between prescribed good play and observed good play exists because of the lack of assumptions about the rationality of the opponents (i. e. the ability of the opponents to be utility maximising). It is shown how a model for a game can be built which is both faithful to the observed common sense behaviour of the subjects of an experimental game, and is also rational (in a Bayesian sense). It is illustrated how the mathematical form of an optimal solution to a game can be found, and then used with an inductive algorithm to determine an explicit optimal strategy. It is argued that the derived form of the optimal solution can be used to gain more insight into the game, and to determine whether an assumed model is realistic. It is also shown that under weak regularity conditions, and assuming that an opponent is playing a strategy from a given class of strategies, S, it is not optimal for the player to adopt any strategy from S, thus compromising the chosen model.