Games with Incomplete Information Played by "Bayesian" Players Part II. Bayesian Equilibrium Points

Part I of this paper has described a new theory for the analysis of games with incomplete information. It has been shown that, if the various players' subjective probability distributions satisfy a certain mutual-consistency requirement, then any given game with incomplete information will be equivalent to a certain game with complete information, called the "Bayes-equivalent" of the original game, or briefly a "Bayesian game." Part II of the paper will now show that any Nash equilibrium point of this Bayesian game yields a "Bayesian equilibrium point" for the original game and conversely. This result will then be illustrated by numerical examples, representing two-person zero-sum games with incomplete information. We shall also show how our theory enables us to analyze the problem of exploiting the opponent's erroneous beliefs. However, apart from its indubitable usefulness in locating Bayesian equilibrium points, we shall show it on a numerical example (the Bayes-equivalent of a two-person cooperative game) that the normal form of a Bayesian game is in many cases a highly unsatisfactory representation of the game situation and has to be replaced by other representations (e.g., by the semi-normal form). We shall argue that this rather unexpected result is due to the fact that Bayesian games must be interpreted as games with "delayed commitment" whereas the normal-form representation always envisages a game with "immediate commitment."