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

TLDR

The paper builds on Part I, which introduced a new theory for analyzing games with incomplete information, and argues that Bayesian games should be viewed as delayed‑commitment games rather than the immediate‑commitment normal form. The study demonstrates that a Nash equilibrium of a Bayesian game corresponds to a Bayesian equilibrium of the original game, illustrates how the theory can exploit opponents’ mistaken beliefs, and shows that the normal‑form representation is often inadequate, requiring alternative forms such as the semi‑normal form. The authors illustrate the theory with numerical examples of two‑person zero‑sum games with incomplete information.

Abstract

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.”