Finite Games with Incomplete Information

In a game of imperfect information players may be uninformed about the moves made by other players. Every one-shot, simultaneous move game is a game of imperfect information. In a game of incomplete information players may be uninformed about certain characteristics of the game or of the players. For instance, a player may have incomplete information about actions available to some other player, or about payoffs of other players. Following Harsanyi [50], we model incomplete information by assuming that every player can be of a number of different types. A type of a player summarizes all relevant information (in particular, actions and payoffs) about that player. Furthermore, it is assumed that each player knows his own type and, given his own type, has a probability distribution over the types of the other players. Often, these probability distributions are assumed to be consistent in the sense that they are the marginal probability distributions derived from a basic commonly known distribution over all combinations of player types.

In this chapter we consider games with finitely many players, finitely many types, and finitely many strategies. These games can be either static (simultaneous, one-shot) or dynamic (extensive form games). A Nash equilibrium in this context is also called ‘Bayesian equilibrium’, and in games in extensive form an appropriate refinement is perfect Bayesian equilibrium. As will become clear, in essence the concepts studied in Chaps. 3 and 4 are applied again.

In Sect. 5.1 we present a brief introduction to the concept of player types in a game. Section 5.2 considers static games of incomplete information, and Sect. 5.3 discusses so-called signaling games, which is the most widely applied class of extensive form games with incomplete information.