Bayesian Game - Perfect Bayesian Equilibrium

Perfect Bayesian Equilibrium

Bayesian Nash equilibrium results in some implausible equilibria in dynamic games, where players take turns sequentially rather than simultaneously. Similarly, implausible equilibria might arise in the same way that implausible Nash equilibria arise in games of perfect and complete information, such as incredible threats and promises. Such equilibria might be eliminated in perfect and complete information games by applying subgame perfect Nash equilibrium. However, it is not always possible to avail oneself of this solution concept in incomplete information games because such games contain non-singleton information sets and since subgames must contain complete information sets, sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Even if a game does have more than one subgame, the inability of subgame perfection to cut through information sets can result in implausible equilibria not being eliminated.

To refine the equilibria generated by the Bayesian Nash solution concept or subgame perfection, one can apply the Perfect Bayesian equilibrium solution concept. PBE is in the spirit of subgame perfection in that it demands that subsequent play be optimal. However, it places player beliefs on decision nodes that enables moves in non-singleton information sets to be dealt more satisfactorily.

So far in discussing Bayesian games, it has been assumed that information is perfect (or if imperfect, play is simultaneous). In examining dynamic games, however, it might be necessary to have the means to model imperfect information. PBE affords this means: players place beliefs on nodes occurring in their information sets, which means that the information set can be generated by nature (in the case of incomplete information) or by other players (in the case of imperfect information).