Harsanyi transformation • Players have private information • Each possibility is called a type. • Nature chooses a type for each player. • Probability distribution common knowledge. • A strategy for each player prescribes a strategy for each possible type.
Bayesian Nash Equilibrium (BNE) • Propose a strategy profile (possibly mixed). • See how players update their believes accordingly (Bayes’ Theorem). • Check that players play a best response given these updated beliefs (Nash condition).
Perfect Bayesian Equilibrium (PBE) • Strategy profile and beliefs for every information set s.t. • The strategies of the game are Nash given all beliefs and strategies; • Where possible, beliefs are obtained from Bayes’ Theorem (i.e. beliefs are rational) • Bayes’ theorem cannot be applied at info sets that are off the equilibrium path (those that occur w.p. 0).