Static games of incomplete information

• given the signals and the prior distribution, players using Bayes' rule calculate a posterior belief: probability distribution over the states
• Harsanyi purification: the probability distributions induced by the pure (Bayesian Nach) equilibria of the perturbed game converge to the distribution of the (mixed Nash) equilibrium of the unperturbed game
• the risk-dominant equilibrium is the one with higher deviation-payoff product. If both deviation-payoff products are equal, neither equilibrium is risk-dominant
• Deviation-payoff product for more than 2 strategies considers the deviation closest to the original payoff.
• Risk dominance is a pairwise criterion
• Bayesian games are strategic situations in which players do not have complete information about the game
• the ex-ante joint distribution of types, unlike the realisation of the type profile, is commonly known: is the common prior assumption
• ex post notions of dominance of equilibrium may be unrealistic as they require a player's strategy choice to be optimal after having learnt the type realisation
• Interim expected payoffs are very natural to base solution concepts on
• for BNE the distinction between interim and ex-ante expected payoffs makes no difference, so can use whichever is more convenient
• Harsanyi purification - the probability distributions over strategies induced by the pure BNE of the perturbed game converge to the distribution of the (mixed Nash) equilibrium of the unperturbed game
• in 2x2 coordination games, as the noise in the players' signals vanishes, IESDS on interim payoffs in the global game forces players to play a specific pure strategy equilibrium - the risk-dominant equilibrium