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