This exercise generalizes Theorems 9.47 (page 354) and 9.53 (page 355) to the case where the prior distributions of the players differ. Let (N, (Ti)i∈N , (pi)i∈N ,S, (st)t∈×i∈N Ti) be a game with incomplete information where each player has a different prior distribution: for each i ∈ N, player i’s prior distribution is pi. For each strategy vector σ, define the payoff function Ui as
and the payoff of player i of type ti by
A strategy vector σ ∗ is a Nash equilibrium if for every player i ∈ N and every strategy σi of player i,
and it is a Bayesian equilibrium if for every player i ∈ N, every type ti ∈ Ti, and every strategy σi of player i,
(a) Prove that a Nash equilibrium exists when the number of players is finite and each player has finitely many types and actions.
(b) Prove that if each player assigns positive probability to every type of every player,
then every Nash equilibrium is a Bayesian equilibrium, and every Bayesian equilibrium is a Nash equilibrium.