Consider the following signaling game. At the outset, Nature moves. Nature can move either left (L) or right (R) with probabilities 0.9 and 0.1, respectively. Player 1, who observes Nature’s move, can propose either deal A or deal B. After 1’s move, player 2 says Yes or No to 1’s proposal. Player 2 sees 1’s proposal, but does not get to observe Nature’s move. As for payoffs, both players get zero if player 2 rejects the offer. If player 2 accepts it, player 1 loses e and player 2 earns e if Nature moved L and the accepted contract is A, or if Nature moved R and the accepted contract is B. The tables are turned otherwise, i.e., player 1 earns e and player 2 loses e if Nature moved L and the accepted contract is B, or if Nature moved R and the accepted contract is A.
(a) Represent graphically the extensive form of this dynamic game of incomplete information.
(b) Find all its separating perfect Bayesian equilibria in pure strategies.
(c) Find all its pooling perfect Bayesian equilibria in pure strategies, specifying beliefs on and off the equilibrium path.