The Beer-Quiche Game Consider the following two-player signaling game. Player 1 is either ‘weak’ or ‘strong’. This is determined by a chance move, resulting in player 1 being ‘weak’ with probability...

The Beer-Quiche Game

Consider the following two-player signaling game. Player 1 is either ‘weak’ or ‘strong’. This is determined by a chance move, resulting in player 1 being ‘weak’ with probability 1=10. Player 1 is informed about the outcome of this chance move but player 2 is not; but the probabilities of either type of player 1 are common knowledge among the two players. Player 1 has two actions: either have quiche (Q) or have beer (B) for breakfast. Player 2 observes the breakfast of player 1 and then decides to duel (D) or not to duel (N) with player 1. The payoffs are as follows. If player 1 is weak and eats quiche then D and N give him payoffs of 1 and 3, respectively; if he is weak and drinks beer, then these payoffs are 0 and 2, respectively. If player 1 is strong, then the payoffs are 0 and 2 from D and N, respectively, if he eats quiche; and 1 and 3 from D and N, respectively, if he drinks beer. Player 2 has payoff 0 from not duelling, payoff 1 from duelling with the weak player 1, and payoff 1 from duelling with the strong player 1.

(a) Draw a diagram modelling this situation.

(b) Compute all the pure strategy Nash equilibria of the game. Find out which of these Nash equilibria are perfect Bayesian equilibria. Give the corresponding beliefs and determine whether these equilibria are pooling or separating, and which ones satisfy the intuitive criterion.