Consider the following signaling game: a man, player 1, goes to a pub and chooses between...

 

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Consider the following signaling game: a man, player 1, goes to a pub and chooses between drinking beer (B) and having a quiche (Q). Player 1 is of either strong (S) or weak (W) type, and type S prefers beer to quiche while type W's preference is in opposite. There is another player, called player 2, who is waiting for player 1 in front of the pub and considering to apply a duel to him. If player 1 is of weak type, player 2 wants to duel with player 1 (since he can win) and he wants to avoid duel were player 1 of strong type. Player 2 chooses between applying a duel (D) or escaping (E). Player 2 can only observe player 1's order in the pub. The prior probability of strong type is 0.9. The game tree is given in the following: 0, D B W Q 2,0- probability 0.11 E -3,0 NH chance 1,0 D -0,0 probability 0.9 B E 3,1 1 (a) (2.5 points) Find a PBE in which both types of player 1 choose beer. (b) (2.5 points) Show that there is no separating equilibrium in which the strong type chooses beer and the weak type chooses quiche. S Consider the following signaling game: a man, player 1, goes to a pub and chooses between drinking beer (B) and having a quiche (Q). Player 1 is of either strong (S) or weak (W) type, and type S prefers beer to quiche while type W's preference is in opposite. There is another player, called player 2, who is waiting for player 1 in front of the pub and considering to apply a duel to him. If player 1 is of weak type, player 2 wants to duel with player 1 (since he can win) and he wants to avoid duel were player 1 of strong type. Player 2 chooses between applying a duel (D) or escaping (E). Player 2 can only observe player 1's order in the pub. The prior probability of strong type is 0.9. The game tree is given in the following: 0, D B W Q 2,0- probability 0.11 E -3,0 NH chance 1,0 D -0,0 probability 0.9 B E 3,1 1 (a) (2.5 points) Find a PBE in which both types of player 1 choose beer. (b) (2.5 points) Show that there is no separating equilibrium in which the strong type chooses beer and the weak type chooses quiche. S

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