W F 2.0 {1} 3,0 B 2.1 Figure 1: Beer-Quiche incremental payoff of 1); also, the Receiver strictly prefers fighting the weak Sender, and not fighting the strong one That all Nash equilibria in this game are pooling is not surprising at all. However there are two classes of pooling equilibria: one in which both types of the Sender order Beer for breakfast, under the threat that Quiche will induce a Fight, and one in which the exact opposite is true-both types order Quiche, under the threat of a Fight if they deviate to Beer. Let us indicate the pure-strategy equilibria by(B, B NF)(the Weak type orders Beer, the Strong type orders Beer, and the receiver play n if he observes Beer and F if he observes Quiche)and( Q, Q, FN) Why exactly do we find the second type of equilibrium unreasonable? A moments reflection is sufficient to conclude that the reasonableness of the equilibrium hinges on the reasonableness of the Receiver's response, which in turn depends on his off- equilibrium beliefs Now, for the Receiver to be willing to choose F after observing B, it must be that he is sufficiently confident that the Sender is Weak. Hence, we might recast our question as follows: Is it reasonable for the receiver to believe that the Sender is Weak after observing Beer Suppose we changed the game so uI B, N, w)=0: then we could simply invoke a dominance argument. That is, we could say In the modified game a Weak sender gets 0 by choosing B; this is strictly less than she can get by choosing Q, regardless of the Receiver's strategy That is, B is a strictly dominated message for the Weak Sender. Hence, if the Receiver thinks that both types of the Sender are rational, he cannot place positive probability on the Weak Sender having chosen B. Thus, the out-of-equilibrium beliefs required to support the equilibrium(Q,Q, FN are unreasonable, and so is the equilibrium itself Note that this argument has a forward induction favor: we consider the out-of- equilibrium choice of B as being intentional, and we attempt to interpret it assuming❜ w {.1} ❍❍ r✛ B 0,1 ❍❨ F ✟✟ 2,0 ✟✙ N Q ✲ r✟ F✟✟✯ 1,1 ❍ N ❍❍❥ 3,0 ❜ s {.9} ❍❍ r✛ B 1,0 ❍❨ F ✟✟ 3,1 ✟✙ N Q ✲ r✟ F✟✟✯ 0,0 ❍ N ❍❍❥ 2,1 Figure 1: Beer-Quiche. incremental payoff of 1); also, the Receiver strictly prefers fighting the weak Sender, and not fighting the strong one. That all Nash equilibria in this game are pooling is not surprising at all. However, there are two classes of pooling equilibria: one in which both types of the Sender order Beer for breakfast, under the threat that Quiche will induce a Fight, and one in which the exact opposite is true—both types order Quiche, under the threat of a Fight if they deviate to Beer. Let us indicate the pure-strategy equilibria by (B, B, NF) (the Weak type orders Beer, the Strong type orders Beer, and the receiver plays N if he observes Beer and F if he observes Quiche) and (Q, Q, FN). Why exactly do we find the second type of equilibrium unreasonable? A moment’s reflection is sufficient to conclude that the reasonableness of the equilibrium hinges on the reasonableness of the Receiver’s response, which in turn depends on his off- equilibrium beliefs. Now, for the Receiver to be willing to choose F after observing B, it must be that he is sufficiently confident that the Sender is Weak. Hence, we might recast our question as follows: Is it reasonable for the Receiver to believe that the Sender is Weak after observing Beer? Suppose we changed the game so u1(B,N,w) = 0: then we could simply invoke a dominance argument. That is, we could say: In the modified game a Weak sender gets 0 by choosing B; this is strictly less than she can get by choosing Q, regardless of the Receiver’s strategy. That is, B is a strictly dominated message for the Weak Sender. Hence, if the Receiver thinks that both types of the Sender are rational, he cannot place positive probability on the Weak Sender having chosen B. Thus, the out-of-equilibrium beliefs required to support the equilibrium (Q,Q,FN) are unreasonable, and so is the equilibrium itself. Note that this argument has a forward induction flavor: we consider the out-of￾equilibrium choice of B as being intentional, and we attempt to interpret it assuming 2