MIT14_126S10_lec06 - 14.126 GAME THEORY MIHAI MANEA...

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Unformatted text preview: 14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Forward Induction in Signaling Games Consider now a signaling game . There are two players, a sender S and a receiver R . There is a set T of types for the sender; the realized type will be denoted by t . p ( t ) denotes the probability of type t . The sender privately observes his type t , then sends a message m M ( t ). The receiver observes the message and chooses an action a A ( m ). Finally both players receive payoffs u S ( t, m, a ) ,u R ( t, m, a ); thus the payoffs potentially depend on the true type, the message sent, and the action taken by the receiver. In such a game we will use T ( m ) to denote the set { t | m M ( t ) } . The beer-quiche game from before is an example of such a game. T is the set { weak, surly } ; the messages are { beer, quiche } ; the actions are { fight,not fight } . As we saw before, there are two sequential equilibria: one in which both types of sender choose beer, and another in which both types choose quiche. In each case, the equilibrium is supported by some beliefs such that the sender is likely to have been weak if he chose the unused message, and the receiver responds by fighting in this case. Cho and Kreps (1987) argued that the equilibrium in which both types choose quiche is unreasonable for the following reason. It does not make any type for the weak type to deviate to ordering beer, no matter how he thinks that the receiver will react, because he is already getting payoff 3 from quiche, whereas he cannot get more than 2 from switching to beer. On the other hand, the surly type can benefit if he thinks that the receiver will react by not fighting. Thus, conditional on seeing beer ordered, the receiver should conclude that the sender is surly and so will not want to fight. Date : March 2, 2010. 2 MIHAI MANEA On the other hand, this argument does not rule out the equilibrium in which both types drink beer. In this case, in equilibrium the surly type is getting 3, whereas he gets at most 2 from deviating no matter how the receiver reacts; hence he cannot want to deviate. The weak type, on the other hand, is getting 2, and he can get 3 by switching to quiche if he thinks this will induce the receiver not to fight him. Thus only the weak type would deviate, so the senders belief (that the receiver is weak if he orders quiche) is reasonable. Now consider modifying the game by adding an extra option for the receiver: paying a million dollars to the sender. Now the preceding argument doesnt rule out the quiche equilibrium either type of sender might deviate to beer if he thinks this will induce the receiver to pay him a million dollars. Hence, in order for the argument to go through, we need the additional assumption that the sender cannot expect the receiver to play a bad strategy....
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MIT14_126S10_lec06 - 14.126 GAME THEORY MIHAI MANEA...

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