ECONS STRATEGY AND GAME THEORY QUIZ #3 (SIGNALING GAMES) ANSWER KEY

Transcription

1 ECONS - STRATEGY AND GAME THEORY QUIZ #3 (SIGNALING GAMES) ANSWER KEY Exercise Mike vs. Buster Consider the following sequential move game with incomplete information. The first player to move is Mike, who privately knows whether he trained hard, or he didn t. Let us assume that the type of training that Mike receives before a fight is not something he can strategically decide, but instead, it depends on his state of mind between the time he signed up the contract for a fight and the time of the fight. In particular, we will assume that the probability that Mike trains hard is given by, and it is /3, as can be seen in the figure. Knowing what kind of physical training he had, Mike decides whether to offer player (Buster ) $1 million dollars if he gives up his right to fight Mike. We will denote Mike s strategies as (that is, offering the bribe) or the (don t offering any bribe to ). After observing whether Mike has offered him any, Buster must decide whether to ight () or Not ight (), without knowing whether Mike has previously trained hard or not. (-, ) /3 the (, -) (1-) (1-) the 1

2 a) Show that there is no Separating PBE where Mike makes the offer when he has trained hard, but does not make such an offer (he chooses the ) when he has not trained hard. To show this, follow the usual steps for finding PBE. (-, ) /3 the (, -) (1-) (1-) the 1. ind Buster beliefs in this Separating PBE (use Bayes rule). After observing the offer from Mike, Buster beliefs are = = 1 = 1 Graphically, Buster believes that if he observes, he must be in the node at the upper left-hand corner of the game tree. After observing the from Mike, Buster beliefs are μμ = = 0 μμ = 0 Graphically, Buster believes that if he observes the, he must be in the node at the lower right-hand corner of the game tree.. ind Buster optimal action (whether to ight or Not ight) after observing that Mike offers him. In addition, find Buster optimal action (whether to ight or Not ight) after observing that Mike does not offers him any bribe ( observes the action the ). After observing that Mike offers, Buster responds with, since he believes to be in the node at the upper left-hand corner of the game and >1.

3 After observing the, Buster responds with, since he believes to be in the node at the lower right-hand corner of the game and 10>1. 3. ind Mike s optimal action when he has trained hard, and when he has not trained hard. If he trained hard, Mike prefers to deviate towards the than selecting (as prescribed in this strategy profile), since >-. We don t even need to check whether Mike chooses the when he didn t trained hard (as prescribed in the separating strategy profile we are testing), since the above argument already shows that this strategy profile cannot be sustained as a PBE. 4. Can this separating PBE be supported from your answer in c)? Obviously, you should obtain that it cannot be supported, but you have to show why from your answers in part c). No, since Mike prefers to deviate towards the when he trained hard. b) ind a Pooling PBE where Mike makes the offer when he has trained hard, and he also makes this offer when he has not trained hard. To show this, follow the usual steps for finding PBE. (-, ) /3 the (, -) (1-) (1-) the 1. ind Buster beliefs in this Pooling PBE (use Bayes rule). 3

4 After observing that Mike offers (in equilibrium), Buster beliefs cannot be updated, and simply coincide with the prior probability distribution, that is = 3 pptttt 3 pptttt ppnnnnnn where pp TTTT denotes the probability that Mike makes the offer after training hard (TH), and similarly pp NNNNNN represents the probability that he makes this offer when he didn t (NTH). Since in this pooling strategy profile pp TTTT = pp NNNNNN = 1, Buster beliefs become = = 3 1 Intuitively, Buster cannot infer any additional information from Mike s type after observing that he offers. 3 After observing that Mike chooses the (which occurs offthe-equilibrium path), Buster beliefs are μμ = = 3 0 and thus must be left undefined, i.e., μμ [0, 1] ind Buster optimal action (whether to ight or Not ight) after observing that Mike offers him. In addition, find Buster optimal action (whether to ight or Not ight) after observing that Mike does not offers him any bribe ( observes the action the ). When Mike offers, expected payoff if fighting is: and if not fighting: Thus, will fight since: EEEE BB ( TTTT) = = 14 3 EEEE BB (NNNN TTTT) = = 1 4

5 EEEE BB ( TTTT) > EEEE BB (NNNN TTTT), ii. ee., 14 3 > 1 When Mike offers the, expected payoff if fighting is: and if not fighting: Thus, will fight only if: EEEE BB ( DDDDDD) = μμ + 10(1 μμ) = 10 1μμ EEEE BB (NNNN DDDDDD) = μμ + (1 μμ) = 1 EEEE BB ( TTTT) > EEEE BB (NNNN TTTT) 10 1μμ 1 μμ 3 4 We then need to divide our following analysis into two cases: 1. Case 1: 33, and Buster chooses to fight after the.. Case : μμ > 33, and Buster chooses not to fight after the. CASE 1: μμ 33. (Buster chooses to fight after observing that Mike chooses the. ) (-, ) /3 the (, -) (1-) (1-) the Let us now check if this pooling strategy profile can be sustained as a PBE in this case (μμ 3 4 ): 5

6 If Mike has trained hard, then he prefers to deviate towards the, where he obtains a payoff of, than selecting as prescribed in this pooling strategy profile, which only yields a payoff of -. We don t even need to check whether Mike chooses when he didn t trained hard (as prescribed in the pooling strategy profile we are testing), since the above argument already shows that this strategy profile cannot be sustained as a PBE when μμ 3. 4 CASE : μμ > 33. (Buster chooses not to fight after observing that Mike chooses Don t take the. ) (-, ) /3 the (, -) (1-) (1-) the Let us now check if this pooling strategy profile can be sustained as a PBE in this case (μμ > 3 ): 4 If Mike has trained hard, then he prefers to select (as prescribed in this strategy profile), where he obtains a payoff of, than deviating towards the, which only yields a payoff of 0. If Mike has not trained hard, then he prefers to deviate towards the, where he obtains a payoff of 0, than selecting (as prescribed in this strategy profile), which yields a lower payoff of -. Hence, this pooling strategy profile cannot be sustained as a PBE when μμ > 3. 4 Concluding, the pooling strategy profile where Mike makes the offer cannot be supported as a PBE of the game, regardless of Buster offthe-equilibrium beliefs, i.e., regardless of the precise value of μμ. 6