HW6-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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Dr. Huanxing Yang Econ 601 Game Theory Homework Assignment 6 Due May 28, Wednesday 1. Chapter 10, Problem 5 In answering part (b), we will also provide the answer to part (a). A strategy for player 1 is a pair of actions; an action when his type is H and an action when his type is L . A strategy for player 2 is a single action since he has only one information set. There are then eight possible strategy pairs, which are shown in Table SOL10.5.1 1 3 1 2 Table SOL10.5.1 # 1's action when H 1's action when L 2's strategy Equilibrium? 1N o 2Y e s i f . 3N o 4N o 5N o 6N o 7N o 8Y e s i f xx a xy a p yx a yy a b b b b p Let's consider each of them. Strategy pair #1 is not an equilibrium as when player 1 is type L , he prefers to choose y and get a payoff of 3 than x and get a payoff of 2. As to strategy pair #2, player 1's strategy is optimal as, given player 2 is to choose action a , player 1's payoff is 3 from x and 2 from y (when he is type H ) and is 3 from y and 2 from x (when he is type L). Action a yields a higher expected payoff than b for player 2 when: p 1 1 p 1 p 3 1 p 0 1 3 p . Turning to strategy pair #3, player 1's strategy is not optimal as it calls for action y when he is type H and we already showed that he prefers x (given player 2 chooses a ). The same argument applies to strategy pair #4. For strategy pair #5, given player 2 chooses b , player 1 prefers y when he is type H (with a payoff of 5) to x (with a payoff of 1). Hence, both strategy pairs #5 and #6 are not equilibria. For strategy pair #7, player 1 prefers y to x when he is type L since
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the former yields 4 and the latter yields 1 (given player 2 is choosing b ). Finally, it should be clear from the above that player 1's strategy in strategy pair #8 is optimal for both types. Player 2's strategy is optimal when: () 1 21 0 11 1 2 pp p p p ×+ − ×≥ ×+ − ×⇒ ≥ The last column on the right in Table SOL10.5.1 answers part (b) and the answer
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HW6-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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