h4answ08 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Econ 805—Homework #4 Answers Prof James Peck Winter 2008 1. Mas-Colell, exercise 8.E.1. Assume that 0 <M< 2 s holds. Answer: First, let us convert the verbal description of the game into a Bayesian game in matrix form. When nature chooses the types of the two armies, it selects one of four payo f matrices. Army 1 either observes that it is strong, so that one of the top matrices is the correct one, or it observes that it is weak, so that one of the bottom matrices is the correct one. Army 2 either observes that it is strong, so that one of the left matrices is the correct one, or it observes that it is weak, so that one of the right matrices is the correct one. army 2: strong army 2: weak attack no attack no army 1: strong attack s, sM , 0 M s, wM , 0 no 0 ,M 0 , 00 0 , 0 army 1: weak attack w,M , 0 w, , 0 no 0 0 , 0 , 0 In order to reduce the number of possible equilibria to consider, notice f rst that, no matter what strategy the opponent is playing, if a weak army is better o f attacking than not attacking, thenastrongarmyisa lsobettero f attacking than not attacking. Letting the strategy of an army be denoted by a vector of what the army does when it is strong, followed by what the army does when it is weak, the only possible Bayesian Nash equilibria are: 1. {(A,A) , (N,N)} 2. {(A,A) , (A,N)} 3. {(A,A) , (A,A)} 4. {(A,N) , (N,N)} 5. {(A,N) , (A,N)} 6. {(A,N) , (A,A)} 7. {(N,N) , (N,N)} 8. {(N,N) , (A,N)} 9. {(N,N) , (A,A)} Because it is always optimal to attack when the rival army never attacks, we can eliminate possibilities (4), (7), and (8) above. Because possibilities (1) and (9) are mirror images of each other in this symmetric problem, and (2) and 1
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(6) are mirror images of each other, we can treat them as the same possibility, so we are essentially down to (1), (2), (3), and (5).
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h4answ08 - Department of Economics The Ohio State...

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