finalansw08

# Finalansw08 - Department of Economics The Ohio State University Final Exam Answers-Econ 805 Prof Peck Winter 2008 1(40 points Consider the

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Department of Economics The Ohio State University Final Exam Answers—Econ 805 Prof. Peck Winter 2008 1. (40 points) Consider the following two player, zero sum game between an attacker and a defender. The attacker must choose which one of three targets to attack, and the defender must choose which one of the three targets to defend. If the defender defends the target that the attacker attacks, both sides receive a payo f of zero. If the defender does not defend the target that the attacker attacks, then the payo f s depend on which target was attacked, as follows: if target 1 is successfully attacked, the payo f s to the attacker and defender are (2 , 2) ;i f target 2 is successfully attacked, the payo f s to the attacker and defender are (1 , 1) ; and if target 3 is successfully attacked, the payo f s to the attacker and defender are ( v, v ) , where we have v< 1 . (a) (10 points) Prove that this game does not have a Nash equilibrium in pure strategies. (b) (20 points) Assuming that there is a mixed strategy Nash equilibrium in which all three targets are attacked with positive probability, f nd the equilib- rium mixing probabilities for the attacker ( p 1 ,p 2 3 ) and the equilibrium mixing probabilities for the defender ( q 1 ,q 2 3 ) . Your answer should be a function of the parameter, v . (c) (10 points) For what values of v w i l ltherebeam ixeds tra tegyNash equilibrium in which all three targets are attacked with positive probability? Answer: (a) Suppose there is a NE in pure strategies. If the attacker is choosing the same target that the defender is defending, then the attacker can dev iatetoad i f erent target and increase his payo f . If the attacker is choosing ad i f erent target from the one that the defender is defending, then the defender can deviate to the attacker’s target and increase her payo f .T h u s , t h e s t r a t e g y pro f le cannot be a NE. (b) If the attacker is attacking each target with positive probability, he must be indi f erent between all three actions. Therefore, we have 2(1 q 1 )=1 q 2 = v (1 q 3 ) and q 1 + q 2 + q 3 =1 .

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## This note was uploaded on 07/17/2008 for the course ECON 805 taught by Professor Peck during the Spring '08 term at Ohio State.

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Finalansw08 - Department of Economics The Ohio State University Final Exam Answers-Econ 805 Prof Peck Winter 2008 1(40 points Consider the

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