This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 1 Prof. William H. Sandholm Department of Economics University of Wisconsin March 12, 2002 Midterm Solutions Economics 713 1. Dave's preferences fail to satisfy the independence axiom. Let q = (0, 1, 0). Then p 3 = 1 2 p 1 + 1 2 q , and p 4 = 1 2 p 2 + 1 2 q . Hence, the independence axiom implies that Dave's preference between p 3 and p 4 must be the same as his preference between p 1 and p 2 , but the question states that this is not the case. (Alternatively, one could show that if the utilities u a , u b , and u c satisfy u a = 1 2 u b + 1 2 u c , they must also satisfy 1 2 u a + 1 2 u b = 3 4 u b + 1 4 u c . This implies that Dave's preferences do not admit an expected utility representation, and so cannot satisfy all three axioms.) 2. The extensive form perfect equilibria of are those strategy profiles which correspond to normal form perfect equilibria of the agent normal form game A ( ). Moreover, it is known that all extensive form perfect equilibria of correspond to sequential equilibria of . But if no player controls more than one information set in , then the agent normal form A ( ) is identical to the reduced normal form G ( ). Therefore, perfect equilibria of G ( ) correspond to perfect equilibria of , and hence to sequential equilibria. 3. All beliefs for player 3 are consistent. Let B , C , and D be the weights placed on B , C , and D in the perturbed strategy profiles. Then ( x ) = B /( B + C D ). Assume that C = D = . If B = , then ( x ) = 1/(1 + ) 1, so ( x ) = 1 is consistent. If (0, 1) and B = 2 /(1 ), then...
View Full Document