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: Prof. William H. Sandholm Department of Economics University of Wisconsin March 13, 2007 Midterm Exam Solutions Economics 713 1. By the oneshot deviation principle, it is enough to check that there is no profitable oneshot deviation in any subgame. There are three classes of subgames corresponding to the three stages specified in i . In stage I, the equilibrium payo ff sequence is (3, 3, 3, ... ); the most profitable oneshot deviation from T is to B , which generates the payo ff sequence (4, 1, 1, 1, ... ). Thus, the deviation is not profitable if 3 (1 ) 4 + 1, and hence if 1 3 . In stage II, the equilibrium payo ff sequence is (1, 1, 1, ... ); the most profitable oneshot deviation from T is to B , which generates the payo ff sequence (3, 0, 0, 0, ... ). Thus, the deviation is not profitable if 1 (1 ) 3, and hence if 2 3 . In stage III, there is no profitable oneshot deviation. Thus, strategy profile is a subgame perfect equilibrium whenever 2 3 . 2. (i) S i is the set of functions of the form s i : T i A i (ii) Assume without loss of generality that p ( t i ) = t i T i p ( t i , t i ) > 0 for all t i T i and i N , and let p ( t i  t i ) = p ( t i , t i ) / p ( t i ). Strategy profile s * is a Nash equilibrium if and only if ( ) X t i T i p ( t i  t i ) u i ( s * i ( t i ) , s * i ( t i )) X t i T i p ( t i  t i ) u i ( a i , s * i ( t i )) for all a i A i , t i T i , and i N . (iii) Multiplying ( ) through by p ( t i ) shows that it is equivalent to ( ) X t i T i p ( t i , t i ) u i ( s * i ( t i ) , s * i ( t i )) X t i T i p ( t i , t i ) u i ( a i , s * i ( t i )) ....
View
Full
Document
This note was uploaded on 08/08/2008 for the course ECON 713 taught by Professor Sandholm during the Spring '08 term at Wisconsin.
 Spring '08
 SANDHOLM
 Economics, Game Theory

Click to edit the document details