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07solution - Prof William H Sandholm Department of...

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Prof. William H. Sandholm Department of Economics University of Wisconsin March 13, 2007 Midterm Exam Solutions – Economics 713 1. By the one-shot deviation principle, it is enough to check that there is no profitable one-shot deviation in any subgame. There are three classes of subgames corresponding to the three stages specified in σ i . In stage I, the equilibrium payo ff sequence is (3, 3, 3, ... ); the most profitable one-shot 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 - δ ) · 4 + δ · 1, and hence if δ 1 3 . In stage II, the equilibrium payo ff sequence is (1, 1, 1, ... ); the most profitable one-shot 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 - δ ) · 3, and hence if δ 2 3 . In stage III, there is no profitable one-shot deviation. Thus, strategy profile σ is a subgame perfect equilibrium whenever δ 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 ( ) t - i T - i p ( t - i | t i ) u i ( s * i ( t i ) , s * - i ( t - i )) 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 ( ) t - i T - i p ( t i , t - i ) u i ( s * i ( t i ) , s * - i ( t - i )) t - i T - i p ( t i , t - i ) u i ( a i , s * - i ( t - i )) . Let T * i ( a i ) = { t i T i : s * i ( t i ) = a i } be the set of player i types who choose action a i under strategy s * . Then t i T * i ( a i ) t - i T - i p ( t i , t - i ) u i ( s * i ( t i ) , s * - i ( t - i )) = a - i A - i ρ * ( a i , a - i ) u i ( a i , a - i ) .
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