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ps4_sol

ps4_sol - ECON 212 Game Theory KyungMin Kim(Teddy Fall 2007...

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Unformatted text preview: ECON 212 Game Theory KyungMin Kim (Teddy) Fall 2007 University of Pennsylvania Suggested Solution for Problem Set #4 1. Let & i ( p i ;p j ) be &rm i ¡s pro&t when &rm i and j set price p i and p j respectively, i = 1 ; 2 ;i 6 = j . a. Suppose ( p & 1 ;p & 2 ) is a price vector which maximizes the joint pro&t. Then ( p & 1 ;p & 2 ) 2 arg max ( p 1 ;p 2 ) & 1 ( p 1 ;p 2 ) + & 2 ( p 1 ;p 2 ) = p 1 & 12 & p 1 + 1 2 p 2 ¡ + p 2 & 12 & p 2 + 1 2 p 1 ¡ = 12( p 1 + p 2 ) & p 2 1 & p 2 2 + p 1 p 2 F.O.C. 12 & 2 p & 1 + p & 2 = 12 & 2 p & 2 + p & 1 = Hence p & 1 = p & 2 = 12 & 1 ( p & 1 ;p & 2 ) = & 2 ( p & 1 ;p & 2 ) = 72 b. Let ( p B 1 ;p B 2 ) be the Bertrand equilibrium price. Then p B i 2 arg max p i & i ( p i ;p B j ) = p i & 12 & p i + 1 2 p B j ¡ F.O.C. 12 & 2 p B 1 + 1 2 p B 2 = 12 & 2 p B 2 + 1 2 p B 1 = Similarly, p B 1 = p B 2 = 8 & 1 ( p B 1 ;p B 2 ) = & 2 ( p B 1 ;p B 2 ) = 64 c. Let H t be the set of public histories by time t (not including t ). That is, H t = ( P 1 ¡ P 2 ) t where P i = R + . Let H = f;g and H = [ t ¡ H t . Each player¡s (pure) strategy is s i : H ! P i . The trigger strategy is formally described by the following strategy pro&le: For all t and for all h t 2 H t , s i ( h t ) = ¢ p & i if t = 0 or h t = (( p & 1 ;p & 2 ) ; ( p & 1 ;p & 2 ) ;:::; ( p & 1 ;p & 2 )) p B i otherwise ;i = 1 ; 2 d. There are essentially two histories, one in which ( p & 1 ;p & 2 ) has been played before and the other in which a deviation occurred at least once. The incentive problem in the latter case is obvious, 1 because a stage-game NE is played independent of time and history. For the strategy pro&le to be incentive compatible in the former history (invoking one shot deviation principle), 72 & (1 ¡ & ) 81 + 64 & because 81 = max p i ¡ i ( p i ;p & j ) = p i & 12 ¡ p i + 1 2 12 ¡ Therefore the strategy pro&le is an equilibrium if and only if...
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ps4_sol - ECON 212 Game Theory KyungMin Kim(Teddy Fall 2007...

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