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solution11b-spring11

# solution11b-spring11 - Econ 354K Game Theory Solutions to...

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1 Econ 354K Game Theory Solutions to Problem Set No. 11b 1. Game of Chicken Minimax: max U 1 (•,C)=1+x; max U 1 (•,S)=0 => min(1+x,0)=0. Therefore minimax is (0,0). IR set: area (0,1+x)--(0,0)—-(1+x,0)-—(1,1) 2 Infinitely Repeated Chicken (a) Subgame perfect NE : Grim strategy: Play C in period 1 and in any period which has a history of everyone playing C. However, in any period in which someone has played S in the past, choose C with probability 1/(1+x) forever thereafter. Current Period Gain from deviation: x Per-Period Future Loss is 1-(1/(1+x))= x/(1+x) , with discounted present value [ δ /(1- δ )]*[x/(1+x)] Deterence requires [ δ /(1- δ )]*[x/(1+x)]>= x => δ >=(1+x)/(2+x) OR, you can think this way: No deviation payoff: 1/(1- δ ) Deviation in the first period, payoff: (1+x)+[1/(1+x)]*[ δ /(1- δ )] Deterence requires 1/(1- δ )>= (1+x)+[1/(1+x)]*[ δ /(1- δ )] Ö δ >=(1+x)/(2+x) (b) Subgame perfect NE : Strategy: Play C in period 1. As long as both players have used the same action in the past, play C in the current period. If different actions were chosen in

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