ps2(2010)_soln_v2

# ps2(2010)_soln_v2 - MS&E 246 Game Theory with...

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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 2 Solutions 1. (20pts)(5pts for identifying the expected payoff at a bid, 10 pts for finding out the prob- abilities that can be observed in an equilibrium/showing that p ( b i ) = 1 100 works leads to the same expected payoff for each bid, 5 pts for arguing/verifying that no other probability distribution can make the player better off) (All calculations are in cents.) Player i mixes between 0 , 1 , 2 ,..., 99 where p ( b i ) denotes the probability that player i bids b i ∈ { , 1 , 2 ,..., 99 } . Player i ’s payoff is then U i ( b i ) = (100- b i ) Prob ( b i > b j )- b i Prob ( b i ≤ b j ) . Consider the following symmetric mixed strategies: p ( b i ) = 1 100 for all b i ∈ , 1 , 2 ,..., 99 for all i ∈ 1 , 2 . In order to verify that the above strategy is a mixed strategy equilibrium in which every bid less than 100 has a positive probability, observe that when player i uses this strategy her payoff would be U i ( b i ) = (100- b i ) b i 100- b i 1- b i 100 = 0 for all b i ∈ { , 1 , 2 ,..., 99 } . Finally, note that playing with some other probability distribution over { 0,1,2,...,99 } cannot make player i better off. 2. (5 pts for each part, 25 pts total) 1. The payoff of each student must be nonnegative, since a student can not download a negative number of files. Each student is allowed a maximum download amount of 100, so 10Π 1 ≤ 100 and 20Π 2 ≤ 100. The total amount that can be downloaded from the university is 150, so 10Π 1 + 20Π 2 ≤ 150. So the set of achievable payoffs is defined by the following inequalities: ≤ Π 1 ≤ 10 ≤ Π 2 ≤ 5 Π 1 + 2Π 2 ≤ 15 1 2. The Pareto efficient pairs are the payoffs that satisfy, 5 ≤ Π 1 ≤ 10 2 . 5 ≤ Π 2 ≤ 5 Π 1 + 2Π 2 = 15 3. The max-min fair payoff pair is (5,5). At any other achievable payoff pair either Π 1 < 5 or Π 2 < 5....
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ps2(2010)_soln_v2 - MS&E 246 Game Theory with...

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