ps5 - is inversely proportional to the total size of the...

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14.12 Game Theory Prof. Muhamet Yildiz Fall 2005 Homework 5 Due on 11/23/2004 (in class) Note: This is a very short homework. I expect that you will submit your solutions on time. But since it is holiday, if you cannot turn in on time, you can submit your solutions when you come back–without any penalty. 1. Consider a two player game with payo f matrix LR X 3, θ 0,0 Y 2,2 θ 2, θ Z 0,0 3, θ where θ { 1 , 1 } is a parameter known by player 2. Player 1 believes that θ = 1 with probability 1/2 and θ =1 with probability 1/2. Everything above is common knowledge. (a) Write this game formally as a Bayesian game. (b) Compute the Bayesian Nash equilibrium of this game. (c) What would be the Nash equilibria in pure strategies (i) if it were common knowl- edge that θ = 1 , or (ii) if it were common knowledge that θ =1 ? 2. (Problem 3 of Homework 1–revisited) In a college there are n students. They are simultaneously sending data over the college’s data network. Let x i 0 bethesizedata sent by student i .Ea chs tud en t i chooses
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Unformatted text preview: is inversely proportional to the total size of the data, so that it takes x i τ ( x 1 , . . . , x n ) minutes to send the message where τ ( x 1 , . . . , x n ) = x 1 + · · · + x n . The payo f of student i is θ i x i − x i τ ( x 1 , . . . , x n ) , where θ i ∈ { 1 , 2 } is a payo f parameter of player i , privately known by himself or herself. For each j 6 = i , independent of θ j , player j assigns probability 1/2 to θ i = 1 and probability 1/2 to θ i = 2 . Everything described so far is common knowledge. (a) Write this game formally as a Bayesian game. (b) Compute the symmetric Bayesian Nash equilibrium of this game. Hint: symmetric means that x i ( θ i ) = x j ( θ j ) when θ i = θ j . In the symmetric equilibrium one of the types will choose zero, i.e., for some θ ∈ { 1 , 2 } , x i ( θ i ) = whenever θ i = θ . The expected value E [ x 1 + · · · + x n ] of x 1 + · · · + x n is E [ x 1 ] + · · · + E [ x n ] . 1...
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