25 06 2023 E TS

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1 June 25 1. Strategies numerical  
2 points  
0 penalty Suppose that player 1 chooses at the beginning x ∈ [0, +∞); then after observing the choice of the first player, player 2 chooses y ∈ [0, +∞). The utility functions of the two players are respectively f(x, y) = −x2 + y2 + 2x, g (x, y) = 8xy2 − y4 + 3x . Use backward induction to find the rational outcome of the game, and write the utility of player 2 (hint: it is an integer). • 153 ✓ Player 2 chooses y to maximise g(x, y), which is a 4th order polynomial in y. Since ∂g(x, y)/∂y = 16xy − 4y3, then there are two maxima at y = ±2√x and a minimum at y = 0. So Player 2 chooses y = 2√x. Then the utility of player 1 is f(x, 2√x) = −x2 + 6x. The maximum is at x = 3. The utility of player 2 is g(3, 2 √ 3) = 8·3·12−122+3 ·3 = 153 2. Value numerical  
2 points  
0 penalty What is the value of this zero-sum game? A =   2 −1 0 −1 1 −2 0 −2 1   • -0.5 ✓ Since maxi minj Aij = −1 and minj maxi Aij = 1, there are no optimal pure strategies. Use the indifference principle. Assume player 2 chooses (p, q, 1 − p − q). Then the utilities for the first player are (2p − q, p + 3q − 2, 1 − p − 3q). The solution of 2p − q = p + 3q − 2 = 1 − p − 3q is p = 0 and q = 1/2. Since the matrix is symmetric, the same holds when we exchange the players. Then both players play (0, 1/2, 1/2) and the value of the game is −1/2. 1 3. Best reply multi  
2 points  
0 penalty  
Single  
Shuffle Consider the game described by the bimatrix  (10, 0) ( −5, −5) (−10, −10) (0 , 10)  . If Player 2 plays (1/2, 1/2) of the following is a best reply of Player 1? (a) (1 , 0) (100%) (b) (1 /2, 1/2) (c) (0 , 1) (d) (1 /3, 2/3) 4. Core truefalse  
2 points Consider the following game ( N, v) with N = {1, 2, . . . ,5}, and v(S) =…