Midterm1review_solutions

Midterm1review_solutions - W3211 Spring 2009 Professor...

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W3211 Spring 2009 Professor Vogel Review for Midterm #1: Solutions Review Questions and solutions: 1. Consider the following payo/ matrix: L R T x 1 ;y 1 x 2 ;y 2 B x 3 ;y 3 x 4 ;y 4 In the following questions, suppose this is a simultaneous move game: (a) What are the weakest conditions under which ( T;L ) is a strict dominant strategy equilibrium? Solution: x 1 > x 3 , x 2 > x 4 , y 1 > y 2 , y 3 > y 4 (b) What are the weakest conditions under which ( T;R ) is a dominant strategy equi- librium? Solution: x 1 x 3 , x 2 x 4 , y 2 y 1 , y 4 y 3 (c) What are the weakest conditions under which ( B;R ) is a pure strategy Nash equilibrium? Solution: (B, R) is a pure strategy Nash equilibrium if no player has an incentive to deviate, given the strategy of the other player. ) y 4 y 3 and x 4 x 2 (d) Suppose ( b ;r ) is a strict mixed strategy Nash equilibrium, where b is the equi- librium probability Player 1 plays B and r is the equilibrium probability Player 2 plays R . Solve for b and r as functions of the x y of b and r for a strict mixed strategy Nash equilibrium to exist? Solution: If ( b ;r ) is a strict mixed strategy Nash Equilibrium, then both players must be indi/erent between playing their respective pure strategies given (1 ± r ) x 1 + r x 2 = (1 ± r ) x 3 + r x 4 (1) (1 ± b ) y 1 + b y 3 = (1 ± b ) y 2 + b y 4 (2) We can rearrange (1) and (2) to get: r = x 1 ± x 3 x 1 + x 4 ± x 2 ± x 3 (3) b = y 1 ± y 2 y 4 + y 1 ± y 2 ± y 3 (4)
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Review for Midterm 1 2 There is a strict mixed strategy Nash Equilibrium if both 0 < b < 1 and 0 < r < 1 hold. (e) What are the weakest conditions under which both ( T;L ) and ( B;R ) are pure strategy Nash equilibria? Solution: From (c), (B, R) is a pure strategy Nash equilibrium if y 4 y 3 and x 4 x 2 . Similarly, (T, L) is a pure strategy Nash equilibrium if y 1 y 2 and x 1 x 3 . For both (T, L) and (B, R) to be pure strategy Nash equilibria, these 4 conditions should hold simultaneously. (f) If ( B;L ) is a pure strategy Nash Equilibrium, can there exist a strict mixed strategy Nash equilibrium? Solution: Yes. If (B, L) is a pure strategy Nash Equilibrium ) x 3 x 1 and y 1 y 2 . If y 4 > y 3 and x 2 > x 4 , then conditions in part (d) still hold and therefore, we can have a strict mixed strategy Nash Equilibrium. (g) If ( B;L ) is a strict dominant strategy equilibrium, can there exist a strict mixed strategy Nash equilibrium? Solution: No, because in order for the players to randomize strictly, they need to be indi/erent between their pure strategies. Now suppose this is a sequential game in which Player 1 2 moves second after observing Player 1 ±s move (h) Write out the payo/ matrix for the sequential move game. Solution:
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Midterm1review_solutions - W3211 Spring 2009 Professor...

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