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Final_2009_soln

# Final_2009_soln - MS&E 246 Ramesh Johari Final Examination...

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MS&E 246 Final Examination Ramesh Johari March 18, 2009 Instructions 1. Take alternate seating. 2. Answer all questions in the blue books provided. If needed, additional paper will be avail- able at the front of the room. Answers given on any other paper will not be counted. 3. Notes, books, calculators, and other aids are not allowed. 4. The examination begins at 7:00 pm, and ends at 10:00 pm. 5. Show your work! Partial credit will be given for correct reasoning. Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) 1

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Problem 1 (30 points) . Answer the following questions. Solution: (a) (10 points) Consider a two player simultaneous move game where player 1 has action set { L, M, R } , and player 2 has actions { l, m, r } . Assume both players receive a payoff of 1 if they play the same action, and zero otherwise. Give two different extensive form represen- tations of this game. Solution: (b) (10 points) True or false : A weakly dominated strategy is never rationalizable. If true, justify your answer; if false, give a counterexample. Solution: FALSE. Consider the game below. Player 2 a b A (0,6) (3,0) Player 1 B (1,5) (5,5) In this game a weakly dominates b; but b is played in the NE (B,b), and hence it must be rationalizable. 2
(c) (10 points) Give an example of a two player game with at least two mixed Nash equilibria (that are not pure Nash equilibria). Solution: Any pair of mixed strategies in the game below is NE. Player 2 a b A (0,0) (0,0) Player 1 B (0,0) (0,0) Problem 2 (30 points) . Consider the following two player game: Player 2 H M L H (0,0) (3,4) (6,0) Player 1 M (4,3) (0,0) (0,0) L (0,6) (0,0) (5,5) (a) (5 points) Are any strategies strictly dominated? Solution: L is strictly dominated by a mixture of H and M; e.g., suppose H is played with probability 0.99, and M with probability 0.01. The resulting payoff is 0.04 if the opponent plays H; 2.97 if the opponent plays M; and 5.94 if the opponent plays L. In all cases the payoff is strictly higher than the payoff obtained by playing L. (b) (5 points) Find all pure Nash equilibria of this game. Solution: (H,M) and (M,H) are the pure NE. (c) (10 points) Find a symmetric mixed Nash equilibrium of this game, i.e., where both players play the same mixed strategy. Solution: By (a), L cannot be played in an NE. So we assume Player 2 plays H with proba- bility p , and M with probability 1 - p . Player 1 must be indifferent between H and M, so we require 4 p = 3(1 - p ) , i.e., p = 3 / 7 . Similarly, if player 1 plays H with probability 3/7, and M with probability 4/7, then player 2 will be indifferent between H and M. So both players using this mixed strategy is a symmetric mixed NE. 3

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(d) (10 points) Now consider an infinitely repeated game, where both players have discount factor δ . Find a sufficiently large value of δ such that there exists a subgame perfect Nash equilibrium where, on the equilibrium path, both players play ( L, L ) in every period.
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Final_2009_soln - MS&E 246 Ramesh Johari Final Examination...

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