SI 563 Homework 3 Oct 5, Determine the set of rationalizable strategies for each of the following games. a) X Y X Y Z

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1 SI 563 Homework 3 Oct 5, 06 Chapter 7 Exercise : ( points) Determine the set of rationalizable strategies for each of the following games. a) U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) X Y U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) b) Z U (,0) (,) (4,) M (3,4) (,) (,3) D (,3) (0,) (3,0) Note: for player strategy D is dominated by a mix strategy of 0.75U and 0.5M /

2 SI 563 Homework 3 Oct 5, 06 X Z U (,0) (4,) M (3,4) (,3) c) Z U (6,3) (5,) (0,) M (0,) (4,6) (6,0) D (,) (3,5) (,8) Note: for player strategy D is dominated by a mix strategy of 0.5U and 0.5M {(U, X)} d) Z U (8,6) (0,) (8,) M (,0) (,6) (5,) D (0,8) (,0) (4,4) Note: for player strategy D is dominated by a mix strategy of 0.5U and 0.5M U (8,6) (0,) M (,0) (,6) e) /

3 SI 563 Homework 3 Oct 5, 06 A (,) (0,0) B (0,0) (3,3) A (,) (0,0) B (0,0) (3,3) f) A (8,0) (4,) B (6,4) (8,5) A (8,0) (4,) B (6,4) (8,5) g) U (3,0) (4,) D (6,4) (8,5) {(8, 5)} 3 /

4 SI 563 Homework 3 Oct 5, 06 Exercise : ( point) Suppose that you manage a firm and are engaged in a dispute with one of your employees. The process of dispute resolution is modeled by the following game, where your employee chooses either to settle or to be tough in negotiation, and you choose either to hire an attorney or to give in. In the cells of the matrix, your payoff is listed second; x is a number that both you and the employee know. Under what conditions can you rationalize selection of give in? Explain what you must believe for this to be the case. you Give in Hire attorney Employee Settle (,) (0,) Be though (3,0) (x,) When x is bigger or equal to 0, for employee strategy Be though dominate strategy Settle. Then you will only choose Hire attorney because is bigger than 0, which is not the case mentioned in the question. When x is smaller than 0, there is no dominate strategies for either of the players. If you choose Hire attorney the expected payoff is always no matter which strategy employee choose. If you choose Give in, your expected payoff is p + 0 ( p) = p if we denote the probability of employee choosing Settle by p. In this case, the only reason you will rationalize selection of give in is x is smaller than 0 AND you believe that the probability that employee play Settle is greater than 0.5 (p > p > 0.5). Note: Since x=0 is the critical value, we won t deduct points if you include 0. 4 /

5 SI 563 Homework 3 Oct 5, 06 Chapter 9 Exercise : ( points) Consider the normal-form game pictured here: a b c w (5,) (3,4) (8,4) x (6,) (,3) (8,8) y (,) (0,) (9,) a) What are the Nash equilibria of this game? b) Which of these equilibria are efficient? a b c w (5,) (3,4) (8,4) x (6,) (,3) (8,8) y (,) (0,) (9,) a) After eliminated the strictly dominated strategies, we find Nash Equilibrium are (w, b) and (y, c) b) The strategy is (y, c) is efficient because no other solution at least better for one agent and not worse for the others. Exercise 3: ( points) Find the Nash equilibria of the games in Exercise of Chapter 7 a) 5 /

6 SI 563 Homework 3 Oct 5, 06 U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) There is no pure Nash equilibria strategy in this game. b) Z U (,0) (,) (4,) M (3,4) (,) (,3) D (,3) (0,) (3,0) Note: For player strategy D is dominated by a mix strategy of 0.5U and0.5m X Z U (,0) (4,) M (3,4) (,3) The Nash equilibrium are (M, X) and (U, Z) c) 6 /

7 SI 563 Homework 3 Oct 5, 06 Z U (6,3) (5,) (0,) M (0,) (4,6) (6,0) D (,) (3,5) (,8) {(U, X)} The Nash equilibria is (U, X) d) Z U (8,6) (0,) (8,) M (,0) (,6) (5,) D (0,8) (,0) (4,4) X Y The Nash equilibrium are (U, X) and (M, Y) U (8,6) (0,) M (,0) (,6) e) X Y A (,) (0,0) B (0,0) (3,3) 7 /

8 SI 563 Homework 3 Oct 5, 06 A (,) (0,0) The Nash equilibrium are (A, X) and (B, Y) B (0,0) (3,3) f) A (8,0) (4,) B (6,4) (8,5) A (8,0) (4,) B (6,4) (8,5) The Nash Equilibrium are (A, X) and (B, Y) g) U (3,0) (4,) D (6,4) (8,5) {(8, 5)} The Nash Equilibria is (D, Y) Chapter Exercise 4: ( points) 8 /

9 SI 563 Homework 3 Oct 5, 06 Compute the mix-strategy equilibria of the following games: A B A (,4) (0,0) p B (,6) (3,7) -p q -q Let p denote the probability that player plays strategy A and q denote the probability that player plays strategy A. For player : q + 0 ( q) = q + 3 ( q) q = 0.75 For player : p ( p) = 0 p + 7 ( p) p = 0. The mix-strategy Nash equilibria is ((0.A, 0.8B), (0.75A, 0.5B)) L M R U (8,3) (3,5) (6,3) p C (3,3) (5,5) (4,8) -p D (5,) (3,7) (4,9) q -q After eliminated dominated strategy L and D, let p denote the probability that player plays strategy U and q denote the probability that player plays strategy M. For player : q ( q) = 5 q + 4 ( q) q = 0.5 For player : p ( p) = 3 p + 8 ( p) p = 0.6 The mix-strategy Nash equilibria is ((0.6U, 0.4C, 0D), (0, 0.5M, 0.5R)) Exercise 6: ( points) Determine all the Nash equilibria (pure-strategy and mixed strategy equilibria) of the following games. a) 9 / H T H (,-) (-,) p T (-,) (,-) -p

10 SI 563 Homework 3 Oct 5, 06 q -q Let p denote the probability that player plays strategy H and q denote the probability that player plays strategy H. For player : q + ( ) ( q) = q + ( q) q = 0.5 For player : p ( ) + ( p) = p + ( ) ( p) p = 0.5 There is no pure-strategy Nash equilibria, and there is a mix-strategy Nash equilibria is ((0.5H, 0.5T), (0.5H, 0.5T)) b) C D C (,) (0,3) D (3,0) (,) After eliminated dominated strategy C and D, there is one pure-strategy Nash equilibria (D, D), and there is no mix-strategy Nash equilibria. c) H D H (,) (3,) p D (3,) (,) -p q -q Let p denote the probability that player plays strategy H and q denote the probability that player plays strategy H. For player : q + 3 ( q) = 3 q + ( q) q = 0.5 For player : p + ( p) = p + ( p) p = 0.5 There is no pure-strategy Nash equilibria, and there is a mix-strategy Nash equilibria is ((0.5H, 0.5D), (0.5H, 0.5D)) d) A B A (,4) (,0) p 0 /

11 SI 563 Homework 3 Oct 5, 06 B (0,8) (3,9) -p q -q Let p denote the probability that player plays strategy A and q denote the probability that player plays strategy A. For player : q + ( q) = 0 q + 3 ( q) q = 0.5 For player : p ( p) = 0 p + 9 ( p) p = 0. There are pure-strategy Nash equilibria (A, A) and (B, B), and there is a mix-strategy Nash equilibria is ((0.A, 0.8B), (0.5A, 0.5B)) /