1\2 L M R 1,-1 | 5,0 0,0 4,4 0,0 0,0 3,3 C 0,5 В 0,0 1. Find all set of rationalizable strategies (Hint: We are allowed to eliminate a pure strategy, if it is strictly dominated by a mixed strategy). 2. Find all set of pure strategy Nash equilibria. 3. Is there a dominant strategy equilibrium for this game?
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- 1. Find all mixed strategy Nash Equilibria for the games below. How many Nash Equilibria do these games have (including both pure and mixed strategy)? Player 2 A) R Player 1 2,2 4,1 E 1.4 3,3 B) Player 2 X y a 4,0 2,1 3,2 Player 1 b 2,2 3,4 0,1 2,2 1,2 0,3 c) Player 3: A Player 2 X y a 2,0,4 1,1,1 1,2,3, Player 1 3,2,3 0,1,0 2,1,3 b 1,0,2 0,0,3 3,1,1 Player 3: B Player 2 X y 2,0,3 4,1,2 1,2,2 a Player 1 b 1,3,2 2,2,2 0,4,0 0,0,0 3,0,3 2,1,02. Consider the following game: u 1 r 6,0 0,6 d 3,2 6,0 Are there any Nash equilibria in pure strategies for this game? Find the mixed strategy Nash equilibrium.1. Consider the following 2 person game H M L H 3,0 2,0 0,2 0,3 3,0 2,0 L 1,0 1,0 0,1 (a) Are there any pure strategy nash equilibria? If not, find the mixed strategy nash equilibrium. (b) In light of your answer above, are there any strategies which are not rationalizable?
- rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.Q2 Consider the following game. (a) Find all pure-strategy Nash equilibria. (b) Find all mixed-strategy Nash equilibria. 2, 4 6, 0 5,1 1,9 A В3. 1 2 J K 2 1 2 0 0 K 3 3 E/ 1.1 2.2 B G 2 -2 a.) How many Subgames does this Game have? b.) Convert this Game Tree into matrix form. c.) Find all Pure Strategy Nash Equilibria of this Game. d.) Find all Subgame Perfect Nash Equilibria of this Game. (Pure and Mixed) 2 2.2
- 2. Find all pure and mixed strategy Nash equilibria of the following game. X Y Z X 4,2 0,0 0,1 0,0 2,4 1,3 Y3. Nash Equilibria Consider the following (normal-form) game. Compute a Nash Equilibrium of this game. U C D L | M R 5,0 1,3 4,0 2,4 1,4 3,5 0,1 1,0 5,01. Find all of the pure strategy Nash Equilibrium to the following simultaneous move game. Show all of your work. Row a b C d e A 9,4 14,8 7,8 15,0 20,18 B 12,10 3,10 6,8 7,4 2,9 Column C 15,7 12,18 20,10 14.2 10,14 D 2,8 4,7 3,12 5.3 3,7 E 15,5 20,12 15,9 9,1 19,20
- 4. Consider the following game. Find the dominant strategy for each player (if any). What is the Nash Equilibrium in pure strategies of this game? Player 2 C R U 10,15 6,27 12,30 M Player 1 20,18 16,15 8,24 D 6,15 18,9 10,12し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?Consider the following game: Player 2 In Out Player 1 In -2,-2 2, 0 Out 0, 2 0, 0 (a) What is the Nash equilibrium of this game, or what are the Nash equilibriaof this game? (b) Does either firm have a dominate strategy (a strategy that is always abest response)? Which? (c) Suppose Player 1 could move before Player 2 and Player 2 could observe Player 1’s move. What do you think would happen?