2. Consider the normal form of the simultaneous game given by table 1. As- sume that both a and b are greater than 0. Table 1: Left Right a,b Up 0,0 Down 0,0 2,2 a) What type of game is this? b) What are the pure strategy Nash equilibrium(s) in the game? c) Re-write the normal form of the game and include the mixed strategy options for both players. Make sure to show the pay-offs. d) What is the mixed strategy Nash equilibrium for the game? e) Draw the best response rules for each plaver and point out the Nash

2. Consider the normal form of the simultaneous game given by table 1. As- sume that both a and b are greater than 0. Table 1: Left Right a,b Up 0,0 Down 0,0 2,2 a) What type of game is this? b) What are the pure strategy Nash equilibrium(s) in the game? c) Re-write the normal form of the game and include the mixed strategy options for both players. Make sure to show the pay-offs. d) What is the mixed strategy Nash equilibrium for the game? e) Draw the best response rules for each plaver and point out the Nash

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.1P

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1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).
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Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 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.
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Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?
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