Consider a modification of the Rock-Paper-Scissors game in which P2 is an expert in body language tells. In particular, assume that P2 can tell whether P1 is about to play R or not. At the same time, P2 cannot tell whether P1 is about to play P or S. In other words, P2 can partially observe P1's choice.

Consider a modification of the Rock-Paper-Scissors game in which P2 is an expert in body language tells. In particular, assume that P2 can tell whether P1 is about to play R or not. At the same time, P2 cannot tell whether P1 is about to play P or S. In other words, P2 can partially observe P1's choice.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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Exercise 3. Consider the following game in normal form. L R U 4, 4 1, 6 D 6, 1-3, –3 (a) Find all the Nash equilibria of this game. (b) Show that the following probability measure is a correlated equilibrium of this game. L R 1 1 U (c) Show that the following probability measure is a correlated equilibrium of this game. L R D (d) Plot the payoff profile of cach equilibrium in (a)-(c) above. Are the payoff profiles in (b) and (c) inside or outside the convex hull of NE payoffs?
. Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player chooses one of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices. The outcome is determined as follows. If both players choose the same strategy, neither player wins or loses anything. Otherwise: • "paper covers rock": if one player chooses paper and the other chooses rock, the player who chose paper wins and is paid 1 by the other player. • "scissors cut paper": if one player chooses scissors and the other chooses paper, the player who chose scissors wins and is paid 1 by the other player. • "rock breaks scissors": if one player chooses rock and the other player chooses scissors, the player who chose rock wins and is paid 1 by the other player.
4. David plays in a game where his strategy (set of possible choices) is denoted by x, which can be any integer between 1 and 7, against Goliath, whose strategy is denoted by y, which can be any integer between 1 and 7. David's and Goliath's utility functions are the same and given by: U(x, y) = (x-2)(y-2). Based on this information, determine whether each of the following statements are true or false. (a) The combination (x, y) = (2, 2) is a Nash equilibrium. (b) This game has no dominant strategy (a choice of integer that gives the highest payoff regardless of what the opponent chooses). (c) This game has only one dominated strategy (a choice of integer that would never be a best response to the opponent's choice of integer).
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7 In bargaining theory, sometimes an individual’s patience level can affect his or her bargaining power. Consider a simple repeated game in which Person 1 makes an offer to purchase a good that Person 2 is selling; Person 2 can either accept this offer or reject it and make a counteroffer. This game continues until one player accepts the other’s offer. Explain how incorporating the rate of time preference (ββ) into such a repeated game might alter the outcome.
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4. Consider the following game, expressed in dollar terms: R $3,$2 $0,$1 D $1,80 $2,$1 (a) Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Compute all Nash equilibria in pure strategies. (5 points) (b) Is there an equilibrium in mixed strategies? If there is, compute the probability p with which player I plays U and the probability q with which player II plays L. Also, compute the amount of utility each player gets in the equilibrium. (5 points) (c) Are any of these Nash equilibria trembling-hand perfect? (5 points) (d) Suppose, next, that this game is played by two utilitarians, for whom u(x,y)=x+y. Compute all Nash equilibria in pure and mixed strategies. (5 points)