4. Consider the following extensive-form version of the Prisoner's Dilemma. In this version, Player 1 takes an action first. Player 2 then learns about Player 1's action and takes an action accordingly. The payoffs are represented as (7₁, 72), where 7₁ is Player 1's payoff and 72 is Player 2's payoff. Player 1 Speak Player 2 Silence Silence Player 2 Speak Speak (-8,-8) (0, -10) (-10,0) Which outcome is most likely to happen in the end of the game? (a) Player 1 plays Speak, Player 2 plays Speak (b) Player 1 plays Speak, Player 2 plays Silence (c) Player 1 plays Silence, Player 2 plays Speak (d) Player 1 plays Silence, Player 2 plays Silence (e) None of the above Answer: 4a. Silence (-1,-1)

4. Consider the following extensive-form version of the Prisoner's Dilemma. In this version, Player 1 takes an action first. Player 2 then learns about Player 1's action and takes an action accordingly. The payoffs are represented as (7₁, 72), where 7₁ is Player 1's payoff and 72 is Player 2's payoff. Player 1 Speak Player 2 Silence Silence Player 2 Speak Speak (-8,-8) (0, -10) (-10,0) Which outcome is most likely to happen in the end of the game? (a) Player 1 plays Speak, Player 2 plays Speak (b) Player 1 plays Speak, Player 2 plays Silence (c) Player 1 plays Silence, Player 2 plays Speak (d) Player 1 plays Silence, Player 2 plays Silence (e) None of the above Answer: 4a. Silence (-1,-1)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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9. In an infinitely repeated Prisoner's Dilemma, a version of what is known as a "tit for tat" strategy of a player 2 is described as follows: ⚫ There are two "statuses" that player i might be in during any period: "normal" and "revenge"; • In a normal status player i cooperates; ⚫ In a revenge status player i defects; • From a normal status, player i switches to the revenge status in the next period only if the other player defects in this period; • From a revenge status player i automatically switches back to the normal status in the next period regardless of the other player's action in this period. Consider an infinitely repeated game so that with probability p that the game continues to the next period and with probability (1-p) it ends. Cooperate (C) Defect (D) Cooperate (C) 4,4 5,0 Defect (D) 0,5 1,1 What is the threshold p* such that when p≥ p* always cooperating by player 2 is a best response to player 1 playing tit for tat and starting in a normal status, but when p < p*…
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1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1…
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).
2. Consider two players playing a simultaneous moves game. These players can be one of the two states of the world, state 1 or state 2, while playing the game. The payoffs-matrix for the two state games are as follows: State 1 P1 U D P2 L R 1,4 1,0 1,6 2,16 0,0 0,24 M State 2 P1 U D P2 L M R 1,0 1,4 1,6 2,16 0,24 0,0 Each players believes that the two states are equally likely, Derive the BNE of the game.
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1. Consider the following game in which each of the two players has two possible actions, cooperate with each other (C) and defect (D, i.e., Don't cooperate with each other). The monetary payoffs to each action profile are given in the following table. Player1 / Player 2 Defect (D) Cooperate (C) 3, 0 Defect (D) 1, 1 Cooperate (C) 0,3 2, 2 However, the above table does not represent the true preferences of the players, as they care about what the other player earns as well as their own earnings. In particular, if mi is the amount of money earned by Player 1 and m2 is the amount of money earned by Player 2, then the payoff to Player 1 is mi + am2 ; where a 2 0. (Of course, when a = 0; the payoffs coincide with the ones in the above table). For example, Player 1's payoff to action profile (C,C) is u1 (C;C) = 2 + 2a; and Player 1's payoff to action profile (C,D) is u1 (C;D ) = 0+ 3a = 3a a) Write down the strategic form of this game for a= 1. Is this game a Prisoners' Dilemma game? b) Find…
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Question 1
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
Refer to Figure a. Charlie and Joe both want to ride shotgun with their mother, so they play a game of rock-paper-scissors to determine who gets to sit in the front seat. In the table, -1 represents a loss, 1 a win and 0 a tie, and Joe's payoff is shown in the upper left-hand corner of each cell, while Charlie's appears in the lower right-hand corner. If Charlie knows Joe always throws scissors, what is Charlie's best response? Joe R0 P 1 S-1 Multiple Choice Rock Paper Scissors R 0 -1 1 -1 0 1 Charlie P Figure a 1 0 -1 1 -1 0 The answer cannot be determined from the information given. S -1 1 0
Question 2 The game of Chicken is played by two teens who speed toward each other on a single lane road. The first to veer off is branded the chicken, whereas the one who doesn’t veer gains peer- group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game are provided in the following table. TEEN 2 TEEN 1 Veer Don't Veer Veer Don't Veer 2,2 3,1 1,3 0,0 a) Draw the extensive form. b) Find the pure-strategy Nash equilibrium or equilibria. c) Compute the mixed-strategy Nash equilibrium. As part of your answer draw the best- response function diagram for the mixed strategies. d) Suppose the game is played sequentially with teen A moving first and committing to this action by throwing away the steering wheel. What are teen B's contingent strategies? Write down the normal and extensive forms for the sequential version of the game. e) Using the normal form for the sequential version of the game, solve for the Nash equilibria. f) Identify the proper…