2. Paul and Stella play a game with three strategies each, T, M, and B for Paul, and L, C, and R for Stella. Both move simultaneously. The payoffs are given by the following form: Stella L R Paul T (8,4) (10, 2) (2,3) M (4,2) (10, 1) (5,7) B (1,4) (10, 3) (9,-4) a. Which strategy is dominated? b. What is the pure-strategy Nash equilibrium? Identify all if there are more than one. c. If Paul moved first, so that Stella observed it, which strategy would Paul choose?

2. Paul and Stella play a game with three strategies each, T, M, and B for Paul, and L, C, and R for Stella. Both move simultaneously. The payoffs are given by the following form: Stella L R Paul T (8,4) (10, 2) (2,3) M (4,2) (10, 1) (5,7) B (1,4) (10, 3) (9,-4) a. Which strategy is dominated? b. What is the pure-strategy Nash equilibrium? Identify all if there are more than one. c. If Paul moved first, so that Stella observed it, which strategy would Paul choose?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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2. Paul and Stella play a game with three strategies each, T, M, and B for Paul, and L, C, and R for Stella. Both move simultaneously. The payoffs are given by the following form: Stella L с R Paul T (8,4) (10, 2) (2,3) M (4,2) (10, 1) (5,7) B (1,4) (10, 3) (9,-4) a. Which strategy is dominated? b. What is the pure-strategy Nash equilibrium? Identify all if there are more than one. c. If Paul moved first, so that Stella observed it, which strategy would Paul choose?
3. Consider the game represented by the payoff matrix below: R S (12, 12) (8,-5) (25, 14) (50, 12) (26,70) (-2.6) (20, 100) (22,75) C (11,0) (3,25) (24, 26) (50,26) D (5,-10) (16,-8) (13,-8) (50,-20) A B a. Find the Nash equilibria of the game using any suitable method. b. Apply Iterated Elimination of Strictly Dominated Actions (IESDA) to this game. Explain the steps as necessary. Do the Nash equilibria found in part a. survive IESDA? c. Apply Iterated Elimination of (Weakly) Dominated Actions (IEDA) to this game by elim- inating all whakly dominated actions in each round. Explain the steps as necessary. Do the Nash equilibria found in part a, survive IEDA
Find the unique Nash equilibrium of a game with the following payoff matrix: Player 2 2 4 1 (6, 7) (13, 8) (9, 9) Player 1 3 (4, 4) (7, 7) (7, 6) (7, 7) (11, 11) (7, 13) О a. (3, 2) O b. (5, 4) О с. (11, 11) O d. (1, 6) O e. There is no unique Nash equilibrium.
GAME UUU B1 Player B B2 A1 7,13 5, 10 A2 3,8 9,16 Player A A3 5,8 4,7 In Game UUU (see table above), assuming players move simultaneously, Player A choosing A1 and Player B choosing B3 is a Nash equilibrium. Player A choosing A3 and Player B choosing B2 is a Nash equilibrium. Both Player A choosing A1 and Player B choosing B1 and Player A choosing A2 and player B choosing B2 are Nash equilibria in pure strategies Player A choosing A1 and Player B choosing B2 is a Nash equilibrium.
2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
Question 1 Consider the following game. Find all Nash equilibria, subgame perfect Nash equilibria, and weak perfect Bayesian equilibria in pure and mixed strategies. a P2 (12, 12) b (12,-8) (7.-3) P1 m d (-3,2) r P2 C (2,7) d (22,17)
(d) Consider a simultaneous-move game between two firms choosing to sell their product at either £6, £7 or £8. The actions and payoffs are given in the matrix below. Firm 2's Prices £6 £7 £8 Firm 1's prices £6 4, 5 3, 5 2, 1 £7 0,4 2, 1 3,0 £8 -1, 1 4, 3 0, 2 What are the Nash equilibria of this game? Game theory is often used by firms competing under an oligopoly as a means of determining their best strategy. Why is game theory a useful tool and which characteristics of an oligopoly make it particularly useful for firms competing in this market structure? One outcome of an oligopoly is that firms may have an incentive to collude. Explain some of the conditions that make collusion more likely to occur and how game theory can explain why collusive agreements often break down.
Consider the payoff matrix below, which describes a symmetric A B A 17,17 0,14 B 14,0 8,8 (a) Find all Nash equilibria of this game, including any mixed strategy equilibria. (b) Draw the expected payoff difference diagram for this game, including all the Nash equilibria you found in (a). is (c) What would you expect to occur most frequently when this game played as a lab experiment? Justify your answer using your answers from (a) and (b).
A game involves two players: player A and player B. Player A has three strategies a1, a2 and a3 while player B has three strategies b1, b2 and b3. Player B b1 b2 b3 a1 -40,30 70,20 -10,120 Player A a2 40,60 80,80 60,20 a3 -30,40 -50,110 150, -70 Assuming that this is a one-time game, answer the following questions: Is there any dominant strategy for each player? What is the secure strategy of each player. What is the Nash equilibrium of the game?
Consider the following game: Player 1 T M B N R (1,2) (1,1) (3,4) (1,-1) L Player 2 (-1,2) (2, -1) (0,1) (4,4) S (3,2) (4,5) (5,4)) (6,1) Q (1,5) (1,2) (4,1) (1,4) a) Does player 1 (the row player) have any dominated strategies. If yes, list all such strategies b) Find all the Pareto Optimal strategy profiles in the game. c) Let 0= (1/2, 0,1/4,1/4). Find BR₂(0).
Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?
C) Two people are suspected of robbing a bank. They are being interrogated in separate rooms. If both stay silent, they can be convicted of a lesser crime and sentenced to only 8 months. If one agreeds to confess that they did it together, he can plea bargain a suspended sentence while the other is convicted to 5 years in prison. If both confess that they did it together, each will receive a 3 year prison term. Create a payoff matrix from the point of view of the first suspect. Find the Nash equilibrium strategies for both suspects in this game.