1. Suppose the following game is repeated infinitely. The players have a common discount factor dE (0,1). C D 0,5 1 С 32 1,1 D 5,0 For high enough values of 6, there is an equilibrium of the infinitely repeated game in which (C, C) is played in the first 4 periods followed by (D. D) in sub- sequent periods. Is this true or false? Explain carefully.

1. Suppose the following game is repeated infinitely. The players have a common discount factor dE (0,1). C D 0,5 1 С 32 1,1 D 5,0 For high enough values of 6, there is an equilibrium of the infinitely repeated game in which (C, C) is played in the first 4 periods followed by (D. D) in sub- sequent periods. Is this true or false? Explain carefully.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

See similar textbooks

Similar questions

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?
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?
see image Use the following setting for questions 1-3. Consider the following static game: C R L 3,1 0,0 5,0 T M 2,1 1,2 3,1 B 1.2 0,1 4,4 Suppose this game is played for two periods. At the beginning of period 2, the players can observe the outcome from period 1. There is no discounting. 1. First, assume only pure-strategy SPNE such that either player's second period strategy DOES NOT depend on the first period outcome. How many such SPNE are in this game? (a) 1 (b) 2 (c) 3 (d) 4 (e) none 2. Consider only SPNE from the previous question. What is the maximum sum of the players' payoffs that can be achieved in the first period? (a) 0 (b) 3 (c) 4 (d) 5 (e) 8 3. Now consider all possible pure-strategy SPNE such that (B,R) is played in the first period. What must be the equilibrium outcome in the second period of such SPNE? (a) (T,L) (b) (M,C) (c) (B,R) (d) (T,R) (e) There are no such SPNE
1. Consider a repeated game in which the following stage game is played in each of two periods and there is no discounting. U Player 1 M D L 5,5 0,0 0,12 Player 2 C 0,0 2,2 0,0 R 12,0 0,0 10, 10 a) Find the Nash equilibrium that maximises the players' final payoffs, i.e. the sum of their payoffs from each period. I b) Discuss the intuition of this equilibrium.
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.
11. What are the subgame perfect equilibria of the following game? Out In T B 7,21 L R L/R 9,3 1,1 1,1 3,9 (a) (InT, L), (OutB, R) and (OutT+OutB, L + R). (b) (InT, L) and (OutB, R). (c) Out and (InT, L). (d) (InT, L), (OutB, R) and (OutT+OutB, L + R). (e) (InT, L). 2
Consider the following two player extensive form game. メ (2,5) (4,) (3,5) a. Find the induced normal form of the given extensive form game using the pure strategies of the players. Find out all the possible Nash Equilibrium of this game. b. From the Nash Equilibrium you are getting in part a., which one is subgame perfect Nash Equilibrium? Explain with appropriate reasons. c. Explain why the NE in part b. is credible. (Use maximum 3 sentences):
Subgames • • • At each node where a player makes a decision, we can think about the "subgame" starting at that node A subgame is the portion of the tree following a particular node How many subgames does our first sequential move example have? • How many subgames does our entry game have? • How many subgames does the game on the right have? • What are its Nash equilibria? Player 1 (1,4) T Player 2 U B (5,2) (3,3) р Player 1 D L 9 (2, 0) Player 2 R (6, 2)
Costco Spend $1B on advertising Spend $1.5B on advertising Spend $1B on advertising $3.5B; $2.7B $4.2; $2.2B Target Spend $1.5B on advertising $3.2B; $3.4B $4B; $3B
Consider the following simultaneous game: Player 1 U D Player 2 L 20,-10 -10, 20 R -10, 20 20,-10 Please indicate whether each of the following statements is true or false. Player 1 has a dominant strategy. This game has a Nash equilibrium. This game has a Nash equilibrium in pure strategies. Player 1's best response is D if player 2 plays R.
2. Market entry following picture. Consider a market entry game with imperfect information as depicted in the Pi A, A4 A2 A3 14 P2 P2 l-p BỊ B2 BỊ B2 B3 B4 B3 B4 4 3 13 7 4 4 5 3 5 10 6 16 3 (a) Write down all strategies of the players. Transform the game into a normal-form. Find all pure strategy Nash equilibria. (b) Find all weak-perfect Bayesian Nash equilibria in pure strategies. Do not forget the systems of beliefs. Argue well if there is none.
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.