6. Consider the following two stage games, the Prisoner's Dilemma (PD, left) and the Revenge game (R, right). When relevant, the discount factor is 8, 0 < 8 < 1. Player 1 M F Player 2 m 4,4 5, -1 f -1,5 1, 1 Player 1 L G Player 2 1 0,0 -1,-4 8 -4, -1 -3, -3 a. Take both games separately as static games. Solve for all pure strategy Nash equilibria (NE) for PD, and then do the same for R. b. Suppose the players play a three-stage game in the following order: R, PD. PD. Find the two pure strategy subgame perfect Nash equilibria (SPNE). c. Suppose the players play a two-stage game in the following order: PD. R. Find a pure strategy SPNE that supports the play of (M,m) in the first round for high enough 8. d. Find the lowest 8 that supports the SPNE you found in part (c).

6. Consider the following two stage games, the Prisoner's Dilemma (PD, left) and the Revenge game (R, right). When relevant, the discount factor is 8, 0 < 8 < 1. Player 1 M F Player 2 m 4,4 5, -1 f -1,5 1, 1 Player 1 L G Player 2 1 0,0 -1,-4 8 -4, -1 -3, -3 a. Take both games separately as static games. Solve for all pure strategy Nash equilibria (NE) for PD, and then do the same for R. b. Suppose the players play a three-stage game in the following order: R, PD. PD. Find the two pure strategy subgame perfect Nash equilibria (SPNE). c. Suppose the players play a two-stage game in the following order: PD. R. Find a pure strategy SPNE that supports the play of (M,m) in the first round for high enough 8. d. Find the lowest 8 that supports the SPNE you found in part (c).

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.6P

See similar textbooks

Similar questions

3. Two players play the following game for infinite times. Cooperate Betray Cooperate 10, 20 -25, 30 Betray 15, -23 -12, -19 For the player to continue to cooperate what would be the ranges of their discount factor, 8, and 82, respectively?
Consider a game in which there is $4 to be divided, and the first mover is only permitted to make one of three proposals: (a) $3 for the first mover and $1 for the second mover, (b) $2 for each, or (c) $1 for the first movier and $3 for the second mover. The second mover is shown the proposal and can either accept, in which case it is implemented, or reject and cause each to earn $0. Show this game in extensive (tree) form. Be sure to show the payoffs for each person, with the first mover listed on the left, for each of the six terminal nodes.
3. Two individuals are planning to start a business. The earnings from a grocery store business would be $30,000 for each, but the business requires two people (one to watch over the stock and the other to act as a cashier). Each individual could instead start an mpesa business. The take from mpesa business is only $3000 but can be done with one person acting alone. i) Write down the payoff matrix for this game ( ii) Solve the Nash equilibrium
5. Consider the following Extended Form game where P1 stands for Player one and P2 stands for player two with A, B and A', B' being their respective strategies. A' (1,1) P2 B' (2,3) P1 A' B (4,5) P2 What is the set of Nash Equilibria? Clearly explain your answer. Full-screen Snip B' (6,7)
i. ii. QUESTION ONE A. A Nash equilibrium is a strategy profile such that every player's strategy is the best response to all the other players. It requires that each player makes a best response and that expectations regarding the play of other players are correct. Below is the table showing strategies and payoff for Player 1 and Player 2. PLAYER 1 R1 R2 R3 R4 C1 0,7 5,2 7,0 6,6 C2 2,5 3,3 2,5 2,2 PLAYER 2 C3 7,0 5,2 0,7 4,4 CA 6,6 2,2 4,4 10,4 REQUIRED; Transform the normal form game above into an imperfect extensive game form Find the Nash equilibrium for the game above using iterative deletion of strictly dominated strategies. Find the Nash equilibrium using brute force or cell by cell inspection.
Consider the following centipede game consisting of two players, Pl and P2. The left/right number each terminal node represents Pl's/P2's payoff, respectively. Then, answer the following questions of [D5] and [M5]-|M8]: P1 G P2 P1 G -(0, 2) D (2, 0) (1, 1) (4, 0) Suppose that P2 chooses G or D randomly. Then, what is the Pl's best response of P1 for the P2's choice? And explain why. We assume that random choice is level-0 in the level-k theory. Then, answer the P2's choice in level-2. (a) D (b) G (c) random choice on (G, D) (d) G with probability 1/3 Answer all the properties of Nash equilibrium and subgame perfect equilibrium which is derives from the backward induction. (a) All subgame perfect equilibria are Nash equilibria in any game. (b) All Nash equilibria are subgame perfect equilibria in any game. (c) There is always a unique Nash equilibrium in any game. (d) There exist pure-strategy Nash equilibria in any game. (e) The Nash equilibrium in prisoners' dilemma game is socially…
4. Correlated EquilibriaConstruct an example (not one from class or the reading) of a Normal form game with a correlated equilibrium that is not a Nash equilibrium.
6. Consider Jake and Al again. They are considering going to Portugal or Sweden once the pandemic ends. Here are their payoffs: Al's choices: Sweden Portugal Jake's choices (8, 4) (4, 9) Sweden (10, 5) Portugal (3, 6) and Al going to d) The strategy (P, S), represents Jake going to e) Is there a dominant strategy in this game? Explain. f) Find the Nash equilibrium (equilibria) in this game, if one or more exists.
4. Consider a normal form game with N= {1,2} and S= {A,B,C,D} x {W,X,Y,Z}. Assume that L1 allows to eliminate D and X, L2 allows to eliminate B and C, and L3 allows to eliminate Z. Provide payoffs that lead to this outcome. State the rationality assumptions explicitly.
Consider the following game : Stag Rabbit Stag 9, 9 0, 8 Rabbit 8, 0 7, 7 The first payoff is that of player 1 and the second that of player 2. a. ) Draw the extensive form of the simultaneous game. Find all the Nash equilibrium. p. Suppose player 1 moves first or we are in a sequential game now. Draw the extensive form in the sequential version. c. What is the subgame perfect Nash Equilibrium (SPNE) in the sequential version? d. ) Explain why it is an SPNE.
Person A English Chinese English 10,10 5,0 Person B Chinese 0,5 5,5 A) Find the Nash equilibria of this two-person game in mixed strategies. B) Is there an equilibrium in which everybody prefers speaking English? If yes, is it stable? C) Is there an equilibrium in where everybody prefers speaking Chinese? If yes, is it stable? D) Is there an equilibrium where some people prefer speaking English, and some prefer speaking Chinese? What is the equilibrium value of p, and is this equilibrium stable?
Game Theory and Strategic Choices – End of Chapter Problem The Old Familiar and The Beehive are the only two bistros in town. Each is trying to decide whether or not it should advertise in the local newspaper. The accompanying payoff table gives their weekly profits under each possible outcome. The Beehive The Beehive does advertise does not advertise The Old Familiar The Old Familiar earns $X in profits. earns $3,500 in profits. The Old Familiar does advertise The Beehive The Beehive earns $Y in profits. earns $2,250 in profits. The Old Familiar The Old Familiar earns $2,000 in profits. earns $2,500 in profits. The Old Familiar does not advertise The Beehive The Beehive earns $4,000 in profits. earns $3,500 in profits. a. Which combination or combinations of X and Y would make a situation in which The Old Familiar does not advertise and The Beehive advertises a Nash equilibrium?