3. Consider the following two-player game:HLDT2,3 0,24,0 1,1(a) What are (pure- and mixed-strategy) Nash equilibria of this game?(b) Suppose the game is repeated twice, and each player's payoff is the sum of thepayoffs they obtain in the two periods. What are the subgame perfect equilibriaof the game?(c) Suppose the game is repeated indefinitely, and each player discounts his/her payoffwith a discount factor 8 € (0, 1). Find a subgame perfect equilibrium in which(H, T) is played in every period on the equilibrium path. Compute the discountfactor & needed for this equilibrium.(d) Suppose the game is repeated indefinitely, and each player discounts his/her payoffwith a discount factor & € (0, 1). Find a subgame perfect equilibrium in which(H,T) and (L,T) are played alternately on the equilibrium path. Compute thediscount factor & needed for this equilibrium.(e) Suppose the game is repeated indefinitely. Player 1 (the row player) discountsher payoff with a discount factor 8₁ € (0,1). Player 2 discounts his payoff witha discount factor d₂ = 0. That is, in any given period, player 2 only cares abouthis payoff in that period. Can you find a subgame perfect equilibrium in which(H,T) is played in every period on the equilibrium path. If no, explain. If yes,compute the discount factor 8₁ needed for this equilibrium.(f) Suppose the game is repeated indefinitely. Player 1 (the row player) discountsher payoff with a discount factor 8₁ € (0,1). Player 2 discounts his payoff witha discount factor d2 0. Can you find a subgame perfect equilibrium in which(H,T) and (L,T) are played alternately on the equilibrium path? If no, explain.If yes, compute the discount factor 81 needed for this equilibrium.(g) Suppose the game is repeated indefinitely. Player 1 (the row player) discountsher payoff with a discount factor d₁ € (0,1). Player 2 discounts his payoff witha discount factor 82 = 0. Can you find a subgame perfect equilibrium in which(H,T) and (L, D) are played alternately on the equilibrium path? If no, explain.If yes, compute the discount factor 8₁ needed for this equilibrium.2(h) Bonus problem: Suppose the game is repeated indefinitely. Player 1 (the rowplayer) discounts her payoff with a discount factor d₁ € (0, 1). Player 2 discountshis payoff with a discount factor d₂ = 0. In addition, player 2 is forgetful: in anygiven period, he can only remember player 1's action in the previous period (heforgets everything else, including his own actions in the past). Player 2's forget-fulness is commonly known. Player 1 never forgets. Can you find an equilibriumin which (L, D) is not played every period on the equilibrium path? Explain.

Given pay-off Player 2 T D Player 1 H 2,3 0,2 L 4,0 1,1

- Consider the following two-player game: H L T D 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game?

- Consider the following two-player game: H L T D 2,3 0,2 4,0 1,1 (a) What are (pure- and mixed-strategy) Nash equilibria of this game? (b) Suppose the game is repeated twice, and each player's payoff is the sum of the payoffs they obtain in the two periods. What are the subgame perfect equilibria of the game?

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?
1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).
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.
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
4. Suppose that Anne and Bob must simultaneously name a number in the set {1,2, ..., 10}. If they name the same number, the each get a payoff of 1; if they name different num- bers, they each get a payoff of 0. (a) Find all (pure and mixed strategy) Nash equilibria of this game. (b) How many are there? Explain why. (Hint: there are a lot of them!)
3. 1 2 J K 2 1 2 0 0 K 3 3 E/ 1.1 2.2 B G 2 -2 a.) How many Subgames does this Game have? b.) Convert this Game Tree into matrix form. c.) Find all Pure Strategy Nash Equilibria of this Game. d.) Find all Subgame Perfect Nash Equilibria of this Game. (Pure and Mixed) 2 2.2
(b) ROWENA up Answer 1: up Answer 2: In this simultaneous move game, the iterated dominance equilibrium is (Rowena's . Colin's left left Answer 3: ). This game has two Nash equilibria one of which is the iterated dominance equilibrium. The other Nash equilibrium, which is not the IDE, is (Rowena's down Colin's right down Up Down Answer 4: Left right 0,0 0,0 COLIN Right 0,0 1,1
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):
Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?
3. Player 1 and Player 2 are going to play the following stage game twice: Player 1 Top Bottom Left 4,3 0,0 Player 2 Middle 0,0 2,1 Right 1,4 0,0 There is no discounting in this problem and so a player's payoff in this repeated game is the sum of her payoffs in the two plays of the stage game. (a) Find the Nash equilibria of the stage game. Is (Top, Left) a Nash of the stage game? (b) Find a subgame perfect Nash equilibrium of the repeated game where the first time they play the stage game Player 1 chooses Top and Player 2 chooses Left.
Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 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.
1. Consider the following 2 person game H M L H 3,0 2,0 0,2 0,3 3,0 2,0 L 1,0 1,0 0,1 (a) Are there any pure strategy nash equilibria? If not, find the mixed strategy nash equilibrium. (b) In light of your answer above, are there any strategies which are not rationalizable?