5. Consider the game below. a. Is this game a version of the Prisoner's Dilemma? Why or why not? b. Consider the infinitely repeated (supergame) version of this game. i. Under what conditions will the trigger strategy = {start with X, punish any strategy other than X with Z for all periods thereafter} be a Nash equilibrium of the supergame? ii. Under what conditions will the trigger strategy = {start with Y, punish any strategy other than Y with Z for all periods thereafter} be a Nash equilibrium of the supergame? iii. Explain the difference to your answers to (i) and (ii).

5. Consider the game below. a. Is this game a version of the Prisoner's Dilemma? Why or why not? b. Consider the infinitely repeated (supergame) version of this game. i. Under what conditions will the trigger strategy = {start with X, punish any strategy other than X with Z for all periods thereafter} be a Nash equilibrium of the supergame? ii. Under what conditions will the trigger strategy = {start with Y, punish any strategy other than Y with Z for all periods thereafter} be a Nash equilibrium of the supergame? iii. Explain the difference to your answers to (i) and (ii).

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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1. Consider the following N-player game. Each agent has a choice of strategy A or B. The state variable is x, the proportion of agents choosing strategy A. For every agent, the utility of choosing strategy A is U(A) = 10 +2x and the utility of choosing strategy B is U(B) = 10+3x. What is the Nash equilibrium to this game? Explain.
9. Little Kona is a small coffee company that is consider- ing entering a market dominated by Big Brew. Each company's profit depends on whether Little Kona enters and whether Big Brew sets a high price or a low price: Little Kona Enter Don't Enter High Price Kona makes $2 million Kona makes zero Big Brew Brew makes $3 million Brew makes $7 million Kona loses $1 million Low Price Kona makes zero Brew makes $1 million Brew makes $2 million
5. Aaron and Betty play the following one-shot game. Aaron Up Middle Down Left 10,0 5,10 1,1 Betty Right 0,5 1,0 2,2 a. Does the game above have any (pure strategy) Nash equilibria? Find any equilibria or explain why an equilibrium doesn't exist. b. Suppose instead that Aaron moves first and Betty moves second. Represent this situation with a game tree and find the subgame perfect Nash equilibrium. Are Aaron and Betty better off playing a simultaneous game or a sequential game in which Aaron moves first? Explain. c.
3. For the friend-foe game, recall that there were 3 Nash equilibria possible, but the equilibria set didn't include the cooperative outcome, for which both players would win. Friend Foe Friend | 500,500 0,1000 Foe 1000,0 0,0 a) If the game is played repeatedly, propose a play strategy that will enforce cooperation. For what values of ô (discount factor) the equilibrium will be (Friend, Friend)?
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.
4. The game to the right is an example of Chicken where T>R>P>S. Consider the following tit-for-tat strategy \Betsy\Susan→ Acquiesce (A) Insist (I) A I R, R Т, Р Р, Т S, S start with A; [play whatever your opponent did in the previous period TFT = Consider a repeated version of this game, where the ending period is finite, but unknown. Under what condition is (TFT, TFT) a Nash equilibrium? b. Is (TFT, TFT) a subgame perfect Nash equilibrium (SPNE)? Note: no illustrations are necessary to justify your answer but you do need to consider potentialities where TFT is paired with some strategy other than TFT. а.
Look at the attached table. A. What are the Nash equilibrium to this game (if there are any)? B. Are any of the Nash equilibrium efficient? If yes, which one(s)? If any are inefficient, explain why.
5. The following problem was first considered by John von Neumann and is a fundamentalresult game theory.A and B play the following game:A writes down either number 1 or number 2, and B must guess which one.If the number that A has written down is i and B has guessed correctly, B receives i units from A.If B makes a wrong guess, B pays 4/5 of a unit to A.First we consider the expected gain of player B.Suppose B guesses 1 with probability p and 2 with probability 1 −p.Let X1 denote B’s gain (or loss) in a game where A has written down 1.Let X2 denote B’s gain (or loss) in a game where A has written down 2.(a) Find the pmf of X1 and X2(b) Find B’s expected gain for these two cases, E[X1] and E[X2].(c) What value of p maximizes the minimum possible value of B’s expected gain?Now consider the expected loss of player ASuppose that A writes down 1 with probability q and 2 with probability 1 −q.Let Y1 be A’s loss (or gain) if B chooses number 1.Let Y2 be A’s loss (or gain) if B…
5. Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game has the following payoff matrix, where the first payoff in each entry is for A and the second for B.(8 points) B Right Left 3,3 5,1 Down 2,2 4,4 a. Find the Nash equilibrium or equilibria. b. Which player, if any, has a dominant strategy? A Up
5. The Stag Hunt is a famous coordination game where two hunters must work together to hunt a stag but each one has the option to go solo and hunt a hare. The following payoff matrix depicts such a game: a. b. Hunter A Stag Hare Stag 5,5 4,0 Hunter B Hare 0,4 4,4 List any pure strategy Nash equilibria if they exist. Does the game have a mixed strategy Nash equilibrium? If so, solve for it.
The following payoff matrix represents a simultaneous-move game between two players: Kay and Jack. Each player has two choices: Black or White. The first number in each cell is the payoff to Kay, and the second number is the payoff to Jack. Jay Black White 50, 30, Black 50 30 Кay 45, 40, White 30 50 Refer to the scenario above. Which is true? a. This game has a dominant strategy equilibrium. b. This game has two dominant strategy equilibria. c. This game has two Nash equilibria. d. This game has one Nash equilibrium.
4. Consider the following game. There are four distinct glasses of wine on a table: three are red wine, and one is white. Two players must pick a glass of wine at the same time. If they choose the exact same glass of wine, then they each earn $1,000. Otherwise, they earn nothing. Draw the payoff matrix. What is this game's Nash equilibrium (or equilibria)? What is this game's focal point? Explain. (20%) 5. Rational ignorance poses a serious problem for democracy, but can be partially resolved by implementing Skin in the Game policies. For example, consider the following law: politicians who vote for war must enlist in the military, or have a family member enlist. How does this law (a) fall under Skin in the Game; and (b) Resolve the problem of Rational Ignorance when it comes to foreign policy? (20%)