3. Examine the following version of the Prisoner's Dilemma: Prisoner 1 Betray Silent Prisoner 2 Betray 0,0 -1,3 Silent 3,-1 1, 1 (a) Verify that the Nash equilibrium is the usual one for the Prisoner's dilemma and that both players have dominant strategies. (b) Suppose the stage game is repeated infinitely many times. Compute the discount factor required for their suspects to be able to cooperate on silent each period. Outline the trigger strategies you are considering for them.

3. Examine the following version of the Prisoner's Dilemma: Prisoner 1 Betray Silent Prisoner 2 Betray 0,0 -1,3 Silent 3,-1 1, 1 (a) Verify that the Nash equilibrium is the usual one for the Prisoner's dilemma and that both players have dominant strategies. (b) Suppose the stage game is repeated infinitely many times. Compute the discount factor required for their suspects to be able to cooperate on silent each period. Outline the trigger strategies you are considering for them.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.6P

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Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.
Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.
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 répeatedly. propose a play strategy that will enforce cooperation. For what valucs of o (discount factor) the equilibrium will be (Friend. Friend)?
Consider the following two-player game with three options for each player. (Payouts are listed for the row player first, then the column player.) player Y layery 3,3 A 1,5 4,4 6,2 K 8,1 3,7 5,2 0,6 1,1 Find a mixed Nash equilibrium for this game. Solution suggestion: Use two variables per player. If p and are the probabilities of selecting the first two strategies, then 1-p-q is the probability of selecting the third strategy. You will need to solve a system of equations.
(a) Design a two-player game with the property that "the payoff for Player1 is Playerl plays a strategy Sl and Player2 plays a strategy S2 is the same as the payoff for Player2 if Player2 plays Sl and Playerl plays S2". Express your game in terms of payoff matrices. Show the above property holds in your given payoff matrices. Give a case to show that there is one Nash equilibrium and such an equilibrium is also a Pareto efficient (justify your answers) in your given payoff matrices. Prove or disprove the above case that is unique. Give another example to show that there are two Nash equilibria in your given payoff matrices.
• (1,0) (1,1) Player 2 G D H (2,0) Action Player 1 Player 1 Action B Player 1 (3,-1) E Player 2 (0,1) (-1,-1) (a) Find all the Subgame Perfect Nash equilibria in this game. (b) Find all the Nash equilibria in this game. (Hint: write the game in strategic form.)
Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.
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.
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.
(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
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.
In equilibrium, what is the probability that player 1 will use the pure strategy E in this game?