3. Consider a repeated version of the following stage game: Player 2 Y 10, 10 В 20, —20 A -20, 20 0,4 Player 1 6, 6 2,4 C 4,0 4,2 3,3 Specifically, the stage game is played twice. For simplicity, assume that players do not discount future payoffs, or in other words, & = 1. (a) Identify all Nash equilibria, pure or mixed, of the stage game. Hint: begin with IESDS on the stage game. Which of these equilibria gives the highest payoff to each player? Which gives the lowest? (b) Verify that there is not an SPNE in which (A, X) is played in the first stage. (Hint: the equi- libria with the highest and lowest payoffs from part (a) serve as the best possible reward and punishment.)

3. Consider a repeated version of the following stage game: Player 2 Y 10, 10 В 20, —20 A -20, 20 0,4 Player 1 6, 6 2,4 C 4,0 4,2 3,3 Specifically, the stage game is played twice. For simplicity, assume that players do not discount future payoffs, or in other words, & = 1. (a) Identify all Nash equilibria, pure or mixed, of the stage game. Hint: begin with IESDS on the stage game. Which of these equilibria gives the highest payoff to each player? Which gives the lowest? (b) Verify that there is not an SPNE in which (A, X) is played in the first stage. (Hint: the equi- libria with the highest and lowest payoffs from part (a) serve as the best possible reward and punishment.)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
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