4. Consider the two-player game with the following matrix form representation A B A 0,0 0,0 B 0,0 0,0 where player 1 is the "row player," player 2 is the "column player," and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile and the right-most number is the utility that player 2 obtains from the corresponding (pure) strategy profile. At the time they choose their strategies, the players are uncertain about and put probability ½ or 0 = 6 and probability on = -8. (a) Find the Nash equilibria of this game. Now suppose that player 1 can acquire information about the value of 0 before choosing between A and B. In particular, player 1 can purchase an information structure at cost 1 that, conditional on 0 = 6, results in signal h with probability and signal with probability, and, conditional on 0 = −8, results in signal h with probability and signal / with probability . Take the (pure) strategy set of player 1 to be {(n, A), (n, B), (i, A, A), (i, A, B), (i, B, A), (i, B, B)}, where, for all y = {A, B}, (n,y) denotes player 1 not purchasing the information structure but choosing Y, and, for all Yh, Y₁ = {A, B}, (i, h, ) denotes player 1 purchasing the information structure and choosing y should signal h occur and should signal l occur. The utilities of player 2 are as in the game above. If player 1 does not purchase an information structure, the player 1 utilities are as in the game above. If player 1 does purchase an information structure, the player 1 utilities equal their values in the game above minus 1. (b) Find the Nash equilibria of this game.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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Can you show me the answers for parts a and b? Thanks.

Transcribed Image Text:4. Consider the two-player game with the following matrix form representation A B A 0,0 0,0 B 0,0 0,0 where player 1 is the "row player," player 2 is the "column player," and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile and the right-most number is the utility that player 2 obtains from the corresponding (pure) strategy profile. At the time they choose their strategies, the players are uncertain about and put probability ½ or 0 = 6 and probability on = -8. (a) Find the Nash equilibria of this game. Now suppose that player 1 can acquire information about the value of 0 before choosing between A and B. In particular, player 1 can purchase an information structure at cost 1 that, conditional on 0 = 6, results in signal h with probability and signal with probability, and, conditional on 0 = −8, results in signal h with probability and signal / with probability . Take the (pure) strategy set of player 1 to be {(n, A), (n, B), (i, A, A), (i, A, B), (i, B, A), (i, B, B)}, where, for all y = {A, B}, (n,y) denotes player 1 not purchasing the information structure but choosing Y, and, for all Yh, Y₁ = {A, B}, (i, h, ) denotes player 1 purchasing the information structure and choosing y should signal h occur and should signal l occur. The utilities of player 2 are as in the game above. If player 1 does not purchase an information structure, the player 1 utilities are as in the game above. If player 1 does purchase an information structure, the player 1 utilities equal their values in the game above minus 1. (b) Find the Nash equilibria of this game.

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