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 0 and put probability / on 0 = 6 and probability ½½ on 0 = −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 1 with probability ½, and, conditional on 0 = −8, results in signal h with probability and signal / with probability 1. 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, v) denotes player 1 not purchasing the information structure but choosing Y, and, for all Yh, Yl Є {A, B}, (i, Yh, 1) denotes player 1 purchasing the information structure and choosing Yh 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, their utilities are as in the game above. If player 1 does purchase an information structure, their utilities equal their values in the game above minus 1. (b) Find the Nash equilibria of this game. E
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 0 and put probability / on 0 = 6 and probability ½½ on 0 = −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 1 with probability ½, and, conditional on 0 = −8, results in signal h with probability and signal / with probability 1. 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, v) denotes player 1 not purchasing the information structure but choosing Y, and, for all Yh, Yl Є {A, B}, (i, Yh, 1) denotes player 1 purchasing the information structure and choosing Yh 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, their utilities are as in the game above. If player 1 does purchase an information structure, their utilities equal their values in the game above minus 1. (b) Find the Nash equilibria of this game. E
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.7P
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Transcribed Image Text:4. Consider the two-player game with the following matrix form representation
A
A
B
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 0 and put probability
0 = 6 and probability / on 0 = −8.
(a) Find the Nash equilibria of this game.
=
on
-8,
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 ½ and signal / with probability ½, and, conditional on 0
results in signal h with probability and signal / with probability ¾. 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, v)
denotes player 1 not purchasing the information structure but choosing Y, and, for all Yh, Y1 Є {A, B},
(i, Yh, Y1) denotes player 1 purchasing the information structure and choosing Yh should signal h occur
and should signal / occur. The utilities of player 2 are as in the game above. If player 1 does not
purchase an information structure, their utilities are as in the game above. If player 1 does purchase
an information structure, their utilities equal their values in the game above minus 1.
(b) Find the Nash equilibria of this game.
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