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Two entrepreneurs, A and B, independently choose whether to sell their product in country X or country Y. Each entrepreneur prefers to be the only one selling in the country she chooses. The size of the market in country X is larger than in country Y, but selling in country X requires an investment of e> 0. The payoffs are given by the following matrix: B X X2-c,2-e 4,4-c 4-c,4 1,1 (a) Identify all dominated strategies for each player for each e > 0. In each case, indicate whether the domination is weak or strict. Solution: For each player, if c = 3 then Y weakly dominates X and if e> 3 then Y strictly dominates X. If e<3 there is no dominated strategy. (b) Find all Nash equilibria of this game for each e > 0, allowing for both pure and mixed strategies. Solution: If e> 3, the only NE is (Y,Y). If e<3, (X,Y) and (Y,X) are NE and so is the mized strategy profile in which each chooses X with probability. If c=3, then any strategy profile in which one player chooses Y (and the other chooses any mized strategy) is a NE.

Two entrepreneurs, A and B, independently choose whether to sell their product in country X or country Y. Each entrepreneur prefers to be the only one selling in the country she chooses. The size of the market in country X is larger than in country Y, but selling in country X requires an investment of e> 0. The payoffs are given by the following matrix: B X X2-c,2-e 4,4-c 4-c,4 1,1 (a) Identify all dominated strategies for each player for each e > 0. In each case, indicate whether the domination is weak or strict. Solution: For each player, if c = 3 then Y weakly dominates X and if e> 3 then Y strictly dominates X. If e<3 there is no dominated strategy. (b) Find all Nash equilibria of this game for each e > 0, allowing for both pure and mixed strategies. Solution: If e> 3, the only NE is (Y,Y). If e<3, (X,Y) and (Y,X) are NE and so is the mized strategy profile in which each chooses X with probability. If c=3, then any strategy profile in which one player chooses Y (and the other chooses any mized strategy) is a NE.

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.7P
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Two entrepreneurs, A and B, independently choose whether to sell their product in country X or country Y. Each entrepreneur prefers to be the only one selling in the country she chooses. The size of the market in country X is larger than in country Y, but selling in country X requires an investment of c> 0. The payoffs are given by the following matrix: A X Y B X 2-c,2-c 4,4-c Y 4-c, 4 1,1 (a) Identify all dominated strategies for each player for each e > 0. In each case, indicate whether the domination is weak or strict. Solution: For each player, if c = 3 then Y weakly dominates X and if c> 3 then Y strictly dominates X. If c<3 there is no dominated strategy. (b) Find all Nash equilibria of this game for each c> 0, allowing for both pure and mixed strategies. Solution: If c> 3, the only NE is (Y,Y). If c <3, (X,Y) and (Y, X) are NE and so is the mixed strategy profile in which each chooses X with probability 3. If c=3, then any strategy profile in which one player chooses Y (and the other chooses any mized strategy) is a NE.
Transcribed Image Text:Two entrepreneurs, A and B, independently choose whether to sell their product in country X or country Y. Each entrepreneur prefers to be the only one selling in the country she chooses. The size of the market in country X is larger than in country Y, but selling in country X requires an investment of c> 0. The payoffs are given by the following matrix: A X Y B X 2-c,2-c 4,4-c Y 4-c, 4 1,1 (a) Identify all dominated strategies for each player for each e > 0. In each case, indicate whether the domination is weak or strict. Solution: For each player, if c = 3 then Y weakly dominates X and if c> 3 then Y strictly dominates X. If c<3 there is no dominated strategy. (b) Find all Nash equilibria of this game for each c> 0, allowing for both pure and mixed strategies. Solution: If c> 3, the only NE is (Y,Y). If c <3, (X,Y) and (Y, X) are NE and so is the mixed strategy profile in which each chooses X with probability 3. If c=3, then any strategy profile in which one player chooses Y (and the other chooses any mized strategy) is a NE.
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