A bag of n players chooses one number between 0 and 10. The winner is the one who chooses the second smallest number. For example, if the selected numbers are [2, 2, 3, 5, 9, 10], the winner will be the one who chose 3. In the event that multiple people choose the second smallest number, the winner is randomly selected from among them. For example, if the selected numbers are [2, 2, 3, 3, 3, 3, 5, 9, 10], each player who chose 3 will win with a probability of 1/4. Even if everyone chooses the same number (meaning there is no second smallest number), the winner is randomly selected from among them. Each player wants to choose the number that will give them the greatest chance of winning. Find the Nash equilibrium.

A bag of n players chooses one number between 0 and 10. The winner is the one who chooses the second smallest number. For example, if the selected numbers are [2, 2, 3, 5, 9, 10], the winner will be the one who chose 3. In the event that multiple people choose the second smallest number, the winner is randomly selected from among them. For example, if the selected numbers are [2, 2, 3, 3, 3, 3, 5, 9, 10], each player who chose 3 will win with a probability of 1/4. Even if everyone chooses the same number (meaning there is no second smallest number), the winner is randomly selected from among them. Each player wants to choose the number that will give them the greatest chance of winning. Find the Nash equilibrium.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.3P

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A bag of n players chooses one number between 0 and 10. The winner is the one who chooses the second smallest number. For example, if the selected numbers are [2, 2, 3, 5, 9, 10], the winner will be the one who chose 3. In the event that multiple people choose the second smallest number, the winner is randomly selected from among them. For example, if the selected numbers are [2,2, 2, 3, 3, 3, 3, 5, 9, 10], each player who chose 3 will win with a probability of 1/4. Even if everyone chooses the same number (meaning there is no second smallest number), the winner is randomly selected from among them. Each player wants to choose the number that will give them the greatest chance of winning. Find the Nash equilibrium for this problem.
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