EXERCISE 74.1 (Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Show that if less than one-third of the citizens' favorite positions are equal to the median favorite position (m), then the game has no Nash equilibrium. Argue as follows. First, show that the game has no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which
EXERCISE 74.1 (Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Show that if less than one-third of the citizens' favorite positions are equal to the median favorite position (m), then the game has no Nash equilibrium. Argue as follows. First, show that the game has no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.5P
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Transcribed Image Text:EXERCISE 74.1 (Electoral competition with three candidates) Consider a variant
of Hotelling's model in which there are three candidates and each candidate has
the option of staying out of the race, which she regards as better than losing and
worse than tying for first place. Show that if less than one-third of the citizens'
favorite positions are equal to the median favorite position (m), then the game has
no Nash equilibrium. Argue as follows. First, show that the game has no Nash
equilibrium in which a single candidate enters the race. Second, show that in any
Nash equilibrium in which more than one candidate enters, all candidates that
enter tie for first place. Third, show that there is no Nash equilibrium in which
two candidates enter the race. Fourth, show that there is no Nash equilibrium in
which all three candidates enter the race and choose the same position. Finally,
show that there is no Nash equilibrium in which all three candidates enter the race
and do not all choose the same position.
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