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Page 19
There is a continuum of citizens, each of whom has a favorite position; the distribution of favorite positions is
given by a density function
f
on [0,1] with
f
(
x
) > 0 for all
. A candidate attracts the votes of those citizens
whose favorite positions are closer to his position than to the position of any other candidate; if
k
candidates choose
the same position then each receives the fraction 1
/k
of the votes that the position attracts. The winner of the
competition is the candidate who receives the most votes. Each person prefers to be the unique winning candidate
than to tie for first place, prefers to tie for first place than to stay out of the competition, and prefers to stay out of
the competition than to enter and lose.
• Exercise 19.1
Formulate this situation as a strategic game, find the set of Nash equilibria when
n
= 2, and show that there is no
Nash equilibrium when
n
= 3.
2.4 Existence of a Nash Equilibrium
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 Spring '10
 VINH

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