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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