Consider a Downsian electoral competition over the line [−1, 1] where there are 11 voters. Each voter has a single-peaked preferences over the policies. Voter 1’s peak is at -1, voter 2’s at -0.8, ..., voter 6’s at 0, voter 7’s at 0.2, ..., and voter 11’s at 1. Suppose there are two candidates, {D, R} and if they get equal votes the tie is broken by a coin toss—that is, both wins the election with probability 1/2. (a) What is the Nash Equilibrium of this game? (b) Now suppose that there is an ideological candidate C that positions himself at -1 whatever the other candidates do. Is there a Nash Equilibrium of the game? If so, what is it? (c) Suppose now that there is another ideological candidate at 1. Is there an equilibrium and what is it?
Consider a Downsian electoral competition over the line [−1, 1] where there are 11 voters. Each voter has a single-peaked preferences over the policies. Voter 1’s peak is at -1, voter 2’s at -0.8, ..., voter 6’s at 0, voter 7’s at 0.2, ..., and voter 11’s at 1. Suppose there are two candidates, {D, R} and if they get equal votes the tie is broken by a coin toss—that is, both wins the election with probability 1/2.
(a) What is the Nash Equilibrium of this game?
(b) Now suppose that there is an ideological candidate C that positions himself at -1 whatever the other candidates do. Is there a Nash Equilibrium of the game? If so, what is it?
(c) Suppose now that there is another ideological candidate at 1. Is there an equilibrium and what is it?
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