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3. Consider the Hotelling-Downs model of spatial competition. Assume there are m = 6 political candidates (or, if you'd like to be more ambitious, let m be arbitrary) who are non-ideological and want to maximize their total number of votes (or points in the Borda context). They simultane- ously select platforms on the interval [0,1]. A continuum of voters' ideal platforms are uniformly distributed on the interval [0,1]. The number of voters is normalized to 1 (that means the number of voters on some line segment is equal to the length of that segment). Voters are sincere and prefer candidates that are closer to their ideal position. If indierent between several candidates, voters randomize with equal probability. Suppose the election is held under Borda. That is, a voter's favorite candidate gets 5 points, second favorite gets 4, third favorite gets 3, ..., and least favorite gets 0.

(i) Suppose all candidates locate at the median voter's platform 1/2. What is each candidate's total number of votes?

(ii) What would be the best potential deviation for some candidate at 1/2?

(iii) How many votes would the deviating candidate earn?

(iv) Thus, is the situation where all candidates locate at 1/2 a Nash equilibrium? Please answer all four