(Citizen-candidates) Consider a game in which the players are the citizens. Any citizen may, at…

(Citizen-candidates) Consider a game in which the players are the citizens. Any citizen may, at some cost c > 0, become a candidate. Assume that the only position a citizen can espouse is her favorite position, so that a citizen’s only decision is whether to stand as a candidate. After all citizens have (simultaneously) decided whether to become candidates, each citizen votes for her favorite candidate, as in Hotelling’s model. Citizens care about the position of the winning candidate; a citizen whose favorite position is x loses |x − x∗| if the winning candidate’s position is x∗. (For any number z, |z| denotes the absolute value of z: |z| = z if z > 0 and |z| = −z if z ∗| − c, where x∗ is the winner’s position; and a citizen with favorite position x who does not become a candidate obtains the payoff −|x − x∗|, where x∗ is the winner’s position. Assume that for every position x there is a citizen for whom x is the favorite position. Show that if b ≤ 2c then the game has a Nash equilibrium in which one citizen becomes a candidate. Is there an equilibrium (for any values of b and c) in which two citizens, each with favorite position m, become candidates? Is there an equilibrium in which two citizens with favorite positions different from m become candidates? Hotelling’s model assumes a basic agreement among the voters about the ordering of the positions. For example, if one voter prefers x to y to z and another voter prefers y to z to x, no voter prefers z to x to y. The next exercise asks you to study a model that does not so restrict the voters’ preferences.