?? EXERCISE 213.1 (Electoral competition with strategic voters) Consider the variant of the game in this section in which (i) the set of possible positions is the set of
numbers x with 0 ≤ x ≤ 1, (ii) the favorite position of at least one citizen is 0
and the favorite position of at least one citizen is 1, and (iii) each citizen’s pref- erences are represented by a payoff function that assigns to each terminal history the distance from the citizen’s favorite position to the position of the candidate in the set of winners whose position is furthest from her favorite position. Under the other assumptions of the previous exercise, show that in every subgame perfect equilibrium in which no citizen’s action is weakly dominated, the position chosen by every candidate who enters is the median of the citizens’ favorite positions. To do so, first show that in any equilibrium each candidate that enters is in the set of winners. Then show that in any Nash equilibrium of any voting subgame in which there are more than two candidates and not all candidates’ positions are the same, some candidate loses. (Argue that if all candidates tie for first place, some citizen can increase her payoff by changing her vote.) Finally, show that in any subgame perfect equilibrium in which either only two candidates enter, or all can- didates who enter choose the same position, every entering candidates chooses the median of the citizens’ favorite positions.