Let's try a variation on the Hotelling model to try to make it more realistic in one particular way. As before, voters are spaced on a line from 0 to 1. We will assume that voters are unwilling to vote for someone who doesn't share their viewpoint. They pay a psychic cost if they vote for someone who is further away on the spectrum. Let's say the cost is t per unit of distance to the candidate for whom they vote. In other words, if they are distance x away from the closest candidate then their psychic cost is tx. A voter gets a payoff of 1 from the act of voting, so a voter's total payoff from voting for someone x units away is 1 tx. If they stay home and don't vote, they will get payoff of 0. Candidates' payoffs are equal to the fraction of voters who vote for them.

(a) In this game, find the minimum value of t for which there exists a Nash equilibrium in which candidates place themselves at 1/4 and 3/4.

(b) Find a Nash equilibrium with 3 candidates. Where do the candidates locate, what is their vote share, and what value(s) of t support the equilibrium?