1. In a congressional district somewhere in the U.S. West a new representative is being elected. The voters all have one-dimensional political views that can be neatly arrayed on a left-right spectrum. We can define the "location" of a citizen’s political views in the following way. The citizen with the most extreme left-wing views is said to be at point 0 and the citizen with the most extreme right-wing views is said to be at point 1. If a citizen has views that are to the right of the views of the fraction x of the state’s population, that citizen’s views are said to be located at point x. Candidates for office are forced to publicly state their own political position simultaneously on the zero-one left-right scale. Voters always vote for the candidate whose stated position is nearest to their own views.

There are two candidates for the congressional seat. Suppose each candidate cares only about getting as many votes as possible. In equilibrium, what will be the two candidates’ positions?

2. In the congressional district described by the previous problem, let us investigate what will happen if the two candidates do not care about the number of votes that they get but only about the amount of campaign contributions that they receive. Therefore each candidate chooses his ideological location in such a way as to maximize the amount of campaign contributions he receives, given the position of the other.

Let us define a left-wing extremist as a voter whose political views lie to the left of the left most candidate, a right-wing extremist as a voter whose political views lie to the right of the rightmost candidate, and a moderate voter as one whose political views lie between the positions of the two candidates. Assume that each extremist contributes to the candidate whose position is closest to his or her own views and that moderate voters make no campaign contributions. The number of dollars that an extremist voter contributes to his or her favorite candidate is proportional to the distance between the two candidates. Specifically, we assume that there is some constant C such that if the left-wing candidate is located at x and the right-wing candidate is located at y, then total campaign contributions received by the left-wing candidate will be $Cx(y − x) and total campaign contributions received by the right-wing candidate will be $C(1 − y)(y − x).”

In equilibrium, what are the two candidates’ positions?