Consider a situation of electoral competition in which candidates choose a position on government spending to allocate to national defense. The alternatives are various percentage of the (fixed) budget (any point in the interval [0; 100]). Voters' preferences are single-peaked, and their ideal percentage is distributed according to the following: 60% of the voters' ideal point is 10 (that is, 60% voters think it's best to spend 10 percent of government budget to defense), 30% of the voters have ideal point of 25, and the rest 10% voters have ideal point of 50. Suppose that all voters care equally about policy differences to the left and right of their ideal point (they have symmetric single-peaked preferences). Voters are sincere and will vote for the candidate with position closest to his/her ideal point, and if two or more candidates have the same position, they split the votes equally.
(i) Suppose that there are two candidates competing for office. What position(s) will the two candidates choose?
(ii) Now suppose that there are three candidates but one of them is non-strategic and takes a fixed position located at 50. For the two strategic candidates, do the position choices in (i) still constitute a Nash equilibrium? (That is, given the position choice of one candidate, is the other candidate's position his/her optimal choice?)