Question 12.55
27. Define 's position in a two-candidate race to be opposition-optimal if, given that the position of is fixed, it maximizes 's vote total. Show that 's opposition-optimal position must be adjacent to 's position and closer to , except when is at the median. (Roughly speaking, being “adjacent” means being a very small distance away.)
27.
If ’s position is not the median, then can do no better than to just settle next to on the side of the median. ’s position should be close enough to ’s position that ’s position is closer to the nearest voter to ’s position that is in between ’s position and the median. This way, maximizes her vote, getting all voters on the side with the median to vote for her. If ’s policy position is at the median, then can do no better than to also select the median (by the median- voter theorem).