parties compete in an uniformly distributed electorate, voters choosing to minimize the distance between his ideal policy and the partys' policy.

Let and be the policies of party 1 and 2. Given party 1 obtains fraction of the votes the utilities are given by :

.

Find the nash equilibrium policies given they get the same fraction of voters if they announce the same policies.

The problem is solved here as the first problem: http://www.sites.carloalberto.org/ge...Final-2017.pdf

I have understood why the policies must be in . But I cant understand why there is no equilibrium at or at , unlike the median voter theorem.

It is stated here it is because each party could increase its fraction of the votes by moving its policy closer to its ideal point. what is the ideal point here? I dont understand this statement.

also why is the replaced by in the final utility function?