$\displaystyle 2$ parties compete in an uniformly distributed electorate, voters choosing to minimize the distance between his ideal policy and the partys' policy.

Let $\displaystyle x_{1}$ and $\displaystyle x_{2}$ be the policies of party 1 and 2. Given party 1 obtains $\displaystyle y$ fraction of the votes the utilities are given by :

$\displaystyle u_{1}=y-(x_{1})^2$

$\displaystyle u_{2}=1-y-(1-{x_{2})^2$.

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 $\displaystyle [0,1]$. But I cant understand why there is no equilibrium at $\displaystyle x_{1}>x_{2}$ or at $\displaystyle x_{1}=x_{2}$, 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 $\displaystyle y$ replaced by $\displaystyle \frac{x_{1}+x_{2}}{2}$ in the final utility function?