State Nash's theorem for a non zero-sum game in the case of two players with two choices.

The role playing game Tixerb involves two players. Before the game begins, each player $i$ chooses a $p_{i}$ with $0 \leqslant p_{i} \leqslant 1$ which they announce. They may change their choice as many times as they wish, but, once the game begins, no further changes are allowed. When the game starts, player $i$ becomes a Dark Lord with probability $p_{i}$ and a harmless peasant with probability $1-p_{i}$. If one player is a Dark Lord and the other a peasant the Lord gets 2 points and the peasant $-2$. If both are peasants they get 1 point each, if both Lords they get $-U$ each. Show that there exists a $U_{0}$, to be found, such that, if $U>U_{0}$ there will be three choices of $\left(p_{1}, p_{2}\right)$ for which neither player can increase the expected value of their outcome by changing their choice unilaterally, but, if $U_{0}>U$, there will only be one. Find the appropriate $\left(p_{1}, p_{2}\right)$ in each case.