PROBLEM
Suppose two players play a coin game. They both have a coin and can choose themselves what the probability is that they will play head $H$. So for player $i$, the chance he gets head is $p_i$, $i = {1,2}$. If both coins turn out to be the same, player $1$ will win, if they differ, player $2$ will win.
Show that $(p_1,p_2)$ = $(\frac{1}{2},\frac{1}{2})$ is the unique Nash Equilibrium of this game.
This is just the question for the normal form game
\begin{matrix} & & Player 2 & \\ & & H & T \\ Player 1&H& (1,-1) & (-1,1) \\ &T& (-1,1) & (1,-1) \end{matrix} (You could take another payoff-distribution as well)
called "matching pennies" and you're asked to find the mixed-strategy equilibrium, which is unique in this case and it is exactly $$(p_1,p_2)= (0.5,0.5)$$ as you can read on Wikipedia. There is no Nash equilibrium in pure strategies. If you have trouble understanding how to find the equilibrium, leave a comment with a question ;-)