How can one use Brouwer's Fixed Point Theorem to prove that the following game F has a solution:
F is defined as N={L,R} Ai=(g,1-g) where g must be positive and smaller than 1, that is, each player plays a completely mixed strategy and has the following payoff matrix:
\begin{array}{|c|c|c|} \hline &LU&LD\\\hline RU&0,0&6,3\\\hline RD&3,6&0,0\\\hline \end{array}
I reached the conclusion that one cannot use such theorem to prove there exists a solution to the game since the plays do not belong to a compact set. Can anyone use it?