Use Brouwer's Fixed Point Theorem to Prove existence of equilibrium(a) with completely mixed strategies 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 
&amp;LU&amp;LD\\\hline
RU&amp;0,0&amp;6,3\\\hline
RD&amp;3,6&amp;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?
Rather than deprive you of the joy of solving your own problem, I will show how one can also use Brouwer's theorem to show that the function $f:\mathbb{R} \to \mathbb{R}$
$$
f(x) = \frac{33+x}{16+x^4}
$$
has a fixed point, even though its range is not compact.

Note that $f(1) = 2$ and $f(2) = 35/32$. As $f$ is monotonically decreasing over $[1,2]$, this also means that $f([1,2]) \subset [1,2]$. Also, $[1,2]$ is compact and $f$ is continuous over this set. So $f: [1,2] \to [1,2]$ will have a fixed point by Brouwer's fixed point theorem.

Similar tricks are used in game theory and general equilibrium theory. By showing that the optimal decisions by some players cannot possibly be outside of some boundaries one can frequently create a compact space over which fixed point theorems are applicable.