Show that the set of $(m \times n)$-matrix games with unique optimal strategies is dense and open.
Let $\mathbb{R}^{nm}$ be a $nm$-vector space of all matrix games and let $M \subset \mathbb{R}^{nm}$ be a set of all matrix games with unique optimal strategies. Also let $A \in M$ be a matrix iff $$\dim{X}=\dim{Y}=0, $$where $X$ is a set of optimal strategies of I and $Y$ -- of II.
Let's start with proof that $M$ is an open set. If the value of a function (or a sum) is non-zero at some point, other сlose enough points to this point are also non-zero. Its true for any continuous function. I think from prove, that $v$ is continuous it follows that we can prove that $M$ is an open set. Am I?
Small reminder. Stratigy $x$ satisfies the condition $$\sum _{i=1}^{m } x_{i}a_{ij} \ge v,$$ where $a_{ij}$ is expected payoff I and $v$ is the value of the game, is called optimal in the sense that there is no strategy which will give him a higher expectation than $v$ against every strategy of II.
In its turn, the common value of maximin of I and minimax of II is called the value of the game. That is, $$v = \max_{x \in X} \min_{y \in Y} K(x,y) = \min_{y \in Y} \max_{x \in X} K(x,y). $$