For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$.

Is there a function $m:\mathbb N\to\mathbb N$ such that for any finite list $A_1,A_2,\ldots,A_k$ of $n\times n$ (real valued) matrices, there exists a subset $J\subset[k]$ with cardinality $m(n)$ such that

$$\min_{x,y\in\Delta(n)}\max_{i\in[k]} xA_i y \ = \min_{x,y\in\Delta(n)}\max_{i\in J} xA_i y\quad ? $$

Moreover, can $m(n)$ be polynomial in $n$?