The game-theoretic value and the spectral radius of a nonnegative matrix

We relate some minimax functions of matrices to some spectral functions of matrices. If $A$ is a nonnegative $n \times n$ matrix, $\upsilon (A)$ is the game-theoretic value of $A$, and $\rho (A)$ is the spectral radius of $A$, then $\upsilon (A) \leq \rho (A)$. Necessary and sufficient conditions for $\upsilon (A) = \rho (A)$ are given. It follows that if $A$ is nonnegative and irreducible and $n > 1$, then $\upsilon (A) < \rho (A)$. Also, if, for a real matrix $A$ and a positive matrix $B$, $\upsilon (A,B) = {\sup _X}{\inf _Y}{X^T}AY/{X^T}BY$ over probability vectors $X$ and $Y$, then for nonnegative, nonsingular $A$ and positive $B$, $\rho (AB) = {[\upsilon ({A^{ - 1}},B)]^{ - 1}}$.