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