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 X n matrix, v(A) is the gametheoretic value of A, and p(A) is the spectral radius of A, then v(A) < p(A). Necessary and sufficient conditions for v(A) = p(A) are given. It follows that if A is nonnegative and irreducible and n > 1, then v(A) < p(A). Also, if, for a real matrix A and a positive matrix B, v(A, B) = supx infy XTAY/X"1 BY over probability vectors X and Y, then for nonnegative, nonsingular A and positive B, p(AB) = [»(A'1^)]-1. The purpose of this paper is to establish some connections between von Neumann's minimax functions of matrices and the spectral radius of nonnegative matrices. Other connections between game-theoretic and linear-algebraic aspects of matrices have been established by Blackwell (1961) and Raghavan (1978). Let Pn = [x e Rn | Xt > o, £xt = l} , p+ = {x e Rn \ Xt > 0, £X¡ = 1} • Where no ambiguity can arise, the subscript n will be dropped. In game theory the minimax theorem (von Neumann, 1928 [1959]) asserts that for any m x n real matrix A, (1) sup inf XTAY= inf sup XTAY, xepm Y€Pn rep,. xePm where T denotes transpose. The common value of both sides of (1) is v(A), the value of A. The spectral radius p(A) of an n x n real matrix A is the maximum of moduli of the eigenvalues of A. Following the terminology of Birkhoff and Varga (1958), who also review much of the background assumed here, the spectral prenorm R(A) of A is the maximum of the real parts of the eigenvalues of A. A matrix A is nonnegative, and we write A > 0, if every element atj of A is greater than or equal to 0. A matrix A is positive, and we write A > 0, if every element is greater than 0. A matrix A is essentially nonnegative if every element not on the main diagonal is nonnegative. An n x n matrix A is irreducible if for every i and j there exists a finite sequence of indices fc(0) = i,..., k(r) = j such that, for h= l,...,r, ak{h_l)Mh) / 0. _ Received by the editors November 30, 1983 and, in revised form, February 23, 1984. 1980 Mathematics Subject Classification. Primary 15A42, 15A48; Secondary 90D05.