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}}
.