In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion.
It states:
Let
X
be a compact convex subset of a linear topological space and
Y
a convex subset of a linear topological space. If
f
is a real-valued function on
X
×
Y
with
f
(
x
,
⋅
)
upper semicontinuous and quasiconcave on
Y
,
∀
x
∈
X
, and
f
(
⋅
,
y
)
lower semicontinuous and quasi-convex on
X
,
∀
y
∈
Y
then,
min
x
∈
X
sup
y
∈
Y
f
(
x
,
y
)
=
sup
y
∈
Y
min
x
∈
X
f
(
x
,
y
)
.
