mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, and in particular

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, Sion's minimax theorem is a generalization of

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

minimax theorem In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem from 1928, which was c ...

, named after

Maurice Sion Maurice Sion (17 October 1927, Skopje – 17 April 2018, Vancouver) was an American and Canadian mathematician, specializing in measure theory and game theory. He is known for Sion's minimax theorem. Biography Sion received from New York Univers ...

. It states: Let

X

be a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

subset of a

linear topological space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

and

Y

a convex subset of a linear topological space. If

f

is a real-valued

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

X\times Y

with :

f(x,\cdot)

upper semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...

and quasi-concave on

Y

\forall x\in X

, and :

f(\cdot,y)

lower semicontinuous and quasi-convex on

X

\forall y\in Y

then, :

\min_\sup_ f(x,y)=\sup_\min_f(x,y).

References

* * Game theory Mathematical optimization Mathematical theorems {{gametheory-stub

See also

References