Minimax theorem

A minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem on this sense is Von Neumann's minimax theorem from 1928, which was considered the starting point of game theory. Since then, several alternative generalizations of Von Neumann's original theorem have appeared in the literature.^[1]

Von Neumann's minimax theorem

Von Neumann's minimax theorem was first published in 1928 by John von Neumann,^[2] who is quoted as saying "As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved".^[3] This theorem is important in game theory as the minimax solution of a zero-sum game is the same as the Nash equilibrium.

Precisely, Von Neumann's minimax theorem states:

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact convex sets. If $f: X \times Y \rightarrow \mathbb{R}$ is a continuous function that is convex-concave, i.e.

f(\cdot,y):X\rightarrow\mathbb{R}

is convex for fixed

y

, and

f(x,\cdot):Y\rightarrow\mathbb{R}

is concave for fixed

x

then,

\min_{x\in X}\max_{y\in Y} f(x,y)=\max_{y\in Y}\min_{x\in X}f(x,y).

References

↑ Du, Ding-Zhu; Pardalos, Panos M., eds. (1995). Minimax and Applications. Boston, MA: Springer US. ISBN 9781461335573.
↑ Von Neumann, J: Zur Theorie der Gesellschaftsspiele Math. Annalen. 100 (1928) 295–320
↑ John L Casti (1996). Five golden rules: great theories of 20th-century mathematics – and why they matter. New York: Wiley-Interscience. p. 19. ISBN 0-471-00261-5.

Minimax theorem

Von Neumann's minimax theorem

References

See also