Minimax theorem

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 considered the starting point of game theory. Since then, several alternative generalizations of von Neumann's original theorem have appeared in the literature.^[1]

Zero-sum Games

In game theory, and more precisely the theory of zero-sum games, the minimax theorem shows that the minimax solution of a zero-sum game is the same as the Nash equilibrium. The theorem was first proven and 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]

Formally, von Neumann's minimax theorem states:

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

${\displaystyle f(\cdot ,y):X\rightarrow \mathbb {R} }$ is convex for fixed $y$, and

${\displaystyle f(x,\cdot ):Y\rightarrow \mathbb {R} }$ is concave for fixed $x$.

Then we have that

${\displaystyle \min _{x\in X}\max _{y\in Y}f(x,y)=\max _{y\in Y}\min _{x\in X}f(x,y).}$

References

1. Du, Ding-Zhu; Pardalos, Panos M., eds. (1995). Minimax and Applications. Boston, MA: Springer US. ISBN 9781461335573.
2. Von Neumann, J: Zur Theorie der Gesellschaftsspiele Math. Annalen. 100 (1928) 295–320
3. 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.

See also

• Sion's minimax theorem
• Parthasarathy's theorem