Minimax theorem

{{short description|Gives conditions that guarantee the max–min inequality holds with equality}}

In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that

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

under certain conditions on the sets $X$ and $Y$ and on the function $f$ .{{Citation |last=Simons |first=Stephen |title=Minimax Theorems and Their Proofs |date=1995 |work=Minimax and Applications |series=Nonconvex Optimization and Its Applications |volume=4 |pages=1–23 |editor-last=Du |editor-first=Ding-Zhu |url=https://link.springer.com/chapter/10.1007/978-1-4613-3557-3_1 |access-date=2024-10-27 |place=Boston, MA |publisher=Springer US |language=en |doi=10.1007/978-1-4613-3557-3_1 |isbn=978-1-4613-3557-3 |editor2-last=Pardalos |editor2-first=Panos M.}} It is always true that the left-hand side is at most the right-hand side (max–min inequality) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928,{{cite journal |last=Von Neumann |first=J. |year=1928 |title=Zur Theorie der Gesellschaftsspiele |journal=Math. Ann. |volume=100 |pages=295–320 |doi=10.1007/BF01448847|s2cid=122961988 }} which is considered the starting point of game theory. Von Neumann 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".{{cite book |author=John L Casti |url=https://archive.org/details/fivegoldenrulesg00cast/page/19 |title=Five golden rules: great theories of 20th-century mathematics – and why they matter |publisher=Wiley-Interscience |year=1996 |isbn=978-0-471-00261-1 |location=New York |page=[https://archive.org/details/fivegoldenrulesg00cast/page/19 19] |url-access=registration}} Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.{{cite book |title=Minimax and Applications |date=1995 |publisher=Springer US |isbn=9781461335573 |editor1-last=Du |editor1-first=Ding-Zhu |location=Boston, MA |editor2-last=Pardalos |editor2-first=Panos M.}}{{Cite journal |last1=Brandt |first1=Felix |last2=Brill |first2=Markus |last3=Suksompong |first3=Warut |year=2016 |title=An ordinal minimax theorem |journal=Games and Economic Behavior |volume=95 |pages=107–112 |arxiv=1412.4198 |doi=10.1016/j.geb.2015.12.010 |s2cid=360407}}

Bilinear functions and zero-sum games

Von Neumann's original theorem was motivated by game theory and applies to the case where

$X$ and $Y$ are standard simplexes: $X = \{ (x_1, \dots, x_n) \in [0,1]^n : \sum_{i = 1}^n x_i = 1 \}$

and

Y = \{ (y_1, \dots, y_m) \in [0,1]^m : \sum_{j = 1}^m y_j = 1 \}

, and

$f(x,y)$ is a linear function in both of its arguments (that is, $f$ is bilinear) and therefore can be written $f(x,y) = x^\mathsf{T} A y$ for a finite matrix $A \in \mathbb{R}^{n \times m}$ , or equivalently as $f(x,y) = \sum_{i=1}^n\sum_{j=1}^m A_{ij}x_iy_j$ .

Under these assumptions, von Neumann proved that

: $\max_{x \in X} \min_{y \in Y} x^\mathsf{T} A y = \min_{y \in Y}\max_{x \in X} x^\mathsf{T} A y.$

In the context of two-player zero-sum games, the sets $X$ and $Y$ correspond to the strategy sets of the first and second player, respectively, which consist of lotteries over their actions (so-called mixed strategies), and their payoffs are defined by the payoff matrix $A$ . The function $f(x,y)$ encodes the expected value of the payoff to the first player when the first player plays the strategy $x$ and the second player plays the strategy $y$ .

Concave-convex functions

File:Saddle point.svg

Von Neumann's minimax theorem can be generalized to domains that are compact and convex, and to functions that are concave in their first argument and convex in their second argument (known as concave-convex functions). Formally, let $X \subseteq \mathbb{R}^n$ and $Y \subseteq \mathbb{R}^m$ be compact convex sets. If $f: X \times Y \rightarrow \mathbb{R}$ is a continuous function that is concave-convex, i.e.

: $f(\cdot,y): X \to \mathbb{R}$ is concave for every fixed $y \in Y$ , and

: $f(x,\cdot): Y \to \mathbb{R}$ is convex for every fixed $x \in X$ .

Then we have that

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

Sion's minimax theorem

Sion's minimax theorem is a generalization of von Neumann's minimax theorem due to Maurice Sion,{{cite journal |last=Sion |first=Maurice |year=1958 |title=On general minimax theorems |journal=Pacific Journal of Mathematics |volume=8 |issue=1 |pages=171–176 |doi=10.2140/pjm.1958.8.171 |mr=0097026 |zbl=0081.11502 |doi-access=free}} relaxing the requirement that X and Y be standard simplexes and that f be bilinear. It states:{{cite journal |last=Komiya |first=Hidetoshi |year=1988 |title=Elementary proof for Sion's minimax theorem |url=http://projecteuclid.org/euclid.kmj/1138038812 |journal=Kodai Mathematical Journal |volume=11 |issue=1 |pages=5–7 |doi=10.2996/kmj/1138038812 |mr=0930413 |zbl=0646.49004}}

Let $X$ be a convex subset of a linear topological space and let $Y$ be a compact convex subset of a linear topological space. If $f$ is a real-valued function on $X\times Y$ with

: $f(\cdot,y)$ upper semicontinuous and quasi-concave on $X$ , for every fixed $y\in Y$ , and

: $f(x,\cdot)$ lower semicontinuous and quasi-convex on $Y$ , for every fixed $x\in X$ .

Then we have that

: $\sup_{x\in X}\min_{y\in Y} f(x,y) = \min_{y\in Y}\sup_{x\in X} f(x,y).$

References

Category:Game theory

Category:Mathematical optimization

Category:Mathematical theorems

Minimax theorem

Bilinear functions and zero-sum games

Concave-convex functions

Sion's minimax theorem

See also

References