The minimax inequality was first proved by John von Neumann. It is the starting point to discuss two-players zero-sum static games theory.

${\bf Theorem ~1:}~~\underline{\rm minimax~inequality,~simple~strategies}$ For any $m\times n$ matrix $A_{i,j}$ we have

(1)$~~\displaystyle{ \max_{1\leq i\leq m} \min_{1\leq j\leq n}A_{i,j} \leq \min_{1\leq j\leq n}\max_{1\leq i\leq m}A_{i,j} }$ (2)$~~\displaystyle{ \max_{1\leq i\leq m} \min_{1\leq j\leq n}A_{i,j} = \min_{1\leq j\leq n}\max_{1\leq i\leq m}A_{i,j} }$ if and only if $A_{i,{\tilde j}}\leq A_{{\tilde i},{\tilde j}} \leq A_{{\tilde i},j}~~~~\forall i,j$ is valid for some $({\tilde i},{\tilde j})$ For a 2 players zero-sum game, the entries of $A_{i,j}$ is interpreted as the payoff when player 1 has chosen the $i^{th}$ strategy while player 2 has chosen the $j^{th}$ strategy. The value $A_{{\tilde i},{\tilde j}}$ is known as the value of the game.

${\bf Proof}$ Since $\displaystyle{ \min_{1\leq j\leq n} A_{i,j}\leq \max_{1\leq i\leq m}A_{i,j}~~~~\forall i,j }~~$ The LHS is independent of $j$ while the RHS is independent of $i$ therefore we obtain $~~\displaystyle{ \max_{1\leq i\leq m} \min_{1\leq j\leq n}A_{i,j} \leq \min_{1\leq j\leq n}\max_{1\leq i\leq m}A_{i,j} }$

${\bf Theorem ~2:}~~\underline{\rm minimax~inequality,~mixed~strategies}$ Let $\displaystyle{S_m=\{x\in {\mathbb R}^m~\vert~x_i\geq 0~\forall i~,~\sum_{i=1}^mx_i=1\}\subseteq {\mathbb R}^{m} }$ For any $m\times n$ matrix $A_{i,j}$ we have

$~~~~\displaystyle{ \max_{x\in S_m} \min_{y\in S_n}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j = \min_{y\in S_n}\max_{x\in S_m}\sum_{i=}^m\sum_{j=1}^nA_{i,j}x_iy_j }$ Here $0\leq x_i\leq 1$ is interpreted as the probability that Player 1 will choose strategy $i$ while $0\leq y_j\leq 1$ is the probability that Player 2 will choose strategy $j$

${\bf Proof}$ For any $x\in S_m$ and any $y\in S_n$ we have $\displaystyle{\max_{x\in S_m}\min_{y\in S_n}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\leq\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j }$ Taking maximum for $x\in S_m$ on both sides, we have $\displaystyle{\max_{x\in S_m}\min_{x\in S_n}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\leq \max_{s\in S_m}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j}$ Taking minimum for $y\in S_n$ on both sides, we have $\displaystyle{v_1=\max_{x\in S_m}\min_{y\in S_n}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\leq v_2=\min_{y\in S_n}\max_{x\in S_m}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j }$ The prove of other half of the inequality takes two steps:

Step 1$~~$ there is a $y\in S_n$ such that $\displaystyle{\sum_{j=1}^nA_{i,j}y_j\leq 0}$ $~~\Rightarrow~~$ is some ${\tilde x}\in S_m$ such that $\displaystyle{\sum_{i=1}^m\left\lgroup \sum_{j=1}^nA_{i,j}y_j\right\rgroup {\tilde x}_i\leq 0 }$ $\Rightarrow~~\displaystyle{\underset{x\in S_n}{\rm max}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\leq 0}$ $~~\Rightarrow~~\displaystyle{v_2=\underset{y\in S_n}{\rm min}\underset{x\in S_m}{\rm max} \sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\leq 0 }~~~~(*1)$ Step 2$~~$ there is a $x\in S_m$ such that $\displaystyle{\sum_{i=1}^mA_{i,j}x_iy_j>0}$ $~~\Rightarrow~~$ is some ${\tilde y}\in S_n$ such that $\displaystyle{\sum_{j=1}^n\left\lgroup \sum_{i=1}^mA_{i,j}x_i\right\rgroup {\tilde y}_j\geq 0 }$ $\Rightarrow~~\displaystyle{\underset{y\in S_n}{\rm min}\sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j\geq 0 }$ $~~\Rightarrow~~\displaystyle{v_1=\underset{x\in S_m}{\rm max}\underset{y\in S_n}{\rm min} \sum_{i=1}^m\sum_{j=1}^nA_{i,j}x_iy_j \geq 0}~~~~(*2)$ Combining (*1) and (*2) we see that either $0\leq v_1$ or $v_2\leq 0$ is the case and $v_1<0<v_2$ cannot be valid. Repeat the same procedure to the matrix ${\tilde A}_{i,j}=A_{i,j}-\lambda$ and we see that $v_1-\lambda <0<v_2-\lambda$ is invalid, i.e. $v_1<\lambda <v_2$ is not valid for any $\lambda$ Since $\lambda$ is arbitrary, we conclude that $v_2\leq v_1$

An entire theory on minimax has already been developed and is one of the major research area in optimization theory. The following contains some good sources for further reference:

Bibliography

1

V.F.Demyanov and V.N.Malozemov, Introduction to Minimax, Keter Publishing House Jerusalem Ltd, 1974.