Editing Minimax theorem (section)

== Bilinear functions and zero-sum games ==
Von Neumann's original theorem<ref name=":0" /> was motivated by game theory and applies to the case where 

* <math>X</math> and <math>Y</math> are [[Simplex|standard simplexes]]: <math display="inline">X = \{ (x_1, \dots, x_n) \in [0,1]^n : \sum_{i = 1}^n x_i = 1 \}
</math> and <math display="inline">Y = \{ (y_1, \dots, y_m) \in [0,1]^m : \sum_{j = 1}^m y_j = 1 \}</math>, and
* <math>f(x,y)</math> is a linear function in both of its arguments (that is, <math>f</math> is [[Bilinear form|bilinear]]) and therefore can be written <math>f(x,y) = x^\mathsf{T} A y</math> for a finite matrix <math>A \in \mathbb{R}^{n \times m}</math>, or equivalently as <math display="inline">f(x,y) = \sum_{i=1}^n\sum_{j=1}^m A_{ij}x_iy_j</math>.

Under these assumptions, von Neumann proved that
: <math>\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. </math>
In the context of two-player [[Zero-sum game|zero-sum games]], the sets <math>X</math> and <math>Y</math> 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|payoff matrix]] <math>A</math>. The function <math>f(x,y)</math> encodes the [[expected value]] of the payoff to the first player when the first player plays the strategy <math>x</math> and the second player plays the strategy <math>y</math>.