Minimax Theorem

The fundamental theorem of Game Theory which states that every Finite, Zero-Sum, two-person Game has optimal Mixed Strategies. It was proved by John von Neumann in 1928.

Formally, let ${\bf X}$ and ${\bf Y}$ be Mixed Strategies for players A and B. Let ${\hbox{\sf A}}$ be the Payoff Matrix. Then

$$\max_X \min_Y {\bf X}^{\rm T}{\hbox{\sf A}}{\bf Y}=\min_Y\max_X {\bf X}^{\rm T}{\hbox{\sf A}}{\bf Y} = v,$$

where $v$ is called the Value of the Game and ${\bf X}$ and ${\bf Y}$ are called the solutions. It also turns out that if there is more than one optimal Mixed Strategy, there are infinitely many.

See also Mixed Strategy

References

Willem, M. Minimax Theorem. Boston, MA: Birkhäuser, 1996.