11. Game Theory

In this chapter, we shall study if not the most practical then certainly an elegant application of linear programming. The subject is called game theory, and we shall focus on the simplest type of game, called the finite two-person zero-sum game, or just matrix game for short. Our primary goal shall be to prove the famous Minimax Theorem, which was first discovered and proved by John von Neumann in 1928. His original proof of this theorem was rather involved and depended on another beautiful theorem from mathematics, the Brouwer Fixed-Point Theorem. However, it eventually became clear that the solution of matrix games could be found by solving a certain linear programming problem and that the Minimax Theorem is just a fairly straightforward consequence of the Duality Theorem.