# Game Theory

• Robert J. Vanderbei
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 196)

## Abstract

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.

## Keywords

Nash Equilibrium Linear Programming Problem Pure Strategy Linear Complementarity Problem Payoff Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliography

1. Chvátal, V. (1983). Linear programming. New York: Freeman.Google Scholar
2. Dresher, M. (1961). Games of strategy: Theory and application. Englewood Cliffs: Prentice-Hall.Google Scholar
3. Gale, D., Kuhn, H., and Tucker, A. (1951). Linear programming and the theory of games. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 317–329). New York: Wiley.Google Scholar
4. Karlin, S. (1959). Mathematical methods and theory in games, programming, and economics (Vols. 1 and 2). Reading: Addison-Wesley.Google Scholar
5. Kuhn, H. (1950). A simplified two-person poker. Annals of Mathematics Studies, 24, 97–103.Google Scholar
6. von Neumann, J. (1928). Zur Theorie der Gesselschaftschpiele. Mathematische Annalen, 100, 295–320.
7. von Neumann, J., and Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.). Princeton: Princeton University Press.Google Scholar