Matrix game (LP for game theory)

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Authors: Nick Dotzenrod and Matt Kweon (ChE 345 Spring 2014)

Steward: Dajun Yue, Fengqi You

Date Presented: Apr. 10, 2014

Linear programming (LP) is a simple yet powerful tool that can be used as an aid in decision making under certainty - that is, the objective, constraints, and any other relevant information about the problem are known. A highly practical application of LP lies in its use in game theory. This page specifically explores how LP can be used to solve a finite two-person zero-sum game, also known as the matrix game, which is one of the simplest form of decision making games.

Contents

Game Theory

Introduction

The objective of game theory is to analyze the relationship between decision-making situations in order to achieve a desirable outcome. The theory can be applied to a wide range of applications, including, but not limited to, economics, politics and even the biological sciences. In essence, game theory serves as means to create a model to represent certain scenarios that have a variety of variables and potential outcomes. With these models developed from game theory, one can determine if assumptions made for a certain scenario are valid or whether additional models should be created that could more accurately assess the current problem. Game theory can be broken into a variety of different "games," each analyzing different situations in which a decision is to be made by one player with other players potentially affecting the process.

History

Johnny.jpg

Many mathematicians would agree that John von Neumann can be considered the Father of Game Theory. John was born in Hungary in 1903 and grew up having a love for math and the sciences. In college, he received degree a degree in chemical engineering and later, a Ph.D. in mathematics from the University of Budapest.

Matrix Game

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Minimax Theorem

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Example

example coming soon!

References

1. S. Tadelis, Game Theory: an Introduction, Princeton University Press, 2013.

2. R. J. Vanderbei, Linear Programming: Foundations and Extensions, Springer, 2008.

3. M. J. Osborne, An Introduction to Game Theory, Oxford University Press, 2004.