Game Theory

An Introduction

E. N. Barron

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118533895
• Edition: 2
• Topic: Business
• Subtopic: Management
• Index: Business

An exciting new edition of the popular introduction to game theory and its applications

The thoroughly expanded Second Edition presents a unique, hands-on approach to game theory. While most books on the subject are too abstract or too basic for mathematicians, Game Theory: An Introduction, Second Edition offers a blend of theory and applications, allowing readers to use theory and software to create and analyze real-world decision-making models.

With a rigorous, yet accessible, treatment of mathematics, the book focuses on results that can be used to determine optimal game strategies. Game Theory: An Introduction, Second Edition demonstrates how to use modern software, such as Maple™, Mathematica®, and Gambit, to create, analyze, and implement effective decision-making models. Coverage includes the main aspects of game theory including the fundamentals of two-person zero-sum games, cooperative games, and population games as well as a large number of examples from various fields, such as economics, transportation, warfare, asset distribution, political science, and biology. The Second Edition features: • A new chapter on extensive games, which greatly expands the implementation of available models • New sections on correlated equilibria and exact formulas for three-player cooperative games • Many updated topics including threats in bargaining games and evolutionary stable strategies • Solutions and methods used to solve all odd-numbered problems • A companion website containing the related Maple and Mathematica data sets and code

A trusted and proven guide for students of mathematics and economics, Game Theory: An Introduction, Second Edition is also an excellent resource for researchers and practitioners in economics, finance, engineering, operations research, statistics, and computer science.

CHAPTER ONE

Matrix Two-Person Games

If you must play, decide upon three things at the start: the rules of the game, the stakes, and the quitting time.

—Chinese proverb

Everyone has a plan until you get hit.

—Mike Tyson, Heavyweight Boxing Champ, 1986–1990, 1996

You can’t always get what you want.

—Mothers everywhere

1.1 The Basics

What is a game? We need a mathematical description, but we will not get too technical. A game involves a number of players1 N, a set of strategies for each player, and a payoff that quantitatively describes the outcome of each play of the game in terms of the amount that each player wins or loses. A strategy for each player can be very complicated because it is a plan, determined at the start of the game, that describes what a player will do in every possible situation. In some games, this is not too bad because the number of moves is small, but in other games, like chess, the number of moves is huge and so the number of possible strategies, although finite, is gigantic. In this chapter, we consider two-person games and give several examples of exactly what is a strategy.

Let’s call the two players I and II. Suppose that player I has a choice of n possible strategies and player II has a choice of m possible strategies. If player I chooses a strategy, say, strategy i, i = 1, ..., n, and player II chooses a strategy j, j = 1, ..., m, then they play the game and the payoff to each player is computed. In a zero sum game, whatever one player wins the other loses, so if aij is the amount player I receives, then II gets −aij. Now we have a collection of numbers {aij}, i = 1, ..., n, j = 1, ..., m, and we can arrange these in a matrix. These numbers are called the payoffs to player I and the matrix is called the payoff or game matrix:

By agreement we place player I as the row player and player II as the column player. We also agree that the numbers in the matrix represent the payoff to player I. In a zero sum game, the payoffs to player II would be the negative of those in the matrix so we don’t have to record both of those. Of course, if player I has some payoff that is negative, then player II would have a positive payoff.

Summarizing, a two-person zero sum game in matrix form means that there is a matrix A = (aij), i = 1, ..., n, j = 1, ..., m of real numbers so that if player I, the row player, chooses to play row i and player II, the column player, chooses to play column j, then...