Invitation to Linear Programming and Game Theory

✍ Scribed by David C. Vella

• Publisher: Cambridge University Press
• Year: 2021
• Tongue: English
• Leaves: 541
• Edition: 1
• Category: Library

✦ Synopsis

Written in a conversational tone, this classroom-tested text introduces the fundamentals of linear programming and game theory, showing readers how to apply serious mathematics to practical real-life questions by modelling linear optimization problems and strategic games. The treatment of linear programming includes two distinct graphical methods. The game theory chapters include a novel proof of the minimax theorem for 2x2 zero-sum games. In addition to zero-sum games, the text presents variable-sum games, ordinal games, and n-player games as the natural result of relaxing or modifying the assumptions of zero-sum games. All concepts and techniques are derived from motivating examples, building in complexity, which encourages students to think creatively and leads them to understand how the mathematics is applied. With no prerequisite besides high school algebra, the text will be useful to motivated high school students and undergraduates studying business, economics, mathematics, and the social sciences.

550 exercises and 110 worked examples
Provides step-by-step instructions for using Wolfram Mathematica and Microsoft Excel to solve linear optimization problems and constant-sum games
Introduces different techniques throughout the text so readers can easily compare them and develop intuition regarding which techniques are appropriate for which type of question
Accommodates readers from different backgrounds by indicating what should be worked through or skipped depending on the reader's previous mathematical experience

✦ Table of Contents

Cover
Contents
Preface
1 Preliminaries
1.1 Mathematical Models
1.2 Systems of Linear Equations
1.3 Elimination and Matrices
1.4 Vectors, Linear Combinations, and Bases
1.5 Three Points of View
1.6 Some Nonlinear Models
2 Matrix Algebra
2.1 Matrices
2.2 Operations on Matrices I
2.3 Operations on Matrices II: Matrix Inversion
2.4 Solving Linear Systems by Matrix Inversion
2.5 Some Applications to Cryptography and Economics
3 Graphical Linear Programming
3.1 Introduction and Graphical Solutions
3.2 The Decision Space: m × 2 Problems
3.3 Convex Sets and the Corner Point Theorem
3.4 Problems with No Solution or Infinitely Many Solutions
3.5 The Constraint Space: 2 × n Problems
4 Sensitivity Analysis and Duality
4.1 Introduction to Sensitivity Analysis
4.2 Changes in the Objective Coefficients
4.3 Marginal Values Associated with Constraints
4.4 Other Changes: Drawbacks of the Graphical Methods
4.5 Duality
5 The Simplex Algorithm
5.1 Standard Form Maximization Problems
5.2 Phase II Pivoting
5.3 Standard Form Minimization Problems
5.4 Solving Linear Programming Problems with a Computer
6 Game Theory
6.1 Introduction
6.2 Dominant Strategies and Nash Equilibrium Points
6.3 Mixed-Strategy Constant-Sum Games
6.4 Solving Mixed-Strategy Games: The Minimax Theorem in the 2 × 2 Case
7 More Game Theory
7.1 Solving Larger Constant-Sum Games
7.2 Solving Constant Sum-Games with Linear Programming
7.3 Using a Computer to Solve Constant-Sum Games
7.4 Variable-Sum Games
8 Sensitivity Analysis, Ordinal Games, and n-Person Games
8.1 Sensitivity Analysis in Game Theory
8.2 Ordinal Games
8.3 Sequential Games: The Theory of Moves
8.4 A Brief Introduction to n-Player Games
8.5 Legislative Voting and Political Power
9 More Linear Programming
9.1 Phase I Pivoting
9.2 Alternate Approach to Minimization Problems
9.3 Sensitivity Analysis and the Simplex Algorithm
Appendix: A Rapid Review of Sets and Probability
A.1 A Review of Sets
A.2 Enumeration
A.3 Experiments, Sample Spaces, and Probability Models
A.4 Uniform Sample Spaces
A.5 Conditional Probability and Independent Events
A.6 Bayes’s Theorem
References
Index