An introduction to linear programming and game theory

"This is quite a well-done book: very tightly organized, better-than-average exposition, and numerous examples, illustrations, and applications."
--Mathematical Reviews of the American Mathematical Society

An Introduction to Linear Programming and Game Theory, Third Edition presents a rigorous, yet accessible, introduction to the theoretical concepts and computational techniques of linear programming and game theory. Now with more extensive modeling exercises and detailed integer programming examples, this book uniquely illustrates how mathematics can be used in real-world applications in the social, life, and managerial sciences, providing readers with the opportunity to develop and apply their analytical abilities when solving realistic problems.

This Third Edition addresses various new topics and improvements in the field of mathematical programming, and it also presents two software programs, LP Assistant and the Solver add-in for Microsoft Office Excel, for solving linear programming problems. LP Assistant, developed by coauthor Gerard Keough, allows readers to perform the basic steps of the algorithms provided in the book and is freely available via the book's related Web site. The use of the sensitivity analysis report and integer programming algorithm from the Solver add-in for Microsoft Office Excel is introduced so readers can solve the book's linear and integer programming problems. A detailed appendix contains instructions for the use of both applications.

Additional features of the Third Edition include:

A discussion of sensitivity analysis for the two-variable problem, along with new examples demonstrating integer programming, non-linear programming, and make vs. buy models

Revised proofs and a discussion on the relevance and solution of the dual problem

A section on developing an example in Data Envelopment Analysis

An outline of the proof of John Nash's theorem on the existence of equilibrium strategy pairs for non-cooperative, non-zero-sum games

Providing a complete mathematical development of all presented concepts and examples, Introduction to Linear Programming and Game Theory, Third Edition is an ideal text for linear programming and mathematical modeling courses at the upper-undergraduate and graduate levels. It also serves as a valuable reference for professionals who use game theory in business, economics, and management science .

Author Notes

PAUL R. THIE , PhD, is Professor Emeritus in the Department of Mathematics at Boston College. Dr. Thie has authored numerous journal articles in the areas of mathematical programming and several complex variables.

GERARD E. KEOUGH , PhD, is Associate Professor and former chair of the Department of Mathematics at Boston College. He has written extensively on operator theory, functional analysis, and the use of technology in mathematics. Dr. Keough is the coauthor of Getting Started with Maple , Second Edition and Getting Started with Mathematica , Second Edition, both published by Wiley.

Preface	p. xi
1 Mathematical Models	p. 1
1.1 Applying Mathematics	p. 1
1.2 The Diet Problem	p. 2
1.3 The Prisoner's Dilemma	p. 5
1.4 The Roles of Linear Programming and Game Theory	p. 8
2 The Linear Programming Model	p. 9
2.1 History	p. 9
2.2 The Blending Model	p. 10
2.3 The Production Model	p. 21
2.4 The Transportation Model	p. 34
2.5 The Dynamic Planning Model	p. 38
2.6 Summary	p. 47
3 The Simplex Method	p. 57
3.1 The General Problem	p. 57
3.2 Linear Equations and Basic Feasible Solutions	p. 63
3.3 Introduction to the Simplex Method	p. 72
3.4 Theory of the Simplex Method	p. 77
3.5 The Simplex Tableau and Examples	p. 85
3.6 Artificial Variables	p. 93
3.7 Redundant Systems	p. 101
3.8 A Convergence Proof	p. 106
3.9 Linear Programming and Convexity	p. 110
3.10 Spreadsheet Solution of a Linear Programming Problem	p. 115
4 Duality	p. 121
4.1 Introduction to Duality	p. 121
4.2 Definition of the Dual Problem	p. 123
4.3 Examples and Interpretations	p. 132
4.4 The Duality Theorem	p. 138
4.5 The Complementary Slackness Theorem	p. 154
5 Sensitivity Analysis	p. 161
5.1 Examples in Sensitivity Analysis	p. 161
5.2 Matrix Representation of the Simplex Algorithm	p. 175
5.3 Changes in the Objective Function	p. 183
5.4 Addition of a New Variable	p. 189
5.5 Changes in the Constant-Term Column Vector	p. 192
5.6 The Dual Simplex Algorithm	p. 196
5.7 Addition of a Constraint	p. 204
6 Integer Programming	p. 211
6.1 Introduction to Integer Programming	p. 211
6.2 Models with Integer Programming Formulations	p. 214
6.3 Gomory's Cutting Plane Algorithm	p. 228
6.4 A Branch and Bound Algorithm	p. 237
6.5 Spreadsheet Solution of an Integer Programming Problem	p. 244
7 The Transportation Problem	p. 251
7.1 A Distribution Problem	p. 251
7.2 The Transportation Problem	p. 264
7.3 Applications	p. 282
8 Other Topics in Linear Programming	p. 299
8.1 An Example Involving Uncertainty	p. 299
8.2 An Example with Multiple Goals	p. 306
8.3 An Example Using Decomposition	p. 314
8.4 An Example in Data Envelopment Analysis	p. 325
9 Two-Person, Zero-Sum Games	p. 337
9.1 Introduction to Game Theory	p. 337
9.2 Some Principles of Decision Making in Game Theory	p. 345
9.3 Saddle Points	p. 350
9.4 Mixed Strategies	p. 353
9.5 The Fundamental Theorem	p. 360
9.6 Computational Techniques	p. 370
9.7 Games People Play	p. 382
10 Other Topics in Game Theory	p. 391
10.1 Utility Theory	p. 391
10.2 Two-Person, Non-Zero-Sum Games	p. 393
10.3 Noncooperative Two-Person Games	p. 397
10.4 Cooperative Two-Person Games	p. 404
10.5 The Axioms of Nash	p. 408
10.6 An Example	p. 414
A Vectors and Matrices	p. 417
B An Example of Cycling	p. 421
C Efficiency of the Simplex Method	p. 423
D LP Assistant	p. 427
E Microsoft Excel and Solver	p. 431
Bibliography	p. 439
Solutions to Selected Problems	p. 443
Index	p. 457

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Author Notes

Table of Contents