Preface for the First Edition
Man is a gaming animal. He must always be trying to get the better in something or other.
—Charles Lamb, Essays of Elia, 1823
Why do countries insist on obtaining nuclear weapons? What is a fair allocation of property taxes in a community? Should armies ever be divided, and in what way in order to attack more than one target? How should a rat run to avoid capture by a cat? Why do conspiracies almost always fail? What percentage of offensive plays in football should be passes, and what percentage of defensive plays should be blitzes? How should the assets of a bankrupt company be allocated to the debtors? These are the questions that game theory can answer. Game theory arises in almost every facet of human interaction (and inhuman interaction as well). Either every interaction involves objectives that are directly opposed, or the possibility of cooperation presents itself. Modern game theory is a rich area of mathematics for economics, political science, military science, finance, biological science (because of competing species and evolution), and so on.1
This book is intended as a mathematical introduction to the basic theory of games, including noncooperative and cooperative games. The topics build from zero sum matrix games, to nonzero sum, to cooperative games, to population games. Applications are presented to the basic models of competition in economics: Cournot, Bertrand, and Stackelberg models. The theory of auctions is introduced and the theory of duels is a theme example used in both matrix games, nonzero sum games, and games with continuous strategies. Cooperative games are concerned with the distribution of payoffs when players cooperate. Applications of cooperative game theory to scheduling, cost savings, negotiating, bargaining, and so on, are introduced and covered in detail.
The prerequisites for this course or book include a year of calculus, and very small parts of linear algebra and probability. For a more mathematical reading of the book, it would be helpful to have a class in advanced calculus, or real analysis. Chapter 7 uses ordinary differential equations. All of these courses are usually completed by the end of the sophomore year, and many can be taken concurrently with this course. Exercises are included at the end of almost every section, and odd-numbered problems have solutions at the end of the book. I have also included appendixes on the basics of linear algebra, probability, Maple, 2 and Mathematica, 3 commands for the code discussed in the book using Maple.
One of the unique features of this book is the use of Maple4 or Mathematica5 to find the values and strategies of games, both zero and nonzero sum, and noncooperative and cooperative. The major computational impediment to solving a game is the roadblock of solving a linear or nonlinear program. Maple/Mathematica gets rid of those problems and the theories of linear and nonlinear programming do not need to be presented to do the computations. To help present some insight into the basic simplex method, which is used in solving matrix games and in finding the nucleolus, a section on the simplex method specialized to solving matrix games is included. If a reader does not have access to Maple or Mathematica, it is still possible to do most of the problems by hand, or using the free software Gambit, 6 or Gams.7
The approach I took in the software in this book is to not reduce the procedure to a canned program in which the student simply enters the matrix and the software does the rest (Gambit does that). To use Maple/Mathematica and the commands to solve any of the games in this book, the student has to know the procedure, that is, what is going on with the game theory part of it, and then invoke the software to do the computations.
My experience with game theory for undergraduates is that students greatly enjoy both the theory and applications, which are so obviously relevant and fun. I hope that instructors who offer this course as either a regular part of the curriculum, or as a topics course, will find that this is a very fun class to teach, and maybe to turn students on to a subject developed mostly in this century and still under hot pursuit. I also like to point out to students that they are studying the work of Nobel Prize winners: Herbert Simon8 in 1979, John Nash, 9 J.C. Harsanyi10 and R. Selten11 in 1994, William Vickrey12 and James Mirrlees13 in 1996, and Robert Aumann14 and Thomas Schelling15 in 2005. In 2007 the Nobel Prize in economics was awarded to game theorists Roger Myerson, 16 Leonid Hurwicz, 17 and Erik Maskin.18 In addition, game theory was pretty much invented by John von Neumann, 19 one of the true geniuses of the twentieth century.
E.N. Barron
Chicago, Illinois
2007
1 In an ironic twist, game theory cannot help with most common games, like chess, because of the large number of strategies involved.
2 Trademark of Maplesoft Corporation.
3 Trademark of Wolfram Research Corp.
4 Version 10.0 or later.
5 Version 8.0.
6 Available from www.gambit.sourceforge.net/.
7 Available from www.gams.com.
8 June 15, 1916–February 9, 2001, a political scientist who founded organizational decision making.
9 See the short biography in the Appendix E.
10 May 29, 1920–August 9, 2000, Professor of Economics at University of California, Berkeley, instrumental in equilibrium selection.
11 Born October 5, 1930, Professor Emeritus, University of Bonn, known for his work on bounded rationality.
12 June 21, 1914–October 11, 1996, Professor of Economics at Columbia University, known for his work on auction theory.
13 Born July 5, 1936, Professor Emeritus at University of Cambridge.
14 Born June 8, 1930, Professor at Hebrew University.
15 Born April 14, 1921, Professor in School of Public Policy, University of Maryland.
16 Born March 29, 1951, Professor at University of Chicago.
17 Born August 21, 1917, Regents Professor of Economics Emeritus at the University of Minnesota.
18 Born December 12, 1950, Professor of Social Science at Institute for Advanced Study, Princeton.
19 See a short biography in the Appendix E and MacRae (1999) for a full biography.