Preface for the Second Edition
This second edition expands the book in several ways. There is a new chapter on extensive games that takes advantage of the open-source software package GAMBIT1 to both draw the trees and solve the games. Even simple examples will show that doing this by hand is not really feasible and that’s why it was skipped in the first edition. These games are now included in this edition because it provides a significant expansion of the number of game models that the student can think about and implement. It is an important modeling experience and one cannot avoid thinking about the game and the details to get it right.
Many more exercises have been added to the end of most sections. Some material has been expanded upon and some new results have been discussed. For instance, the book now has a section on correlated equilibria and a new section on explicit solutions of three player cooperative games due to recent results of Leng and Parlar (2010). The use of software makes these topics tractable. Finding a correlated equilibrium depends on solving a linear programming problem that becomes a trivial task with Maple™/Mathematica® or any linear programming package.
Once again there is more material in the book than can be covered in one semester, especially if one now wants to include extensive games. Nevertheless, all of the important topics can be covered in one semester if one does not get sidetracked into linear programming or economics. The major topics forming the core of the course are zero sum games, nonzero sum games, and cooperative games. A course covering these topics could be completed in one quarter.
The foundation of this class is examples. Every concept is either introduced by or expanded upon and then illustrated with examples. Even though proofs of the main theorems are included, in my own course I skip virtually all of them and focus on their use. In a more advanced course, one might include proofs and more advanced material. For example, I have included a brief discussion of mixed strategies for continuous games but this is a topic that actually requires knowledge of measure theory to present properly. Even knowing about Stieltjes integrals is beyond the prerequisites of the class. Incidentally, the prerequisites for the book are very elementary probability, calculus, and a basic knowledge of matrices (like multiplying and inverses).
Another new feature of the second edition is the availability of a solution manual that includes solutions to all of the problems in the book. The new edition of the book contains answers to odd-numbered problems. Some instructors have indicated that they would prefer to not have all the solutions in the book so that they could assign homework for grades without making up all new problems. I am persuaded by this argument after teaching this course many times.
All the software in the book in both Maple and Mathematica is available for download from my website:
My classes in game theory have had a mix of students with majors in mathematics, economics, biology, chemistry, but even French, theology, philosophy, and English. Most of the students had college mathematics prerequisites, but some had only taken calculus and probability/statistics in high school. The prerequisites are not a strong impediment for this course.
I have recently begun my course with introducing extensive games almost from the first lecture. In fact, when I introduce the Russian Roulette and 2 × 2 Nim examples in Chapter 1, I take that opportunity to present them in Gambit in a classroom demonstration. From that point on extensive games are just part of the course and they are intermingled with matrix theory as a way to model a game and come up with the game matrix. In fact, demonstrating the construction using Gambit of a few examples in class is enough for students to be able to construct their own models. Exercises on extensive games can be assigned from the first week. In fact, the chapter on extensive games does not have to be covered as a separate chapter but used as a source of problems and homework. The only concepts I discuss in that chapter are backward induction and subgame perfect equilibrium that can easily be covered through examples.
A suggested syllabus for this course may be useful:
Naturally, instructors may choose from the many peripheral topics available in the book if they have time, or for assignment as extra credit or projects. I have made no attempt to make the book exhaustive of topics that should be covered, and I think that would be impossible in any case. The topics I have chosen I consider to be foundational for all of game theory and within the constraints of the prerequisites of an undergraduate course. For further topics, there are many excellent books on the subject, some of which are listed in the references.
As a final note on software, this class does not require the writing of any programs. All one needs is a basic facility with using software packages. In all cases, solving any of the games in Maple or Mathematica involves looking at the examples and modifying the matrices as warranted. The use of software has not been a deterrent to any student I have had in any game theory class, and in fact, the class can be designed without the use of any software.
In the first edition I listed some of the game theorists who have been awarded the Nobel Prize in Economics. In 2012, Lloyd Shapley and Alvin Roth, both pioneers in game theory and behavioral economics, were awarded the Nobel prize, continuing the recognition of the contributions of game theory to economics.
Acknowledgment: I am very grateful to everyone who has contacted me with possible errors in the first edition. I am particularly grateful to Professor Kevin Easley at Truman State, for his many suggestions, comments, and improvements for the book over the time he has been teaching game theory. His comments and the comments from his class were invaluable to me. I am grateful to all of those instructors who have adopted the book for use in their course and I hope that the second edition removes some of the deficiencies in the first and makes the course better for everyone.
As part of an independent project, I assigned my student Zachary Schaefer the problem of writing some very useful Mathematica programs to do various tasks. The projects ranged from setting up the graphs for any appropriate, sized matrix game, solving any game with linear programming by both methods, automating the search for Nash equilibria in a nonzero sum game, and finding the nucleolus and Shapley value for any cooperative game (this last one is a real tour de force). All these projects are available from my website. Zachary did a great job.
I also thank the National Science Foundation for partial support of this project under grant 1008602.
I would be grateful for notification of any errors found.
E.N. Barron
Chicago, Illinois
2012
1 McKelvey RD, McLennan AM, Turocy TL. Gambit: software tools for game theory, Version 0.2010.09.01; 2010. Available at: http://www.gambit-project.org (accessed on 2012 Nov 15), and which can also be obtained from website www.math.luc.edu/∼enb.