Game Theory E-Book

E. N. Barron

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Series in Operations Research and Management Science
• Sprache: Englisch

Seitenzahl: 963

Veröffentlichungsjahr: 2013

Beschreibung

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(TM), 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.

Leseprobe

Contents

Cover

Wiley Series in Operations Research and Management Science

Title Page

Copyright

Dedication

Preface for the Second Edition

Preface for the First Edition

Acknowledgments

Introduction

Chapter One: Matrix Two-Person Games

1.1 The Basics

1.2 The von Neumann Minimax Theorem

1.3 Mixed Strategies

1.4 Solving 2 × 2 Games Graphically

1.5 Graphical Solution of 2 × m and n × 2 Games

1.6 Best Response Strategies

Bibliographic Notes

Chapter Two: Solution Methods for Matrix Games

2.1 Solution of Some Special Games

2.2 Invertible Matrix Games

2.3 Symmetric Games

2.4 Matrix Games and Linear Programming

2.5 Appendix: Linear Programming and the Simplex Method

2.6 Review Problems

2.7 Maple/Mathematica

Bibliographic Notes

Chapter Three: Two-Person Nonzero Sum Games

3.1 The Basics

3.2 2 × 2 Bimatrix Games, Best Response, Equality of Payoffs

3.3 Interior Mixed Nash Points by Calculus

3.4 Nonlinear Programming Method for Nonzero Sum Two-Person Games

3.5 Correlated Equilibria

3.6 Choosing Among Several Nash Equilibria (Optional)

Bibliographic Notes

Chapter Four: Games in Extensive Form: Sequential Decision Making

4.1 Introduction to Game Trees—Gambit

4.2 Backward Induction and Subgame Perfect Equilibrium

Bibliographic Notes

Chapter Five: N-Person Nonzero Sum Games and Games with a Continuum of Strategies

5.1 The Basics

5.2 Economics Applications of Nash Equilibria

5.3 Duels (Optional)

5.4 Auctions (Optional)

Bibliographic Notes

Chapter Six: Cooperative Games

6.1 Coalitions and Characteristic Functions

6.2 The Nucleolus

6.3 The Shapley Value

6.4 Bargaining

6.5 Maple/Mathematica

Bibliographic Notes

Chapter Seven: Evolutionary Stable Strategies and Population Games

7.1 Evolution

7.2 Population Games

Bibliographic Notes

Appendix A: The Essentials of Matrix Analysis

Appendix B: The Essentials of Probability

B.1 Discrete Random Variables

B.2 Continuous Distributions

B.3 Order Statistics

Appendix C: The Essentials of Maple

C.1 Features

C.2 Functions

C.3 Some Commands Used in This Book

Appendix D: The Mathematica Commands

D.1 The Upper and Lower Values of a Game

D.2 The Value of an Invertible Matrix Game with Mixed Strategies

D.3 Solving Matrix Games by Linear Programming

D.4 Interior Nash Points

D.5 Lemke–Howson Algorithm for Nash Equilibrium

D.6 Is the Core Empty?

D.7 Find and Plot the Least Core

D.8 Nucleolus and Shapley Value Procedure

D.9 Plotting the Payoff Pairs

D.10 Bargaining Solutions

D.11 Mathematica for Replicator Dynamics

Appendix E: Biographies

E.1 John von Neumann

E.2 John Forbes Nash

Problem Solutions

Solutions for Chapter 1

Solutions for Chapter 2

Solutions for Chapter 3

Solutions for Chapter 4

Solutions for Chapter 5

Solutions for Chapter 6

Solutions for Chapter 7

References

Index

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Barron • Game Theory: An Introduction, Second Edition

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Library of Congress Cataloging-in-Publication Data:

Barron, E. N. (Emmanual N.), 1949– Game theory: an introduction / Emmanuel N. Barron. – Second edition. pages cm Includes bibliographical references and index. ISBN 978-1-118-21693-4 (cloth) 1. Game theory. I. Title. QA269.B27 2013 519.3–dc23 2013008270

ISBN: 9781118216934

To Christina, Michael, andAnastasia; and Fotini and Michael

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:

1. Chapters 1, 2 (3 weeks)

(a) Upper and lower values, mixed strategies, introduction to game trees and Gambit

(b) Expected payoffs, minimax theorem, graphical method

(c) Invertible matrix games, symmetric games, linear programming method

2. Chapter 3 (2 weeks)

(a) Nonzero sum two-person games, pure and mixed Nash equilibrium

(b) Best responses, equality of payoffs

(c) Calculus method, Lemke-Howson, correlated equilibrium

3. Chapter 4 (1 week)

(a) Extensive form games

(i) Trees in Gambit, information sets, examples

(ii) Subgame perfect equilibrium, backward induction, examples

(b) Exam 1

4. Chapter 5 (2 weeks)

(a) Pure Nash equilibrium for games with a continuum of strategies

(b) Selected examples: Cournot, Stackelberg, Traveler’s paradox, Braess’ paradox, War of Attrition

5. Chapter 6 (3 weeks)

(a) Cooperative games

(i) Characteristic functions, imputations, core, least core

(ii) Nucleolus, Shapley value.

(b) Bargaining, Nash solution, threats

(c) Exam 2

6. Chapter 7 (1 week)

(a) Evolutionary stable strategies

(b) Population games and stability

(c) Review

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

[email protected]

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.

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, Illinois2007

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.

Acknowledgments

I am very grateful to Susanne Steitz-Filler, Kris Parrish, and all the support staff of John Wiley & Sons, as well as Amy Hendrickson at Texnology, Inc. I am also grateful to the reviewers of this book.

E.N. Barron

Introduction

Mostly for the Instructor

My goal is to present the basic concepts of noncooperative and cooperative game theory and expose students to a very important application of mathematics. In addition, this course introduces students to an understandable theory created by geniuses in many different fields that even today has a low cost of entry.

My experience is that the students who enroll in game theory are primarily mathematics students interested in applications, with about one-third to one-half of the class majoring in economics or other disciplines (such as biology or biochemistry or physics). The modern economics and operations research curriculum requires more and more mathematics, and game theory is typically a required course in those fields. For economics students with a more mathematical background, this course is set at an ideal level. For mathematics students interested in a graduate degree in something other than mathematics, this course exposes them to another discipline in which they might develop an interest and that will enable them to further their studies, or simply to learn some fun mathematics. Many students get the impression that applied mathematics is physics or engineering, and this class shows that there are other areas of applications that are very interesting and that opens up many other alternatives to a pure math or classical applied mathematics concentration.

Game theory can be divided into two broad classifications: noncooperative and cooperative. The sequence of main topics in this book is as follows:

1. Two-person zero sum matrix games

2. Nonzero sum games, both bimatrix, and with a continuum of strategies, Nash and correlated equilibria

3. Cooperative games, covering both the nucleolus concept and the Shapley value

4. Bargaining with and without threats

5. Evolution and population games and the merger with stability theory

This is generally more than enough to fill one semester, but if time permits (which I doubt) or if the instructor would like to cover other topics (duels, auctions, economic growth, evolutionary stable strategies, population games), these are all presented at some level of depth in this book appropriate for the intended audience. Game theory has a lot of branches and these topics are the main branches, but not all of them. Combinatorial game theory is a branch that could be covered in a separate course but it is too different from the topics considered in this book to be consistent. Repeated games and stochastic games are two more important topics that are skipped as too advanced and too far afield.

This book begins with the classical zero sum two-person matrix games, which is a very rich theory with many interesting results. I suggest that the first two chapters be covered in their entirety, although many of the examples can be chosen on the basis of time and the instructor’s interest. For classes more mathematically oriented, one could cover the proofs given of the von Neumann minimax theorem. The use of linear programming as an algorithmic method for solving matrix games is essential, but one must be careful to avoid getting sucked into spending too much time on the simplex method. Linear programming is a course in itself, and as long as students understand the transformation of a game into a linear program, they get a flavor of the power of the method. It is a little magical when implemented in Maple or Mathematica, and I give two ways to do it, but there are reasons for preferring one over the other when doing it by hand. I skip the simplex method.

The generalization to nonzero sum two-person games comes next with the foundational idea of a Nash equilibrium introduced. It is an easy extension from a saddle point of a zero sum game to a Nash equilibrium for nonzero sum. Several methods are used to find the Nash equilibria from the use of calculus to full-blown nonlinear programming. A short introduction to correlated equilibrium is presented with a conversion to a linear programming problem for solution. Again, Maple/Mathematica is an essential tool in the solution of these problems. Both linear and nonlinear programs are used in this course as a tool to study game theory, and not as a course to study the tools. I suggest that the entire chapter be covered.

It is essential that the instructor cover at least the main points in Chapters 1–7. Chapter 5 is a generalization of two-person nonzero sum games with a finite number of strategies (basically matrix games) to games with a continuum of strategies. Calculus is the primary method used. The models included in Chapter 5 involve the standard economic models, the theory of duels, which are just games of timing, and the theory of auctions. An entire semester can be spent on this one chapter, so the instructor will probably want to select the applications for her or his own and the class’s interest. Students find the economic problems particularly interesting and a very strong motivation for studying both mathematics and economics.

When cooperative games are covered, I present both the theory of the core, leading to the nucleolus, and the very popular Shapley value. Students find the nucleolus extremely computationally challenging because there are usually lots of inequalities to solve and one needs to find a special condition for which the constraints are nonempty. Doing this by hand is not trivial even though the algebra is easy. Once again, Maple/Mathematica can be used as a tool to assist in the solution for problems with four or more players, or even three players. In addition, the graphical abilities of software permit a demonstration of the actual shrinking or expanding of the core according to an adjustment of the dissatisfaction parameter. On the other hand, the use of software is not a procedure in which one simply inputs the characteristic function and out pops the answer. A student may use software to assist but not solve the problem.1 Chapter 6 ends with a presentation of the theory of bargaining in which nonlinear programming is used to solve the bargaining problem following Nash’s ideas. In addition, the theory of bargaining with optimal threat strategies is included. This serves also as a review section because the concepts of matrix games are used for safety levels, saddle points, and so on.

Chapter 7 serves as a basic introduction to evolutionary stable strategies and population games. If you have a lot of biology or biochemistry majors in your class, you might want to make time for this chapter. The second half of the chapter does require an elementary class in ordinary differential equations. The connection between stability, Nash equilibria, and evolutionary stable strategies can be nicely illustrated with the assistance of the Maple/Mathematica differential equation packages, circumventing the need for finding the actual solution of the equations by hand. Testing for stability is a calculus method. One possible use of this chapter is for projects or extra credit.

My own experience is that I run out of time with a 14-week semester after Chapter 6. Too many topics need to be skipped, but adjusting the topics in different terms makes the course fresh from semester to semester. Of course, topics can be chosen according to your own and the class’s interests. On the other hand, this book is not meant to be exhaustive of the theory of games in any way.

The prerequisites for this course have been kept to a minimum. This course is presented in our mathematics department, but I have had many economics, biology, biochemistry, business, finance, political science, and physics majors. The prerequisites are listed as a class in probability, and a class in linear algebra, but very little of those subjects are actually used in the class. I tell students that if they know how to multiply two matrices together, they should do fine; and the probability aspects are usually nothing more than the definitions. The important prerequisite is really not being afraid of the concepts. I have had many students with a background of only two semesters of calculus, no probability or linear algebra, or only high school mathematics courses. Students range from sophomores to graduate students (but I have even taught this course to freshman honors students). As a minor reference I include appendixes on linear algebra, probability, a more detailed explanation of some of the Maple commands, and a translation of the major procedures in the text to Mathematica.

The use of software in this class is also optional, but then it is like learning multivariable calculus without an ability to graph the surfaces. It can be done, but it is more difficult. Why do that when the tool is available? That may be one of the main features of this book, because before the technology was available, this subject had to be presented in a very mathematical way or a very nonmathematical way. I have tried to take the middle road, and it is not a soft class. On the other hand, the new Chapter 4 on extensive games absolutely requires the use of GAMBIT because the interesting models are too complex to do by hand. I feel that this is appropriate considering that GAMBIT is free and runs on all machines. It is also possible for an instructor to skip this chapter entirely if so desired.

There are at least two important websites in game theory. The first is

gametheory.net,

which is a repository for all kinds of game theory stuff. I especially like the notes by T. Ferguson at UCLA, H. Moulin at Rice University, W Bialis at SUNY Buffalo, and Y. Peres, at the University of California, Berkeley. The second site is

www.gambit.sourceforge.net,

which contains the extremely useful open-source software GAMBIT, which students may download and install on their own computers. Gambit is a game-solving program that will find the Nash equilibria of all N-person matrix games with any number of strategies. It may also be used to solve any zero sum game by entering the matrix appropriately. Students love it. Finally, if a user has Mathematica, there is a cooperative game solver available from the Wolfram website, known as TuGames, written by Holgar Meinhardt, that can be installed as a Mathematica package. TuGames can solve any characteristic function cooperative game, and much more. There is also a MatLab package written by Professor Meinhardt for solving cooperative games. The MatLab package is available at http://www.mathworks.com/matlabcentral/fileexchange/35933-mattugames.

As mentioned in the preface, I will post on my website various Maple and Mathematica programs for this course.

I would be grateful for any notification of errors, and I will post errata on my website

I will end this introduction with an intriguing quote that Professor Avner Friedman included in his book Differential Games that he had me read for my graduate studies. The quote is from Lord Byron: “There are two pleasures for your choosing; the first is winning and the second is losing.” Is losing really a pleasure? I can answer this now. The answer is no.

1 On the other hand, the recent result of Leng and Parlar (2010) has explicit formulas for the nucleolus of any three player game. The software doing the calculation is a black box.