Combinatorial Games

Combinatorial games are games of pure strategy involving two players, with perfect information and no element of chance. Starting from the very basics of gameplay and strategy, the authors cover a wide range of topics, from game algebra to special classes of games. Classic techniques are introduced and applied in novel ways to analyze both old and

It is wonderful to see advanced combinatorial game theory made accessible. Siegel's expertise and enjoyable writing style make this book a perfect resource for anyone wanting to learn the latest developments and open problems in the field. —Erik Demaine, MIT Aaron Siegel has been the major contributor to Combinatorial Game Theory over the last decade or so. Now, in this authoritative work, he has made the latest results in the theory accessible, so that the subject will achieve the place in mathematics that it deserves. —Richard Guy, University of Calgary Combinatorial game theory is the study of two-player games with no hidden information and no chance elements. The theory assigns algebraic values to positions in such games and seeks to quantify the algebraic and combinatorial structure of their interactions. Its modern form was introduced thirty years ago, with the publication of the classic Winning Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy, and interest has rapidly increased in recent decades. This book is a comprehensive and up-to-date introduction to the subject, tracing its development from first principles and examples through many of its most recent advances. Roughly half the book is devoted to a rigorous treatment of the classical theory; the remaining material is an in-depth presentation of topics that appear for the first time in textbook form, including the theory of misère quotients and Berlekamp's generalized temperature theory. Packed with hundreds of examples and exercises and meticulously cross-referenced, Combinatorial Game Theory will appeal equally to students, instructors, and research professionals. More than forty open problems and conjectures are mentioned in the text, highlighting the many mysteries that still remain in this young and exciting field. Aaron Siegel holds a Ph.D. in mathematics from the University of California, Berkeley and has held positions at the Mathematical Sciences Research Institute and the Institute for Advanced Study. He was a partner at Berkeley Quantitative, a technology-driven hedge fund, and is presently employed by Twitter, Inc.

Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication.

The aim of Cooperative Games on Combinatorial Structures is to analyze conflict situations in which two or more players can make coalitions and obtain prizes and penalties. This approach to situations of competition and cooperation was given in the seminal treatise by John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior. Cooperative game theory has succeeded in providing many applications of game theory. In this volume, games defined on combinatorial structures will be analyzed, i.e. a set system over a set of players. In many situations the author will work in a closure space. Examples of closure operators are the spanning operator of linear algebra and all convex hull operators. Chapters 1-4 constitute a review of mathematical concepts from Cooperative Game Theory, Graph Theory, Linear and Integer Programming, Combinatorial Optimization, Discrete Convex Analysis and Computational Complexity. The table of contents is a short guide to the topics and methods covered in this book. In Chapters 11 and 12, several notebooks are presented with the system Mathematica by Wolfram in the contexts of the packages DiscreteMath (Skiena) and Cooperative (Carter). There will also be found in the book several research projects. These are intended to offer new ideas that the reader should consider with caution. This book will be of interest to graduate students with some experience in game theory or mathematical programming and professional researchers in game theory, operational research and its applications in economic theory, and the political and social sciences. In addition, it will be especially useful for professionals who are interested in models for understanding and managing conflicts: management and operational research scientists, political and military scientists, and professional negotiators.

In this book applications of cooperative game theory that arise from combinatorial optimization problems are described. It is well known that the mathematical modeling of various real-world decision-making situations gives rise to combinatorial optimization problems. For situations where more than one decision-maker is involved classical combinatorial optimization theory does not suffice and it is here that cooperative game theory can make an important contribution. If a group of decision-makers decide to undertake a project together in order to increase the total revenue or decrease the total costs, they face two problems. The first one is how to execute the project in an optimal way so as to increase revenue. The second one is how to divide the revenue attained among the participants. It is with this second problem that cooperative game theory can help. The solution concepts from cooperative game theory can be applied to arrive at revenue allocation schemes. In this book the type of problems described above are examined. Although the choice of topics is application-driven, it also discusses theoretical questions that arise from the situations that are studied. For all the games described attention will be paid to the appropriateness of several game-theoretic solution concepts in the particular contexts that are considered. The computation complexity of the game-theoretic solution concepts in the situation at hand will also be considered.

Based on lectures presented at the AMS Short Course on Combinatorial Games, held at the Joint Mathematics Meetings in Columbus in August 1990, the ten papers in this volume will provide readers with insight into this exciting field. Because the book requires very little background, it will likely find a wide audience that includes the amateur interested in playing games, the undergraduate looking for a new area of study, instructors seeking a refreshing area in which to give new courses at both the undergraduate and graduate levels, and graduate students looking for a variety of research topics.

This volume is dedicated to the work of three leading mathematicians in combinatoric game theory, Elwyn Berlekamp, John Conway, and Richard Guy and includes 20 contributions from colleagues reflecting on their work.

This second edition of Lessons in Play reorganizes the presentation of the popular original text in combinatorial game theory to make it even more widely accessible. Starting with a focus on the essential concepts and applications, it then moves on to more technical material. Still written in a textbook style with supporting evidence and proofs, the authors add many more exercises and examples and implement a two-step approach for some aspects of the material involving an initial introduction, examples, and basic results to be followed later by more detail and abstract results. Features Employs a widely accessible style to the explanation of combinatorial game theory Contains multiple case studies Expands further directions and applications of the field Includes a complete rewrite of CGSuite material