Review of manuscript “Introduction to the analysis of many agent systems with
competition and cooperation” by V.N. Kolokoltsov and O.A. Malafeyev.
Reviewer: Geert Jan Olsder
As clearly stated in the Preface, the book essentially consists of two parts;
- Chapters 1-8 (pages 1-109) give an introduction of game theory to the novice.
Though some mathematics is used, it is never more than at high school level.
Usually common sense suffices to understand and follow the more analytical parts
of (or: the analytical reasoning in) the text.
- Chapters 9-11 (pages 110-253) require a mathematical background, in general of
(at least) B.Sc. level.
The book shows that game theory --in more modern terminology, “multi-agent decision
making”-- is omnipresent. A unique feature of the book is that this omnipresence is made
clear by brief, mostly informal, excursions to various fields, such as literature, philosophy,
history, politics, and folk stories. I enjoyed reading this embedding in a broader setting.
In much more depth, applications to –specifically-- economics, biology and finance are
given. The book covers a broad range of subjects, broader than what one sees in other
game theory books. Such subjects, somewhat out the main stream, are for instance
auctions, hawks and doves in population biology, social choice theory, the Braess
paradox, and even quantum games (this is the first time that I have seen this in a book).
Of course, the more traditional subjects, such as the prisoners’ dilemma, the battle of the
sesxes, and duopoly models are not missing.
In the first part of the book, first certain ideas and features of phenomena of possible
conflicts are given by means of examples, which are subsequently given a more rigorous
basis. It shows how many “best solutions” are possible, how rich the set of “possible best
solutions” is. Various definitions of “best” are possible and it very much depends on the
specific application, and the players or agents concerned, what the outcome of a game
will be. This all is clearly explained; the reader (newcomer to the field) will feel
immersed in this rich field of decision making. It will show to him/her the arguments as
to why our society is so complex, why so many intricacies exist. The book will give a
better understanding of this “complexity”. This first part of the book is interesting reading
for readers with different backgrounds, such as businessmen, managers, politicians, and,
of course, economists.
In the second part of the book one should distinguish chapters 9 and 10 from chapter 11.
For a proper understanding of Chapters 9 and 10 one needs some knowledge of ordinary
differential equations and probability theory. Chapter 9 deals with the more mathematical
fundaments and Chapter 10 deals with certain applications. The application on rainbow
options appears for the first time in a book (I believe) and is an example of how game
theory has gotten a clear role in financial analysis.
As the title of Chapter 11 already suggests, more advanced analysis is necessary here to
follow the text. The subjects are partly determined by the scientific interests of the
authors (such as games in the context of idempotent mathematics). Chapter 11
specifically could be the starting point for original research. Many references are given to
this extent.
Many examples are given throughout the text, as well as some exercises (sometimes
solutions are provided) on which the reader can test his/her knowledge of the subject.
As already said, game theory is omnipresent and therefore it is a huge subject. Therefore
not all topics can be covered (not in any book). Subjects only briefly touched upon, or not
treated at all, in the book under review, are pursuit-evasion games, Stackelberg games
(and their ramifications), open- versus closed-loop solutions in dynamic games, repeated
games. But, as should be clear, the authors are not at all to be blamed for this: any
author(s) of a game theory book faces the dilemma as to what to include and what not.