Why a book on Evolutionary Game Theory? Why this book? I presume that most game theorists interested in evolution will already own this book as it is unique in its coverage of topics and also well written. Therefore I have directed my short review to the more applied social scientist who is less interested in game theory for its own sake but more interested in, for example, social simulation.
To this target audience (but also to game theorists and others with an interest in game theory) I can warmly recommend the book as it is a well written technical compendium of recent research which is comprehensive in its chosen area: evolutionary stability and its relationship to evolutionary dynamics.
Before expounding this view in greater detail, I offer the reader some thoughts on the focus of evolutionary game theory and its possible connections to social simulation.
The review is organised as follows. Two short sections motivate the evolutionary approach to game theory and tentatively suggest what an applied social scientist might gain from studying this field. The next two sections briefly discuss the topics covered in the book as well as some important topics which the author chose to omit. The last section concludes with an overall appraisal of the book.
For a surprisingly long period of time, game theorists forgot about Nash's statistical population interpretation of his equilibrium concept (presented in his unpublished doctoral thesis). Instead, they devised ever more sophisticated (normatively motivated) theories or definitions of rational behaviour. Unsurprisingly (with the benefit of hindsight) this approach fails on two accounts. Firstly, the rationality assumptions became so stringent and demanding that the predictive (positive) value of the theory is doubtful. Secondly, even in a purely normative framework, there has been little success solving the equilibrium selection problem. The 1980s saw a crucial new development on this front with the publication of John Maynard Smith's seminal work Evolution and the Theory of Games. Maynard Smith envisaged randomly drawn members from populations of pre-programmed players meeting and playing strategic games. A biological (or social) selection process would then change the proportions of the different populations of pre-programmed "types". The concept of an evolutionary stable strategy (ESS) was then developed to describe fixed points in such selection processes. At the same time, dynamic concepts were perfected which explicitly modelled the evolution of such populations.
The relationship between these two approaches and their comparison to classical concepts of game theory is the focus of Weibull's book.
In my view, one of the great benefits of evolutionary game theory is that it has shifted the focus away from ex-post theories - an equilibrium is a point from which one does not move but nobody explains how one gets there in the first place - to dynamical theories which explicitly model how one gets to where one is. The painful lesson from this shift in approach is that one cannot expect to obtain general theories in which historical and institutional factors can be ignored. On the other hand, classical game theory is still the main workhorse of economics and not without reason. Evolutionary game theory has still not developed far enough to provide applied researchers with a sufficiently sophisticated enough toolset to analyse their problems. Experiments have shown that people do not appear to use Kohlberg/Mertens strategic stability (or other such strong normative concepts) as their primary techniques for analysing strategic situations. Instead, behaviour is to a large extent conditioned by rules of thumb which have evolved in society over longer periods of time. However people do not function totally automatically either. If the carrot in front of their noses is sufficiently large, made of platinum and they have time to reason, many people will break with this conditioning and display surprisingly rational behaviour. The simple behavioural models underlying current evolutionary and learning theory do not do justice to these surprising "bursts of reason". In short, I believe that evolutionary game theory is here to stay (in one form or other) but still has a very long way to go before it is applicable to a wide range of important questions.
In my personal (albeit biased) view, the best simulations are those which just peek over the rim of theoretical understanding, displaying mechanisms about which one can still obtain causal intuitions. If simulations are produced for models which are orders of magnitude more complicated than those susceptible to formal analysis then the causes underlying the results will be difficult to interpret (at least for theoretically biased people). Ideally, simulations should be made for models of which simplified versions can be analysed analytically. In such situations, the simulation results can extend the limited knowledge of formal theory while still retaining some of its intellectual rigour.
For example, the selection algorithm underlying most simple Genetic Algorithms works as follows: a new generation is generated by selecting members of the current generation in probabilistic proportion to the their fitness. (More technical details can be found in Goldberg 1989). It is straightforward to show that this stochastic process converges to deterministic replicator dynamics as the population size goes to infinity. Therefore analysing the behaviour of replicator dynamics for the underlying problem (or even static evolutionary stability conditions if a dynamic approach is not feasible) will give the researcher valuable information about the possible behaviour of his simulation algorithm. On the other hand, theoretical results for stochastic selection dynamics with high levels of mutation (as one must have in a simulation, lest one wait an infinite period of time for important mutations to crop up) are scarce, to say the least. Here simulations can help to suggest fruitful ways of pursuing interesting new theoretical results (even if these are of a non general nature and geared towards the specific question in mind).
In this context, I would direct the attention of the reader to the interesting discussions in sections 4.4.2 and 4.4.3 of the Weibull's book. These link replicator dynamics to imitation based behaviour. (These issues will be discussed further in the next section.)
The book's focus is on the classical setup in evolutionary game theory with large (infinite) populations in which players are matched to play a normal form game. Emphasis is put on evolutionary stability criteria like the classical ESS and their relationship to deterministic dynamics. The books cited in the next section have originated from (economic) game theorists' interests. These have evolved from learning models to large population settings and ultimately converged on biological concepts. By contrast, Weibull's approach is more geared towards taking biological concepts (or motivations) and looking at them from an (economic) game theorist's perspective. The frequent and helpful examples are kept simple and do not distract the reader from the underlying concepts. However, the cost of this is that part of the readership will miss applications more closely related to their own fields of interest.
Chapter 1 starts off with a short overview of the elements of non-cooperative game theory needed for what follows. (These include properties of Nash equilibria and normal form refinements.) As the author himself states, the treatment will most likely appear terse to a reader with little or no experience in game theory.
Chapter 2 introduces the classical stability criterion of the ESS, its weaker versions (neutral stability) and various set valued concepts. The chapter concludes with a discussion of the interesting topic of pre-play communication (cheap talk) and how evolution leads to efficiency in games with this feature.
Chapter 3 discusses continuous time replicator dynamics and standard results about the extinction of dominated strategies over time. The chapter continues by presenting the relationship between the stability properties of replicator dynamics and the static stability concepts discussed in the previous chapter.
Chapter 4 exemplifies the differences between discrete time and continuous time versions of replicator dynamics and discusses more general classes of dynamics such as monotone dynamics. In the very interesting sections 4.4.2 and 4.4.3, Weibull presents newer research on imitation models, which motivate replicator dynamics in terms of the imitative behavior of agents in large but finite populations. As discussed in the previous section, I believe this area to be of particular importance for comparing simulation analysis to theoretical results. Alas, Weibull restricts attention to his own joint research with Jonas Björnerstedt. Further relevant results can also be found in Schlag (1998) as well as Fudenberg and Levine (1998) and Samuelson (1997).
While chapters 2, 3 and 4 concentrate on single population models, in which members play a symmetric game, Chapter 5 extends the analysis to multi-population settings, that is, asymmetrical games in which each player is represented by a population.
Chapter 6 concludes the book with a brief review of the theory for ordinary differential equations.
Due to the huge existing (and rapidly growing) literature on evolutionary game theory and related fields, it would be a daunting (and extremely difficult) task to include all relevant topics. The author was wise enough not to sacrifice the clarity and lucidity of his presentation for a broader perspective (which would also necessitate the inclusion of additional technical material). However, there are fields which nicely complement the topics covered in the book. For the sake of the potential reader, I would like to describe two recent areas of development which the author chose not to cover - but which I believe to be of interest to social scientists involved in simulation.
Stochastic (Non Regular) Dynamics: Weibull not only focuses on deterministic dynamics but on regular dynamics. Regularity rules out mechanisms like the best response dynamics deeply cherished by many researchers. Parallel to large population evolutionary models, an interesting and fruitful field of research has emerged under the title of learning in games. Learning models tend to deal with behavioural strategies which are slightly more complicated than those used in evolutionary models. Their focus also tends to be learning in small population (two person) settings. However, part of this literature - pioneered by Young (1993a, 1993b) and Kandori, Mailath and Rob (1993) - also deals with large populations and provides a useful complement to the classical evolutionary approach which emerged from theoretical biology. However, the intersection between the two fields necessitates further mathematical apparatus, such as the use of perturbed Markov processes. One of the recurring results in this field is the selection of risk-dominant equilibria which contrasts with the broad tendency of evolutionary processes towards efficiency. Recent textbook references dealing with these and related issues include Samuelson (1997), Fudenberg and Levine (1998) and Young (1998).
Evolution in extensive form games: There have been several recent advances in analysing evolutionary processes in more complicated extensive form games. Amongst others, a notable contribution is recent research by Nöldeke and Samuelson, which is extensively discussed in Samuelson (1997). Samuelson's book also includes an important discussion of drift - an umbrella term for various perturbations of the selection process not specifically modelled - and shows how these can influence equilibrium selection. (See also Binmore and Samuelson 1995).
As mentioned before, the author's aim was not to provide a complete compendium of evolutionary game theory but to focus on an important section of the literature to which the author himself has made major contributions. The result is a lucid and rigorous technical coverage. In my opinion, an introductory course in game theory is necessary to appreciate the book in full. It would serve admirably as a classroom text for first year graduate students in economics. (While I have not personally used it in teaching, a colleague of mine has to his great satisfaction.) It also serves well as a reference book for researchers. Concepts are carefully explained and frequently accompanied by figures and examples. Mathematical rigour is of a high priority to the author and even the smallest claim is proved. Readers who manage to work through the book in all its depth will be rewarded with a thorough understanding of the underlying mathematical structures. Nonetheless, interested readers will find themselves delving into the original articles as well as the literature referred to in the previous section.
Given the book's focus I cannot find any serious omissions (save perhaps my comments regarding the absence of a more representative discussion of imitation games in chapter 4) . Nor would I suggest dropping any material. Complementing the book with one or more titles mentioned in the previous section is the ideal solution to the issue of coverage in my view.
In short, this book is a "must have" for anyone seriously interested in evolutionary game theory and an important reference book for social scientists doing evolutionary simulation.
BINMORE K. and L. Samuelson 1995. Evolutionary Drift and Equilibrium Selection, ELSE Working Paper, University College London.
FUDENBERG and D. Levine. 1998. The Theory of Learning in Games, The M.I.T. Press, Cambridge, MA.
GOLDBERG D. 1989. Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA.
KANDORI M., G. Mailath and R. Rob 1991. Learning, Mutation and Long-run Equilibria in Games, Econometrica, 61:29-56.
SAMUELSON L. 1997. Evolutionary Games and Equilibrium Selection, The M.I.T. Press, Cambridge, MA.
SCHLAG K. 1998. Why Imitate, and if so, How? Journal of Economic Theory, 78:130-156.
YOUNG P. 1993a. The Evolution of Conventions, Econometrica, 61:57-84.
YOUNG P. 1993b. An Evolutionary Model of Bargaining. Journal of Economic Theory, 59:145-168.
YOUNG P. 1998. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions, Princeton University Press, Princeton, NJ.
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