Evolutionary Game Theory.

Ortmann, Andreas

What is evolutionary game theory? It is an attempt to discard the heroic knowledge and rationality assumptions of standard noncooperative (eductive) game theory. It gets its name from the use of evolutionary models which enable us to track the distribution of actions in repeated encounter games. Essentially, evolutionary models are systems of deterministic or stochastic differential or difference equations (dynamical systems), which are derived from the payoff matrix of the same constituent game that eductive game theory takes as point of departure.

Over the past seven or eight years, evolutionary game theory has gained rapid acceptance among game theorists. Its major selling point has been the seeming ability to explain experimental results of repeated encounter games with multiple equilibria.

In his new book, Weibull focuses on continuous evolutionary models of the deterministic variety. For this set of models he offers, often drawing on his own work and that of his collaborators, a well-written, mathematically elegant, and self-contained treatment that has a good chance indeed of becoming a staple of the literature.

The book is divided into six chapters. Chapter One reviews key concepts and results from eductive game theory, such as mixed strategy payoff functions and their geometric interpretation, Nash equilibria, and refinements such as perfection, properness, strict perfection, essentiality, and strategic stability. It also introduces symmetric two-player games which are the main menu of chapters Two through Four, and a nice classification of symmetric 2 x 2 games which is employed throughout the remainder of the text.

Chapter Two discusses evolutionary stability criteria. The notion of an evolutionarily stable strategy (ESS) was at the center of Maynard Smith's exploration of the applicability of game theory to biology [1] and is a refinement of the Nash equilibrium concept, which it augments by a robustness condition that prevents mutants from upsetting the prevailing equilibrium. The ESS is quasi-dynamic in nature, i.e., while it is concerned with evolution, it does not model the evolutionary process explicitly. Chapter Two discusses the ESS and a number of other evolutionary stability criteria (evolutionarily stable sets, equilibrium evolutionarily stable sets); it also offers an intriguing discussion of "cheap talk" (costless pre-play communication) and its ability to produce Pareto efficient outcomes.

Chapters Three and Four present replicator and other selection dynamics, respectively, for random matchings of pairs of individuals who are drawn from a large population; their interactions are modelled as a symmetric two-player game in normal form. These chapters include discussions of such important issues as the long-run survival of weakly, strictly, and iteratively strictly dominated strategies, and the mapping between stationary states of the dynamical system on the one hand, and aggregate Nash equilibrium behavior (the static solution concepts discussed in Chapter One) and evolutionary stability criteria (the quasidynamic solution concepts discussed in Chapter Two) on the other. Throughout Chapter Four, Weibull discusses the relation between replicator and other selection dynamics.

Chapter Five is nearly twice as long as the other chapters individually, and extends the simplistic set-up of symmetric pairwise random matchings in one-population models to introduce (symmetric and asymmetric) multipopulation models. Such models are of obvious interest to economists. In many environments of strategic uncertainty economic agents have distinct roles. A classic example is the roles that buyers and sellers play in markets where goods and services of adjustable quality are traded. Unfortunately, the generalization from one-population to multipopulation models is not straightforward. For example, while mixed strategy Nash equilibria are (strategically) stable in one-population models, they tend to be unstable in the associated multipopulation models. Using the classification of symmetric 2 x 2 games he introduced earlier, Weibull illustrates this result for his running examples. Overall, Chapter Five makes it very clear that the application of evolutionary models is an art as much as it is a science, and that there are important pitfalls to navigate if one is to move from one-population to multipopulation models.

The last chapter, titled "Elements of the Theory of Ordinary Differential Equations," is an appendix of sorts for those that need a refresher on ODEs. The chapter introduces the three general properties necessary for the solution of ODEs; it also discusses invariance, stationarity, and stability concepts including the direct Lyapunov method. Weibull assumes that the reader has "familiarity with standard notions in mathematics (basic set theory, topology, and calculus) at about the level achieved after the first year of graduate studies in economics" [p. xiv] - a quite appropriate warning. His book is neither an introductory text nor beach-chair reading.

I mentioned at the beginning of this review that evolutionary game theory has gained rapid acceptance among game theorists, since it appears to explain experimental results of repeated encounter games with multiple equilibria. I also mentioned that Weibull focuses on continuous models of the deterministic variety, which are derived from the payoff matrix of the same constituent game that eductive game theory departs from.

Truth be told, the derivation of a dynamical system from the payoff matrix of the underlying game is tricky business. Relevant issues include the choice of the appropriate model function (replicator, rate of change, etc.) which requires important assumptions about behavior and the matching protocol; they also include the choice of a dynamic, i.e., the particular way (continuous or discrete, deterministic or stochastic) the dynamical system is assumed to evolve from one state of the world to another. These choices affect - often dramatically - the fixed points, limit cycles, etc. of the dynamical system to be constructed. Because evolutionary models are increasingly being used to explain experimental results, additional important modelling issues arise which are germane to the use of evolutionary models in experimental economics. For example, experiments typically involve small numbers of subjects; implicit nearly always in evolutionary models is the assumption of a large population (or large populations) of interacting agents. Because experiments typically involve small numbers of subjects, their idiosyncracies do not easily cancel out; evolutionary models often assume away heterogeneity of that type. Also, experiments are by their very nature discrete; evolutionary models often come in continuous form.

The author's focus on continuous evolutionary models of the deterministic variety means that, with the exception of some cursory discussion of the consequences of switching from continuous to discrete formulations of dynamical systems, these modelling issues do not get addressed. This is partially a choice Weibull made, but it also reflects the progress evolutionary game theory had made at the time of the writing of the book. The problem is that the modelling issues enumerated above open the door for "model mining." If evolutionary game theory wants to become truly applicable to the domain that helped fuel its rapid growth, it eventually will have to customize its models to the specifics of experimental practices.

In sum, Evolutionary Game Theory is an excellent compendium, as of early 1994, of a selected set of core results in evolutionary game theory. Since then, evolutionary game theory has made significant progress. There is now a wealth of stochastic evolutionary models, models incorporating heterogeneous beliefs, and models that discard the assumption of random matching. Researchers have also discovered the value of computer simulations to explore the robustness of their models. The readability and elegance of the present version of this book makes the reader hope that Weibull ultimately will expand it. In the meantime, kudos to the author for a fine job indeed.

Andreas Ortmann Bowdoin College and Max Planck Institute for Psychological Research

Reference

1. Maynard Smith, John. Evolution and the Theory of Games. Cambridge: Cambridge University Press, 1982.