Evolutionary Games and Equilibrium Selection.

By Larry Samuelson.

Cambridge, MA: The MIT Press, 1997. Pp. ix, 309. $40.00.

This book is an overview of the recent economics literature on evolutionary games with a special emphasis on Larry Samuelson's own, wide-ranging research. Although most of the results have already appeared in journal articles, they are here put together and elaborated upon in a very convincing and coherent way. In fact, one of the nicest features of this book is that it gives us a clear picture of the underlying motivation and views of a leading researcher on evolutionary game theory. The book should be of great interest to anyone interested in this area and, more in general, game theory.

The evolutionary approach to games has been one of the hot topics of the 1990s in economic theory. Stemming from the application of noncooperative game theory to biology, evolutionary game theory postulates that players (e.g., genes or animals) are programmed to play a given strategy. Under the assumption that strategies yielding higher payoffs reproduce faster (payoffs measure reproductive fitness), biologists have studied the evolution of populations of players over time. In particular, they have focused on the study of strategies that are stable in the sense of surviving in the long run. Interestingly, only strategies corresponding to a Nash equilibrium, the fundamental concept of noncooperative game theory, have been found to be evolutionarily stable. However, not all Nash equilibria are evolutionarily stable.

By replacing the assumption of faster biological reproduction of the fittest with adaptation by players towards strategies yielding higher payoffs, economists have looked at evolutionary game theory as providing a framework for modelling boundedly rational behavior. Two main questions have attracted much attention. First, to what extent do evolutionary models yield outcomes in which agents appear to act rationally? Second, to what extent do evolutionary models help in selecting among Nash equilibria? Providing answers to these two questions is the main running theme of this book.

The first three chapters lay the foundations upon which the book is built. Chapter 1 is a very well-written introduction in which Samuelson explains his research philosophy and general views on game theory. Chapter 2 contains a brief survey of the evolutionary approach to games. The notion of an evolutionarily stable strategy (and some of its variants) is introduced first, and its relationship with standard equilibrium concepts is discussed. Then the chapter looks at the basic dynamic model in evolutionary biology, the replicator dynamics, and discusses its relationship with Nash equilibrium and evolutionarily stable strategies.

Chapter 3 introduces an explicit model of boundedly rational behavior. In such a model, players have inertia and tend to adopt the same strategy they have chosen in the past; players occasionally (with small probability) compare realized payoffs to an aspiration level, and abandon their present strategy if its realized payoff falls below the aspiration level. This chapter shows how the stochastic dynamics associated with this model can be well approximated, along any finite time path, by the deterministic replicator dynamics. Thus, a model originally introduced in the biological literature is validated as a reasonable way of representing boundedly rational behavior.

Chapter 3 also introduces the distinction between the long run and the ultralong run. The long run refers to the steady states of the deterministic dynamics that approximate the behavior of the stochastic system over arbitrarily long, but finite, periods of time. The ultralong run is described by the stationary distribution of the stochastic process. When the stochastic system is governed by small probability events, it may take an arbitrarily long time to converge to the stationary distribution; that is, the system may spend a long period of time in, or close to, states to which the stationary distribution of the process assigns probability zero. These states correspond to equilibria of the approximating deterministic system, and their relative permanence is captured in a long-run analysis.

Chapters 4, 5, and 6 study the long run, that is, systems of deterministic differential equations. Chapter 4 shows that, while the evolutionary pressure embodied in a wide class of deterministic dynamics eliminates strictly dominated strategies, it does not eliminate weakly dominated strategies. This result is robust to the introduction of drift, additional dynamic pressure that moves the system in the interior of the state space. Drift makes sure that strategies that have disappeared from the population can always reappear. Chapter 6 expands on the study of perturbed deterministic dynamics by looking at drift in a more general way.

Chapter 5 studies the ultimatum game. This is a two-moves, two-players game in which the first mover proposes how to divide a sum of money and the second mover can only accept or reject the proposed division. The game has a unique subgame-perfect equilibrium in which the first mover gets all the money (accepting any offer is a weakly dominant strategy for the second mover). The ultimatum game has received great attention in experimental studies. The evidence points to outcomes that differ from the subgame-perfect equilibrium; the first mover typically offers a sizeable share of the money to her opponent, and the second mover frequently rejects stingy offers, thus leaving money on the table. Samuelson argues that evolutionary models direct attention to Nash equilibria that are not subgame-perfect. Here the argument is a little stretched; the prediction of the evolutionary model discussed in this chapter seems hardly consistent with the experimental evidence. Others have stressed that players' preferences and motivations may not be well captured by simply looking at their monetary rewards. This seems especially plausible in a typical experimental set-up, where the monetary stakes are not terribly high. I personally sympathize with this latter approach, but I must acknowledge that Samuelson has a point in claiming that it explains too much; all sorts of behaviors can be justified by simply appealing to the right preferences.

Chapters 7, 8, and 9 study the ultralong run. Chapter 7 summarizes the techniques for analyzing the stationary distribution of finite state space Markov chains due to Freidlin and Wentzell (1984). It then confirms results from Chapter 4 concerning the evolutionary resilience of weakly dominated strategies by showing that weakly dominated strategies may have positive mass in the stationary distribution. Chapter 8 extends the analysis to simple extensive form games. First, it shows that the evolutionary process directs attention to the self-confirming equilibria. Then it shows that forward induction equilibrium outcomes of some simple games have positive mass in the stationary distribution.

In a model for selecting among strict Nash equilibria of 2 x 2 coordination games, Kandori, Mailath, and Rob (1993) and Young (1993) showed that the stationary distribution puts positive mass only on the risk dominant equilibrium. Chapter 9 looks at a variant of this model in which players' adaptation toward best replies is noisy. It shows that in such a case the stationary distribution sometimes puts all mass on the risk dominant equilibrium and sometimes on the Pareto dominant equilibrium. Chapter 10 contains some brief conclusions.

In the end, it seems that Samuelson's answers to whether evolutionary models yield outcomes in which agents appear to act rationally and whether they help to build a general theory of how to select among Nash equilibria are "not completely" and "not a lot." I do not view these as discouraging answers. In my opinion, the most important feature of evolutionary game theory is that it provides economists and, more in general, social scientists with a way to model boundedly rational behavior and to study its implications.

References

Freidlin, M. I., and A.D. Wentzell. 1984. Random perturbations of dynamical systems. New York: Springer-Verlag.

Kandori, M., G. J. Mailath, and R. Rob. 1993. Learning, mutation, and long-run equilibria in games. Econometrica 61: 29-56.

Young, P. 1993. The evolution of conventions. Econometrica 61:57-84.

Claudio Mezzetti University of North Carolina