# Population Games and Evolutionary Dynamics

William H. Sandholm
Pages: 616
https://www.jstor.org/stable/j.ctt5hhbq5

1. Front Matter
(pp. i-iv)
(pp. v-xvi)
3. Series Foreword
(pp. xvii-xviii)
Ken Binmore

The MIT Press series on Economic Learning and Social Evolution reflects the widespread renewal of interest in the dynamics of human interaction. This issue has provided a broad community of economists, psychologists, biologists, anthropologists, and others with a sense of common purpose so strong that traditional interdisciplinary boundaries have begun to melt away.

Some of the books in the series will be works of theory. Others will be philosophical or conceptual in scope. Some will have an experimental or empirical focus. Some will be collections of papers with a common theme and a linking commentary. Others will have an expository...

4. Preface
(pp. xix-xxviii)
5. 1 Introduction
(pp. 1-18)

This book describes an approach to modeling recurring strategic interactions in large populations of small anonymous agents. The approach is built upon two basic elements. The first, called apopulation game, describes the strategic interaction that is to occur repeatedly. The second, called arevision protocol, specifies the myopic procedure that agents employ to decide when and how to choose new strategies. Starting with a population game and a revision protocol, one can derive dynamic processes, both deterministic and stochastic, that describe how the agents’ aggregate behavior changes over time. These processes are known asevolutionary game dynamics.

This introductory...

6. I Population Games
• 2 Population Games
(pp. 21-52)

The examples discussed in chapter 1 are among the many important strategic interactions exhibiting the following properties:

1.The number of agents is large.

2.Individual agents are small. Any one agent’s behavior has little or no effect on other agents’ payoffs.

3.Agents interact anonymously. Each agent’s payoffs only depend on opponents’ behavior through the distribution of their choices.

While these three properties are basic, two additional restrictions of a more technical nature will sharpen the focus on the environments of most interest here. For the distribution of opponents’ choices mentioned in property 3 to exist, there must be...

• 3 Potential Games, Stable Games, and Supermodular Games
(pp. 53-116)

Chapter 2 provided a general definition of population games and characterized their Nash equilibria geometrically. Still, since any continuous mapFfrom the state spaceXto Rndefines a population game, population games with even a moderate number of strategies can be difficult to analyze.

This chapter introduces three important classes of population games: potential games, stable games, and supermodular games. From an economic point of view, each definition places constraints on the sorts of externalities agents impose on one another through their choices in the game, and each encompasses a variety of interesting applications. From a mathematical point...

7. II Deterministic Evolutionary Dynamics
• 4 Revision Protocols and Evolutionary Dynamics
(pp. 119-138)

The theory of population games developed in the previous chapters provides a simple framework for describing strategic interactions among large numbers of agents. Having explored these games’ basic properties, we now turn to modeling the behavior of the agents who play them.

Traditionally, predictions of behavior in games are based on some notion of equilibrium, typically Nash equilibrium or some refinement thereof. These notions are founded on the assumption of equilibrium knowledge, which posits that each player correctly anticipateshowhis opponents will act. The equilibrium knowledge assumption is difficult to justify, and in contexts with large numbers of agents it is...

• 5 Deterministic Dynamics: Families and Properties
(pp. 139-176)

In the model of stochastic evolution introduced in chapter 4, a large society of agents recurrently play a population gameFby applying a revision protocol ρ. Through an informal appeal to the law of large numbers (made formal in chapter 10), we argued that aggregate behavior in the society can be described by a differential equation on the set of social statesX. This differential equation, themean dynamic, describes the inflows and outflows of agents to and from each available strategy:

$\dot x_i^p = \sum\limits_{j \in S^P} {x_j^p} \rho _{ji}^p\left( {{F^p}(x),{x^p}} \right) - x_j^p\sum\limits_{j \in S^P} {\rho _{ij}^p} \left( {{F^p}(x),{x^p}} \right)$.

Each Revision protocol ρ can be viewed as defining a map from population gamesFto...

• 6 Best Response and Projection Dynamics
(pp. 177-218)

This chapter continues the procession of deterministic evolutionary dynamics. In the first two sections, the step from payoff vector fields to evolutionary dynamics is traversed through a traditional game-theoretic approach, by employing best response correspondences and perturbed versions thereof. The third section follows a geometric approach, defining an evolutionary dynamic via closest point projections of payoff vectors.

Thebest response dynamicembodies the assumption that revising agents always switch to their current best response. Because the best response correspondence is discontinuous and multivalued, the basic properties of solution trajectories under the best response dynamic are quite different from those studied...

8. III Convergence and Nonconvergence of Deterministic Dynamics
• 7 Global Convergence of Evolutionary Dynamics
(pp. 221-270)

The preceding chapters introduced a variety of classes of evolutionary dynamics and exhibited their basic properties. Links were established between the rest points of each dynamic and the Nash equilibria of the underlying game, and were shown to be are valid regardless of the nature of the game at hand. This connection is expressed in its strongest form by dynamics satisfying Nash stationarity (NS), under which rest points and Nash equilibria coincide.

Still, once one specifies an explicitly dynamic model of behavior, the most natural approach to prediction is not to focus immediately on equilibrium points, but to determine where...

• 8 Local Stability under Evolutionary Dynamics
(pp. 271-318)

Chapter 7 analyzed classes of games in which evolutionary dynamics converge to equilibrium from all or most initial conditions. Although games from many applications lie in these classes, at least as many interesting games do not.

In cases where global convergence results are not available, one can turn instead to analyses of local stability. If a society somehow finds itself playing a particular equilibrium, how can one tell whether this equilibrium will persist in the face of occasional small disturbances in behavior? This chapter introduces a refinement of Nash equilibrium—the notion of anevolutionarily stable state(ESS)—that captures...

• 9 Nonconvergence of Evolutionary Dynamics
(pp. 319-364)

The analysis of global behavior of evolutionary dynamics in chapter 7 focused on combinations of games and dynamics generating global or almost global convergence to equilibrium. The analysis there demonstrated that global payoff structures—in particular, the structure captured in the definitions of potential, stable, and supermodular games—make compelling evolutionary justifications of the Nash prediction possible. Onthe other hand, when one moves beyond these classes of well-behaved games, it is not clear how often convergence will occur. This chapter investigates nonconvergence of evolutionary dynamics for games, describing a variety of environments in which cycling or chaos offer the best...

9. IV Stochastic Evolutionary Models
• 10 Stochastic Evolution and Deterministic Approximation
(pp. 367-396)

In Parts II and III of this book, we investigated the evolution of aggregate behavior under deterministic dynamics. We provided foundations for these dynamics in chapter 4. There we showed that any revision protocol ρ and population gameFdefine a mean dynamic$\dot x = {V_F}(x)$, a differential equation that describes expected motion under the stochastic process implicitly defined by ρ andF. The focus on this deterministic equation was justified by an informal appeal to a law of large numbers; since all randomness in the evolutionary model is idiosyncratic, it should be averaged away in the aggregate as long as the...

• 11 Stationary Distributions and Infinite-Horizon Behavior
(pp. 397-450)

The central result of chapter 10 established that over finite time spans, when the population size is sufficiently large, the stochastic evolutionary process$\left\{ {X_t^N} \right\}$follows a nearly deterministic path, closely shadowing a solution trajectory of the corresponding mean dynamic (M). But over longer time spans—that is, if we fix the population sizeNand consider the process at large values oft—the random nature of the process must assert itself. In particular, if the process is generated by afull support revision protocol, one that always assigns positive probabilities to transitions to all neighboring states in${X^N}$, then$\left\{ {X_t^N} \right\}$must...

• 12 Limiting Stationary Distributions and Stochastic Stability
(pp. 451-540)

Chapter 11 began the study of the infinite-horizon behavior of stochastic evolutionary processes generated by full support revision protocols. Any such process admits a unique stationary distribution, which describes the behavior of the process over very long time spans regardless of the initial population state. Even when the underlying game has multiple strict equilibria, the stationary distribution is often concentrated in the vicinity of just one of them if the noise level η is small or the population sizeNis large. In these cases, the population state so selected provides a unique prediction of infinite horizon play.

The analyses...

10. References
(pp. 541-564)
11. Notation Index
(pp. 565-574)
12. Index
(pp. 575-589)