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Population Games and Evolutionary Dynamics

时间:2010-07-04 19:17:54  来源:  作者:

by William H. Sandholm

To be published by MIT Press in 2010.


Overview

Population games provide a general model of strategic interactions among large numbers of agents; network congestion, multilateral externalities, and natural selection are among their many applications. As the direct assumption of equilibrium play seems difficult to justify in these games, behavior is most naturally modeled as a dynamic adjustment processes. To accomplish this, one can begin with an explicit stochastic description of how individual agents make decisions. When the number of agents is large enough and the time horizon of interest not too long, the evolution of aggregate behavior is well approximated by solutions to ordinary differential equations. We discuss various classes of population games in which these deterministic evolutionary dynamics lead to equilibrium play, and also consider simple examples in which more complicated limit behavior occurs. If one is interested in behavior over very long time spans, one studies the stochastic evolutionary processes directly, focusing on their ergodic and large deviations properties.

Many of the mathematical techniques used in evolutionary game theory are not part of the standard economics curriculum. To make evolution more accessible to graduate students in economics and other fields, the book includes detailed appendices on topics in multivariate calculus, convex analysis, dynamical systems, and stochastic processes that are essential for understanding evolutionary models. The book also includes many color figures, created using the shareware program Dynamo, to present population games and evolutionary dynamics in an intuitive, geometric fashion.

A detailed table of contents and Chapter 1 can be downloaded here. The remaining chapters have been removed from the website at the publisher's request.

 

Table of Contents

Chapter 1: Introduction

Population Games
Evolutionary Dynamics
Remarks on History, Motivation, and Interpretation
Part I: Population Games

Chapter 2: Population Games

Introduction
Population Games
Examples
The Geometry of Population Games and Nash Equilibria
Affine Spaces, Tangent Spaces, and Orthogonal Projections
Chapter 3: Potential Games, Stable Games, and Supermodular Games

Introduction
Full Potential Games
Potential Games
Stable Games
Supermodular Games
Multivariate Calculus
Affine Calculus
Part II: Deterministic Evolutionary Dynamics

Chapter 4: Revision Protocols and Evolutionary Dynamics

Introduction
Revision Protocols and Mean Dynamics
Examples
Evolutionary Dynamics
Ordinary Differential Equations
Chapter 5: Deterministic Dynamics: Families and Properties

Introduction
Principles for Evolutionary Modeling
Desiderata for Revision Protocols and Evolutionary Dynamics
Families of Evolutionary Dynamics
Imitative Dynamics
Excess Payoff Dynamics
Pairwise Comparison Dynamics
Multiple Revision Protocols and Combined Dynamics
Chapter 6: Best Response and Projection Dynamics

Introduction
The Best Response Dynamic
Perturbed Best Response Dynamics
The Projection Dynamic
Differential Inclusions
The Legendre Transform
Perturbed Optimization
Part III: Convergence and Nonconvergence

Chapter 7: Global Convergence of Evolutionary Dynamics

Introduction
Potential Games
Stable Games
Supermodular Games
Dominance Solvable Games
Limit and Stability Notions for Deterministic Dynamics
Stability Analysis via Lyapunov Functions
Cooperative Differential Equations
Chapter 8: Local Stability under Evolutionary Dynamics

Introduction
Non-Nash Rest Points of Imitative Dynamics
Local Stability in Potential Games
Evolutionarily Stable States
Local Stability via Lyapunov Functions
Linearization of Imitative Dynamics
Linearization of Perturbed Best Response Dynamics
Matrix Analysis
Linear Differential Equations
Linearization of Nonlinear Differential Equations
Chapter 9: Nonconvergence of Evolutionary Dynamics

Introduction
Conservative Properties of Evolutionary Dynamics
Games with Nonconvergent Evolutionary Dynamics
Chaotic Evolutionary Dynamics
Survival of Dominated Strategies
Three Classical Theorems on Nonconvergent Dynamics
Attractors and Continuation
Part IV: Stochastic Evolutionary Models

Chapter 10: Stochastic Evolution and Deterministic Approximation

Introduction
The Stochastic Evolutionary Process
Finite Horizon Deterministic Approximation
Extensions
The Exponential and Poisson Distributions

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