Quantitative Sociodynamics

Second Edition

Dirk Helbing

Quantitative Sociodynamics

Stochastic Methods and Models of SocialInteraction Processes

Second Edition

123

Prof. Dr. Dirk HelbingETH ZurichSwiss Federal Institute of TechnologyChair of Sociology, in particular of Modeling

and SimulationCLU E 1Clausiusstrasse 508092 ZurichSwitzerlanddhelbing@ethz.ch

ISBN 978-3-642-11545-5 e-ISBN 978-3-642-11546-2DOI 10.1007/978-3-642-11546-2Springer Heidelberg Dordrecht London New York

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Preface

When I wrote the book Quantitative Sociodynamics, it was an early attempt tomake methods from statistical physics and complex systems theory fruitful for themodeling and understanding of social phenomena. Unfortunately, the first editionappeared at a quite prohibitive price. This was one reason to make these chaptersavailable again by a new edition. The other reason is that, in the meantime, many ofthe methods discussed in this book are more and more used in a variety of differentfields. Among the ideas worked out in this book are:

• a statistical theory of binary social interactions,1

• a mathematical formulation of social field theory, which is the basis of socialforce models,2

• a microscopic foundation of evolutionary game theory, based on what is knowntoday as ‘proportional imitation rule’, a stochastic treatment of interactions inevolutionary game theory, and a model for the self-organization of behavioralconventions in a coordination game.3

It, therefore, appeared reasonable to make this book available again, but at a moreaffordable price. To keep its original character, the translation of this book, which

1 D. Helbing, Interrelations between stochastic equations for systems with pair interactions. Phys-ica A 181, 29–52 (1992); D. Helbing, Boltzmann-like and Boltzmann-Fokker-Planck equations asa foundation of behavioral models. Physica A 196, 546–573 (1993).2 D. Helbing, Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behav-ioral models. Physica A 196, 546–573 (1993); D. Helbing, A mathematical model for the behaviorof individuals in a social field. Journal of Mathematical Sociology 19(3), 189–219 (1994); D. Hel-bing and P. Molnár, Social force model for pedestrian dynamics. Physical Review E 51, 4282–4286(1995).3 D. Helbing, A mathematical model for behavioral changes by pair interactions. Pages 330–348in: G. Haag, U. Mueller, and K. G. Troitzsch (eds.) Economic Evolution and Demographic Change.Formal Models in Social Sciences (Springer, Berlin, 1992); D. Helbing (1996) A stochastic behav-ioral model and a ‘microscopic’ foundation of evolutionary game theory. Theory and Decision40, 149–179; D. Helbing (1998) Microscopic foundation of stochastic game dynamical equations.Pages 211–224 in: W. Leinfellner and E. Köhler (eds.) Game Theory, Experience, Rationality(Kluwer Academic, Dordrecht); D. Helbing (1991) A mathematical model for the behavior ofpedestrians. Behavioral Science 36, 298–310 (1991).

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is based on my PhD thesis, has not been changed. For readers whose excitementfor mathematical details is limited, I would suggest to start reading the ‘Introduc-tion and Summary’. Chapter 2 offers a synopsis of the technical parts. The furtherchapters elaborate mathematical methods that are important to capture typical fea-tures of socio-economics systems such as time-dependence, non-linear interaction,and randomness. These are essential prerequisites to model social and economicself-organization phenomena.

Finally, I would like to thank Peter Felten and Pratik Mukerji for supporting thepreparation of this book.

Zurich, September 2010 Dirk Helbing

Preface of the First Edition

This book presents various new and powerful methods for a quantitative anddynamic description of social phenomena based on models for behavioural changesdue to individual interaction processes. Because of the enormous complexity ofsocial systems, models of this kind seemed to be impossible for a long time. How-ever, during the recent years very general strategies have been developed for themodelling of complex systems that consist of many coupled subsystems. Thesemainly stem from statistical physics (which delineates fluctuation affected processesby stochastic methods), from synergetics (which describes phase transitions, i.e.self-organization phenomena of non-linearly interacting elements), and from chaostheory (which allows an understanding of the unpredictability and sensitivity ofmany non-linear systems of equations).

Originally, these fascinating concepts have been developed in physics and math-ematics. Nevertheless, they have often proved their explanatory power in variousother fields like chemistry, biology, and economics. Meanwhile they also receive anincreasing attention in the social sciences. However, most of the related work in thesocial sciences is still carried out on a qualitative level though in complex systemssome phenomena can only be adequately understood by means of a mathematicaldescription.

Up to now, there is not very much literature available that deals with a quanti-tative theory of social processes. Apart from the approaches of game theory, pio-neering models have been developed by WEIDLICH and HAAG concerning sponta-neous behavioural changes due to indirect interactions [285, 289, 292]. The workpresented in this book was stimulated by both approaches which first seemed to beincompatible. However, the main focus was put on the concept of direct pair inter-actions. This was first applied to the description of dynamic pedestrian behaviour[123, 125] but it can be extended to the delineation of a variety of other socialprocesses, e.g. of opinion formation.

It turned out that the concepts of indirect interactions and pair interactions canbe unified in one general model. Surprisingly, by incorporating decision theoret-ical approaches a very fundamental model resulted which seems to open newperspectives in the social sciences. It includes as special cases many establishedmodels, e.g. the logistic equation for limited growth processes, the gravity modelfor exchange processes, some diffusion models for the spreading of information,

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the game dynamical equations for cooperation and competition processes, andLEWIN’s social field theory that describes behavioural changes by dynamic forcefields. Therefore, this model is well-founded and could be viewed as some kind ofmetatheory. In addition, it implies many new results. Consequently, in the courseof time a large amount of material was accumulated concerning the foundation,generalization, and unification of behavioural models most of which is includedin this book. Except for the omitted chapter on pedestrian behaviour, this book isessentially the translation of an expanded version of my PhD thesis [132] which haswon two research prizes. It includes new results and many fruits of discussions Ihad in connection with presentations at a number of international conferences (e.g.the STATPHYS 18 in Berlin, 1992, and the XIIIth World Congress of Sociology inBielefeld, 1994).

‘Quantitative Sociodynamics’ is directed to the following readership: First, tonatural scientists, especially physicists and mathematicians who are interested in acomprehensive overview of stochastic methods and/or their application to interdis-ciplinary topics. Second, to social scientists who are engaged in mathematical soci-ology, decision theory, rational choice approaches, or general system theory. More-over, this book supplies a number of relevant methods and models for economists,regional scientists, demographers, opinion pollsters, and market researchers.

It is assumed that the reader possesses basic mathematical knowledge in calcu-lus and linear algebra. Readers who are not interested in mathematical details maycontinue with Part II after having read the different introductions preceding it. More-over, those who are interested in the basic ideas only may skip the formulas duringreading. The underlying terms, principles, or equations can always be looked up inPart I if necessary. An additional help is provided by the detailled ‘Index’ and the‘List of Symbols’ which contains supplementary information about mathematicaldefinitions and relations. Readers interested in an introduction to calculus, linearalgebra, and complex analysis can, for example, consult some of the References[20, 140, 164, 165, 246, 266]. A more detailled discussion of stochastic methods,synergetics, and chaos theory can be found in [84, 106, 107, 135, 151, 183, 254].

Here I want to thank W. WEIDLICH for teaching me his fascinating ideas, forhis encouragement, his numerous useful comments, and the freedom he gave meconcerning my work. Moreover, I am grateful to A. DIEKMANN, W. EBELING,N. EMPACHER, M. EIGEN, G. HAAG, H. HAKEN, I. PRIGOGINE, R. REINER,P. SCHUSTER, F. SCHWEITZER, K. TROITZSCH, and many others for stimulatingdiscussions and/or the inspiration by their work. I am very much indebted to R.CALEK for his tireless engagement in connection with the translation of the Germanmanuscript and to A. DRALLE as well as R. Helbing for their many useful advicesconcerning the final corrections. Moreover, I would like to thank M. SCHANZ

for his help regarding the numerical calculation of the LIAPUNOV exponents inSect. 10.3.4. Finally, I am pleased about the support, motivation, and understandingof my friends, relatives, and collegues.

Stuttgart, 1995 Dirk Helbing

Contents

1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantitative Models in the Social Sciences . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The Logistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 The Gravity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 The Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.5 Decision Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.6 Final Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 How to Describe Social Processes in a Mathematical Way . . . . . . . . 61.2.1 Statistical Physics and Stochastic Methods . . . . . . . . . . . . . 71.2.2 Non-linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Dynamic Decision Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Modelling Dynamic Decision Behavior . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Questioning Transitive Decisions and Homo Economicus 182.2.2 Probabilistic Decision Theories . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Are Decisions Phase Transitions? . . . . . . . . . . . . . . . . . . . . . 232.2.4 Fast and Slow Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.5 Complete and Incomplete Decisions . . . . . . . . . . . . . . . . . . 252.2.6 The Red-Bus-Blue-Bus Problem . . . . . . . . . . . . . . . . . . . . . 262.2.7 The Freedom of Decision-Making . . . . . . . . . . . . . . . . . . . . 272.2.8 Master Equation Description of Dynamic Decision

Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.9 Mean Field Approach and Boltzmann Equation . . . . . . . . . 292.2.10 Specification of the Transition Rates

of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Fields of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 The Logistic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.2 The Generalized Gravity Model and Its Application

to Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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2.3.3 Social Force Models and Opinion Formation . . . . . . . . . . . 332.3.4 The Game-Dynamical Equations . . . . . . . . . . . . . . . . . . . . . 352.3.5 Fashion Cycles and Deterministic Chaos . . . . . . . . . . . . . . . 372.3.6 Polarization, Mass Psychology, and Self-Organized

Behavioral Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Part I Stochastic Methods and Non-linear DynamicsOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Master Equation in State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Derivation from the MARKOV Property . . . . . . . . . . . . . . . . 523.2.2 External Influences (Disturbances) . . . . . . . . . . . . . . . . . . . . 533.2.3 Internal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.4 Derivation from Quantum Mechanics . . . . . . . . . . . . . . . . . 57

3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3.2 Non-negativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.3 The LIOUVILLE Representation . . . . . . . . . . . . . . . . . . . . . . . 653.3.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.5 Convergence to the Stationary Solution . . . . . . . . . . . . . . . . 67

3.4 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.1 Stationary Solution and Detailed Balance . . . . . . . . . . . . . . 683.4.2 Time-Dependent Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4.3 ‘Path Integral’ Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Mean Value and Covariance Equations . . . . . . . . . . . . . . . . . . . . . . . . 79

4 BOLTZMANN-Like Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Subdivision into Several Types of Subsystems . . . . . . . . . . . . . . . . . . 874.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4.1 Non-negativity and Normalization . . . . . . . . . . . . . . . . . . . . 884.4.2 The Gaskinetic BOLTZMANN Equation . . . . . . . . . . . . . . . . . 884.4.3 The H-Theorem for the Gaskinetic BOLTZMANN Equation 914.4.4 Solution of the Gaskinetic BOLTZMANN Equation . . . . . . . 94

4.5 Comparison of Spontaneous Transitions and Direct Interactions . . . 954.5.1 Transitions Induced by Interactions . . . . . . . . . . . . . . . . . . . 954.5.2 Exponential Function and Logistic Equation . . . . . . . . . . . . 964.5.3 Stationary and Oscillatory Solutions . . . . . . . . . . . . . . . . . . 97

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5 Master Equation in Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Transitions in Configuration Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.1 Spontaneous Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.2 Pair Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.3 Mean Value and Covariance Equations . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 Corrections and Higher Order Interactions . . . . . . . . . . . . . . . . . . . . . 1075.5 Indirect Interactions and Mean Field Approaches . . . . . . . . . . . . . . . 1115.6 Comparison of Direct and Indirect Interactions . . . . . . . . . . . . . . . . . 111

5.6.1 Differences Concerning the Covariance Equations . . . . . . . 1115.6.2 Differences Concerning the Mean Value Equations . . . . . . 112

6 The FOKKER-PLANCK Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.1 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.3.3 The LIOUVILLE Representation . . . . . . . . . . . . . . . . . . . . . . . 1216.3.4 Non-negativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.5 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.3.6 Convergence to the Stationary Solution . . . . . . . . . . . . . . . . 121

6.4 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4.1 Stationary Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4.2 Path Integral Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4.3 Interrelation with the SCHRÖDINGER Equation . . . . . . . . . . 124

6.5 Mean Value and Covariance Equations . . . . . . . . . . . . . . . . . . . . . . . . 1256.5.1 Interpretation of the Jump Moments . . . . . . . . . . . . . . . . . . 126

6.6 BOLTZMANN-FOKKER-PLANCK Equations . . . . . . . . . . . . . . . . . . . . 1276.6.1 Self-Consistent Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 LANGEVIN Equations and Non-linear Dynamics . . . . . . . . . . . . . . . . . . 1357.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.3 Escape Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.4 Phase Transitions, LIAPUNOV Exponents, and Critical

Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.5 Routes to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.5.1 RUELLE-TAKENS-NEWHOUSE Scenarioand LIAPUNOV Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.5.2 Period Doubling Scenario and Power Spectra . . . . . . . . . . . 147

Part II Quantitative Models of Social ProcessesOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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8 Problems and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.1.1 System and Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1.2 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1.3 Subpopulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1.4 Socioconfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568.1.5 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.2 Problems with Modelling Social Processes . . . . . . . . . . . . . . . . . . . . 1578.2.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2.2 Individuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.2.3 Stochasticity and Disturbances . . . . . . . . . . . . . . . . . . . . . . . 1598.2.4 Decisions and Freedom of Decision-Making . . . . . . . . . . . 1608.2.5 Experimental Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.2.6 Measurement of Behaviours . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9 Decision Theoretical Specification of the Transition Rates . . . . . . . . . . 1679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.2.1 The Multinomial Logit Model . . . . . . . . . . . . . . . . . . . . . . . 1689.2.2 Entropy Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1709.2.3 FECHNER’S Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1729.2.4 Utility and Distance Function . . . . . . . . . . . . . . . . . . . . . . . . 173

9.3 Pair Interaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1769.3.1 Special Applications in the Social Sciences . . . . . . . . . . . . 183

9.4 Properties of the Utility Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.4.1 Stationary Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.4.2 Contributions to the Utility Function . . . . . . . . . . . . . . . . . . 186

10 Opinion Formation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18710.2 Indirect Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10.2.1 A Period Doubling Route to Chaos . . . . . . . . . . . . . . . . . . . 19110.2.2 A RUELLE-TAKENS-NEWHOUSE Route to Chaos . . . . . . . . 191

10.3 Direct Pair Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19110.3.1 Kinds of Pair Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19210.3.2 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.3.3 Influence of the Interaction Frequencies . . . . . . . . . . . . . . . 20110.3.4 Period Doubling Scenarios and Chaos . . . . . . . . . . . . . . . . . 205

10.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.5 Spatial Spreading of Opinions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

10.5.1 Opinion Spreading by Diffusion . . . . . . . . . . . . . . . . . . . . . . 21810.5.2 Opinion Spreading by Telecommunication . . . . . . . . . . . . . 220

Contents xiii

11 Social Fields and Social Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22511.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22611.3 The Social Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

11.3.1 Comparison with LEWIN’s ‘Social Field Theory’ . . . . . . . . 23311.4 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

11.4.1 Imitative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.4.2 Avoidance Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

12 Evolutionary Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24712.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24712.2 Derivation of the Game Dynamical Equations . . . . . . . . . . . . . . . . . . 248

12.2.1 Payoff Matrix and Expected Success . . . . . . . . . . . . . . . . . . 24812.2.2 Customary Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24912.2.3 Fields of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24912.2.4 Derivation from the BOLTZMANN-Like Equations . . . . . . . 250

12.3 Properties of Game Dynamical Equations . . . . . . . . . . . . . . . . . . . . . 25312.3.1 Non-negativity and Normalization . . . . . . . . . . . . . . . . . . . . 25312.3.2 Formal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25312.3.3 Increase of the Average Expected Success

in Symmetrical Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25412.3.4 Invariant of Motion for Antisymmetrical Games . . . . . . . . 25612.3.5 Interrelation with the LOTKA-VOLTERRA Equations . . . . 25712.3.6 Limit Cycles and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

12.4 Stochastic Version of the Game Dynamical Equations . . . . . . . . . . . 26012.4.1 Self-Organization of Behavioural Conventions for the

Case of Two Equivalent Competing Strategies . . . . . . . . . 263

13 Determination of the Model Parameters from Empirical Data . . . . . . 27513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27513.2 The Case of Complete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27513.3 The Case of Incomplete Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

13.3.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28213.3.2 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

13.4 Migration in West Germany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28913.4.1 First Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29013.4.2 Second Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29213.4.3 Comparison of the WEIDLICH-HAAG Model

and the Generalized Gravity Model . . . . . . . . . . . . . . . . . . 29513.4.4 Third Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

13.5 Evaluation of Empirically Obtained Results . . . . . . . . . . . . . . . . . . . . 30013.5.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30013.5.2 Decomposition of the Utility Functions with Respect

to Explanatory Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30113.5.3 Prognoses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

xiv Contents

13.6 Examples for Decompositions of Utility Functions . . . . . . . . . . . . . . 30213.6.1 Purchase Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30213.6.2 Voting Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.6.3 Gaps in the Market and Foundations of New Parties . . . . . 308

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

List of Symbols

Notation and Conventions

x = (x1, . . . , xi , . . . , xn)tr ≡

⎛⎜⎜⎜⎜⎜⎜⎝

x1...

xi...

xn

⎞⎟⎟⎟⎟⎟⎟⎠

: Vector; as subscript or super-

script not printed in boldface! Vectors xi for which the equation∑i ci xi = 0 is only solvable for ci ≡ 0 are called linearly indepen-

dent, otherwise linearly dependent. A set {xi } of linearly indepen-dent vectors xi build a basis of a set M of vectors if each vectorsx ∈ M is representable as linear combination x = ∑

i ci xi withsuitable numbers ci . Given that the vectors xi of a basis are nor-malized and pairwise orthogonal to each other, the basis is calledorthonormal. Cf. [20], pp. 55ff., 139, 292ff.

xi = (x)i : Component i of the vector x .

(x1, . . . , xN )tr =

⎛⎜⎝

x1...

xN

⎞⎟⎠.

x · y =∑

i

xi yi : Scalar product (inner product), dot product) of the

vectors x and y. If we have x · y = 0, the vectors x and y areorthogonal, i.e. perpendicular to each other. Cf. [20], pp. 130ff.,290ff.

xv

xvi List of Symbols

M ≡(

Mi j

):=

⎛⎜⎜⎜⎜⎜⎜⎝

M11 · · · M1 j · · · M1n.... . .

......

Mi1 · · · Mi j · · · Min...

.... . .

...

Mn1 · · · Mnj · · · Mnn

⎞⎟⎟⎟⎟⎟⎟⎠

: (Quadratic) matrix.

Mi j = (M)i j : Matrix elements (components) of the matrix M .

M−1 ≡(

M−1i j

): Inverse of the matrix M . Important relations: M M−1 =

1 = M−1 M and [L M]−1 = M−1L−1.

M tr ≡(

M ji

): Transpose of the matrix M ≡

(Mi j

). Important relation:

[L M]tr = M trL tr.

L + M =(

Li j + Mi j

): Sum of two matrices.

L M =(∑

k

Lik Mkj

): Product of two matrices; normally we have

L M �= M L .

Mx =(∑

j

M1 j x j , . . . ,∑

j

Mi j x j , . . . ,∑

j

Mnj x j

)tr: Multiplication

of the matrix M with a vector x (which has the meaning of a trans-formation of vector x). The vectors xi which satisfy Mxi = λi xi

are called eigenvectors to the (possibly complex) eigenvalues λi .Provided that all eigenvalues λi are greater than zero, the matrix Mis called ‘positive definite’.

Mostly the n eigenvalues of the matrix M are different from eachother. In this case the representation M = S−1 DS is possible where

D =(λiδi j

)is a diagonal matrix and S a transformation matrix.

The columns of S−1 are eigenvectors of M . If the matrix M is sym-metrical (i.e. M = M tr), all its eigenvalues λi are real and S can bechosen as rotation matrix. In comparison with orthogonal matriceswhich satisfy the relation S−1 = Str, rotation matrices additionallyfulfil the relation |S| = 1. Cf. [20], pp. 147ff., 212ff., 226ff.

f (..) Function.

f (k)(x) kth approximation of the function f (x). For a convergent sequenceof functions we have lim

k→∞ f (k)(x) = f (x).

f∗(x), f ∗(x) Estimate for f (x).

fe(x), f e(x) Empirical value of the model quantity f (x).

List of Symbols xvii

p....(.|..) Transition probability. Important relations: p....(.|..) ≥ 0 and∑.

p....(.|..) = 1.

P .... (.) Probability. Important relations: P .... (.) ≥ 0 and∑.

P .... (.) = 1.

P(..|.) = P(.., .)/P(.) : Conditional probability. Important relations: P(..|.) ≥0 and

∑..

P(..|.) = 1. In the special case P(.., .) = P(..) · P(.) or

P(..|.) = P(..) the variables . and .. are called ‘statistically indepen-dent’.

w....(..|..) Transition rate. Important relation: w....(..|..) ≥ 0.

ν.... (.) Rate. Important relation: ν.... (.) ≥ 0.

A Operator of a quantum mechanical observable.

AXY = (A)XY = 〈X |A|Y 〉 : Components of the operator A in matrix repre-sentation.

AX Eigenvalue of the operator A to the eigenvector |X〉 (i.e. we haveA|X〉 = AX |X〉).

B Tetradic operator (superoperator).

BXY X ′Y ′ = (B)XY X ′Y ′ : Components of the tetradic operator B.

Frequently Occuring Symbols

0 m-dimensional zero vector (null vector). Important relations: x+0 = xand c0 = 0.

0 Zero matrix (null matrix). Important relations: M + 0 = M , 0M = 0 =M0, and 0x = 0.

0 Null operator. Important relations: A+ 0 = A and 0A = 0 = A0.

0 Tetradic null operator. Important relations: B + 0 = B, 0B = 0 = B0,and 0A = 0.

1 Identity matrix. Important relations: 1M=M=M1 and 1x= x .

1 Identity operator (unity). Important relations: 1A = A = A1 and1|X〉 = |X〉.

1 Tetradic identity operator (unity). Important relations: 1B = B = B1and 1A = A. Cf. (3.58).

a, a, b, c Types of subsystems, behavioural types, subpopulations.

xviii List of Symbols

A Number of distinguished types a of subsystems.

Aab(i, j; t) = rab(t)Eab(i, j; t) : Payoff of strategy i for an individual of sub-population a when confronted with strategy j in interactions withindividuals of subpopulation b. Cf. (12.2b).

Cm(t) Relative central moments. Cf. (12.70).

dim Γ Dimension of the Γ -space.

Da(t) Unit of distance. Cf. (9.38b).

Da(x ′, x; t) = D′a(x ′, x; t)/Da(t) : Scaled distance (dissimilarity) of twobehaviours x and x ′ for spontaneous behavioural changes from theperspective of the individuals of subpopulation a. Important relation:Cf. (9.37b).

D′a(x ′, x; t) = eS′a(x ′,x;t) = D′a(x, x ′; t) ≥ 0 : Distance of two behaviours x andx ′ for spontaneous behavioural changes from the viewpoint of theindividuals of subpopulation a, measure for the incompatibility of xand x ′.

Da(x ′, x; t) = Da(x, x ′; t) ≥ 0 : Distance (dissimilarity) of two behaviours xand x ′ in pair interactions from the perspective of the individuals ofsubpopulation a. Important relation: Cf. (9.61b).

e = 2.71828 . . . : EULER constant.

ex = exp x =∞∑

k=0

xk

k! : Exponential function.

Ea(i, t) Expected success of strategy i for an individual of subpopulation a.Cf. (12.2a).

Ea(i, n; t) Configuration-dependent expected success of strategy i for an indi-vidual of subpopulation a.

Eab(i, j; t) Success of strategy i for an individual of subpopulation a in interac-tions with individuals of subpopulation b who pursue strategy j .

f (ω) =+∞∫

−∞dt eiωt

y(t0 + t) : FOURIER transform of function y(t0 + t).

Important relations:

y(t0+ t) = 1

2π

+∞∫

−∞dω e−iωt

f (ω) (inverse FOURIER transformation)

and

⎛⎝+∞∫

−∞dteiωt

y1(t)

⎞⎠

(+∞∫−∞

dt eiωty2(t)

)

=+∞∫−∞

dt eiωtt∫

0dt ′ y1(t ′) y2(t − t ′) (convolution theorem). Cf. [218].

List of Symbols xix

f (ω) = | f (ω)|2 : Power spectrum of the function y(t).

f kab(t) Interaction frequency of kind k for an individual of subpopulation a in

pair interactions with individuals of subpopulation b. Cf. (10.28).

G(u) =∞∫

0

dt e−utP(t0 + t) : LAPLACE transform of P(t0 + t). Important

relation: P(t0+ t) = 1

2π i

c+i∞∫

c−i∞du eut

G(u) (inverse LAPLACE transfor-

mation). Cf. [218].

G(u) LAPLACE transform of the statistical operator ρ(t).

Important relation:

⎛⎝∞∫

0

dt e−utρ1(t)

⎞⎠

(∞∫0

dt e−utρ2(t)

)

=∞∫

0

dt e−utt∫

0

dt ′ ρ1(t′) ρ2(t − t ′) (convolution theorem). Cf. (3.59)

and [218].

h = h/(2π) where h = 6.6262 · 10−34Js is PLANCK’S quantum of action.

H HAMILTON operator of a system.

i = √−1 : Imaginary unit.

i , j (Behavioural) strategies, (expressed) opinions.

I, J, K , L (Vectorial) subscripts each of which combines various sub-scripts/superscripts, e.g. I = (α, i)tr.

Im(x) Imaginary part of a complex number x = Re(x)+ i Im(x).

k = 1.38062 · 10−23JK−1 : BOLTZMANN constant.

kI (Z , t) Scaled first jump moments (drift coefficients). Cf. (6.14b).

K (t) LIAPUNOV function. Cf. (3.109), (6.25).

Ka(x, t) = (Ka1(x, t), . . . , Kai (x, t), . . . , Kam(x, t))tr : Vector of the effectivedrift coefficients Kai ; social force that acts on an individual of subpop-ulation a who shows behaviour x .

Kai (x, t) Effective drift coefficients in γa-space. Cf. (6.64b).

limx→y

f (x) = f (y) : Limit of f (x) as x approaches y.

ln x = loge x : Natural logarithm of x . Important relations: ln(xy) = ln x+ln y, ln x

a = a ln x , andd

dxln x = 1

x.

xx List of Symbols

log x = log10 x : Decadic logarithm of x .

logy x = ln x

ln y: Logarithm of x to the base y.

L LIOUVILLE matrix. Cf. (3.100).

m = dim γa : Dimension of the γa-space.

ml1...lM (X, t) lth jump moments in Γ -space (l = l1 + · · · + lM ). Cf.(6.2a).

max (n, t) First jump moments (drift coefficients) in Γ ′-space. Cf.

(5.15).

max

bx ′(n, t) Second jump moments (diffusion coefficients) in

Γ ′-space. Cf. (5.24).

mai1xi1......

aimxim(n, t) mth jump moments in Γ ′-space. Cf. (5.38).

m I (X, t), m(1)(X, t) First jump moments (drift coefficients) in Γ -space.

m I J (X, t), m(2)(X, t) Second jump moments (diffusion coefficients) inΓ -space.

max(x, y) =⎧⎨⎩

x if x ≥ y

y if y > x: Maximum of the numbers x and y.

a mod b a modulo b. Important relation: x ≡ a mod b if and only ifx = a + zb with an integer z.

M = dim Γ : Dimension of the state space Γ .

Ml1...lM (Z , t) Scaled lth jump moments (l = l1 + · · · + lM ). Cf. (6.12b).

n = (n1, . . . , n A)tr : Configuration of a system that consistsof many elements.

na = (nax1, . . . , na

xS)tr.

nx =∑

a

nax : Number of subsystems (of arbitrary type) that

are in state x .

nax Occupation number, i.e. number of subsystems of type a

that are in state x .

na′1x ′1......

a′kx ′k

a1x1......

akxk

= (. . . , (na′1x ′1+1), . . . , (na1

x1−1), . . . , (na′kx ′k+1), . . . , (nak

xk−1), . . . )tr.

N =∑a,x

nax : Number of subsystems of a considered system.

List of Symbols xxi

Na =∑

x

nax : Number of subsystems of type a.

O(xn) represents terms f (x) the absolute value | f (x)| of which con-verges, for x → 0, to zero like the function |x |n , i.e. thereexists a constant C so that | f (x)| ≤ C |x |n .

pa(x ′|x; t) x ′ �=x= wa(x ′|x; t)Δt : Probability for a subsystem of type a tochange from state x to state x ′ during a time period Δt .

pab(x ′, y′|x, y; t) = wab(x ′, y′|x, y; t)/νab(t) : Probability of an individualof subpopulation a to exchange the behaviour x for x ′ inan interaction with an individual of subpopulation b whochanges the behaviour from y to y′. Cf. (9.50).

pkab(x

′|x; t) x ′ �=x= f kab(t)R

a(x ′|x; t) : Probability with which an individualof subpopulation a exchanges the behaviour x for x ′ in aninteraction of kind k with an individual of subpopulation b.

P(n, t) Probability of configuration n to occur at the time t .

P(t) Vector with the components P(X, t).

P(x, t) =∑

a

Na

NPa(x, t) : Probability of a subsystem to be in state

x ; proportion of subsystems that are in state x .

P(X, t) = ρX X (t) : Probability of state X to occur at the time t . Inthe case of a continuous state space: Probability density. Cf.(3.52).

P(X, t |X0, t0) Transition probability from state X0 to state X during thetime period (t − t0).

P(Xn, tn; Xn−1, Joint probability to find the system in states X0, . . . , Xn atthe times t0, . . . , tn .tn−1; . . . ; X0, t0)

P(Xn, tn|Xn−1, Conditional probability to find the considered system in stateXn at the time tn given that it was in states X0, . . . , Xn−1 atthe times t0, . . . , tn−1.

tn−1; . . . ; X0, t0)

P(a, x; t) = Na

NPa(x, t) : Probability to find a subsystem of type a in

state x . In the case of a continuous state space: Probabilitydensity.

Pλ(X) Eigenfunction to eigenvalue λ.

xxii List of Symbols

P0(X) Stationary probability distribution.

Pa(x, t)Na�1= 〈na

x 〉Na

: Probability of a subsystem of type a to be in state

x ; (relative) proportion of subsystems of type a that are instate x .

Prob(M) Probability with which a value (event) x is element of theset M.

qI J (Z , t) Scaled second jump moments (diffusion coefficients). Cf. (6.14c).

Qai j (x, t) Effective diffusion coefficients in γa-space. Cf. (6.64c).

r Location.

rab(t) Relative frequency of interactions of an individuals of subpopula-tion a with individuals of subpopulation b. Cf. (12.1).

Ra(x ′|x; t) x ′ �=x= wa(x ′|x; t)/νa(t) : Readiness of individuals of subpopulationa to spontaneously change their behaviour from x to x ′. Cf. (9.1b).

Ra(x ′|x; t) Readiness of an individual of subpopulation a to exchange thebehaviour x for x ′ due to a pair interaction with another individual.Cf. (9.62a).

Ra(x ′|x; n; t) Configuration-dependent readiness Ra .

Re(x) Real part of a complex number x = Re(x)+ i Im(x).

sin x Sine of x .

S, Sa Number of states x belonging to the γa-space.

S′a(x ′, x; t) = S′a(x, x ′; t) : Symmetrical part of the scaled conditional utilityU ′a(x ′|x; t); subjective effort of a behavioural change from x to x ′for an individual of subpopulation a. Cf. (9.30b).

Sa(x ′, x; t) = ln Da(x ′, x; t) = Sa(x, x ′; t) = Ga(x ′, x; t) + Ga(x, x ′; t) :(Scaled) effort of a behavioural change from x to x ′ (or vice versa)for individuals of subpopulation a.

t , t ′, t ′′ Times (points in time).

t0 Initial time.

T , T Time periods.

T ′, T ′a Absolute temperature in K = temperature in ◦C+ 273.15 K.

List of Symbols xxiii

T exp(

L, t, t0)= 1 +

t∫

t0

dt ′L(t ′) +t∫

t0

dt ′t ′∫

t0

dt ′′ L(t ′)L(t ′′) + . . . : Time-

ordered exponential function. (T indicates DYSON’s chronolog-

ical operator, here.) Important relations:d

dtT exp

(L, t, t0

)=

L(t)T exp(

L, t, t0)

. For the time-independent case L(t) ≡ L , by

carrying out the time integrations, one obtains T exp(

L, t, t0)=

∞∑k=0

Lk(t − t0)k

k! = exp(

L(t − t0))

.

Tr(A) =∑

Y

〈Y |A|Y 〉 : Trace of the operator A.

u Argument (independent variable) of a LAPLACE transform.

Ua(x, t) (Scaled) utility of behaviour x for spontaneous behaviouralchanges. For reasons of uniqueness one usually demands∑

x

Ua(x, t)!= 0.

Ua(x, n; t) Configuration-dependent utility Ua of behaviour x .

U a(x, t) Utility of behaviour x for individuals of subpopulation a in pair

interactions. Normally one demands∑

x

U a(x, t)!= 0.

U∗a (x ′|x; t) = Ua(x ′|x; t) + εa(x ′|x; t) : Estimated conditional utility of aspontaneous behavioural change to behaviour x ′ for an individualof subpopulation a who pursues behaviour x .

Ua(x ′|x; t) Known part of the conditional utility U∗a (x ′|x; t).U ′a(x ′|x; t) Scaled known conditional utility. Cf. (9.13).

v, w Velocities.

V Volume.

Va(x, t) Social field for individuals of subpopulation a with behaviour x .Cf. (11.16a).

Va(x ′|x; t) Objective advantage of a behavioural change from x to x ′ for anindividual of subpopulation a. Important relation: Va(x ′|x; t) ≥0. Cf. (9.27).

w(n′|n; t) Configurational transition rate with which a transition from con-figuration n to configuration n′ occurs. Cf. (5.34), (5.50b).

w(X ′|X; t) X ′ �=X= limΔt→0

P(X ′, t +Δt |X, t)

Δt: Transition rate from state X to

state X ′. Cf. (3.22b), (3.91).

xxiv List of Symbols

wa(x ′|x; t) = wa(x ′|x; t) Individual transition rate for spontaneous transitionsfrom state x to state x ′ for systems of type a.

wa(x ′|x; n; t) Effective configuration-dependent individual transitionrate. Cf. (5.15b).

wa(x ′|x; n; t) Configuration-dependent individual transition rate.

wa(x ′|x; t) Effective transition rate from state x to state x ′ by spon-taneous transitions and interactions for a subsystem oftype a. Cf. (4.22b).

wab(x ′, y′|x, y; t) Na�1= Nbwab(x ′, y′|x, y; t) : Pair interaction rate for asubsystem of type a with subsystems of type b due towhich the element of type a changes from state x to x ′and the element of type b from state y to y′. Cf. (4.20b).

wab(x ′, y′|x, y; t) Individual pair interaction rate (transition rate for pairinteractions) of two concrete subsystems of types a andb due to which the element of type a exchanges thestate x for x ′ and the element of type b exchanges thestate y for y′.

wab(x ′, y′|x, y; n; t) Configuration-dependent pair interaction rate wab.

wt−t ′(X ′|X; t ′) Transition rate of the generalized master equation. Cf.(3.87).

x , x ′, y, z States of elements.

xα State of element α.

X = (x1, . . . , xN )tr : System state in Γ -space.

X (t) Most probable state of a system at time t .

X0 Initial state at time t0.

X I Component I of the state vector X .

Z = εX with ε = 1/Ω : Scaled state vector X .

Greek Symbols

α, β, γ Subsystems; individuals.

γa γa-space: Set of all possible states x for subsystems of type a.

Γ State space (LIOUVILLE space), set of all possible states X of the consid-ered system.

Γ ′ Configuration space. Cf. (5.2).

List of Symbols xxv

δ(x − y) DIRAC delta function. Important relation:y1∫

y0

dy f (y)δ(x − y) = f (x) ·⎧⎨⎩

1 if y0 ≤ x ≤ y1

0 otherwise.

δxy =⎧⎨⎩

1 if x = y

0 otherwise: KRONECKER symbol.

Δt Small time period.

ΔV Volume element.

Δx ′ = x ′ − x .

Δ j i f = f ( j)− f (i).

Δx f (x) =∑

i

(∂

∂xi

)2

f (x) : LAPLACE operator applied to the function f (x).

ε Small number. Normally one assumes ε � 1.

εa(x ′|x; t) Unknown part of the conditional utility U∗a (x ′|x; t) that can only beestimated.

κ Control parameter.

λi , λ Eigenvalues (which are possibly complex numbers).

λi (κ) LIAPUNOV exponent. Cf. (7.43).

νa(t) = 1

Da(t)Δt: Rate which is a measure for the frequency of sponta-

neous behavioural changes of individuals of subpopulation a.

νa(t) = νaa(t) : Meeting rate of an individual of subpopulation a with otherindividuals of the same subpopulation.

νa(x ′|x; t) Rates that are a measure for the overall frequency of interactions ofan individual of subpopulation a with other individuals. Cf. (9.62c).

νab(t) Meeting rate of an individual of subpopulation a with individuals ofsubpopulation b.

νab(t) = νab(t)/Nb.

νkab(t) = νab(t) f k

ab(t) : Interaction rate of kind k between an individual ofsubpopulation a and individuals of subpopulation b.

ξ(t) Fluctuation at the time t .

π = 3.1415926 . . . : Pi.

ρ(a, x; t) = N P(a, x; t) = Na Pa(x, t) : Density of subsystems which are oftype a and in state x .

xxvi List of Symbols

ρ(t) Statistical operator. Cf. Sect. 3.2.4.1.

σI J (t) = 〈X I X J 〉 − 〈X I 〉〈X J 〉 : Covariances in Γ -space. The correlation of tworandomly distributed quantities X I and X J is measured with the correla-

tion coefficient rI J := σI J√σI IσJ J

. Important relation: −1 ≤ rI J ≤ 1. For

rI J = 0 the quantities X I and X J are called uncorrelated.

σ ax

bx ′ = 〈na

x nbx ′ 〉 − 〈na

x 〉〈nbx ′ 〉 : Covariances of the occupation numbers na

x .

τ ≡ τ(t) : Transformed time.

ω = 2π/T : (Angular) frequency of oscillation.

ω0 (Angular) frequency of the fundamental mode of oscillation.

Ω System size. Cf. (6.9).

Operators and Special Symbols

x := y, y =: x x is defined by y.

x=y x corresponds to y.

x!= y x must be equal to y.

xa=b= y x ist equal to y on condition a = b.

x ≡ y x is identical with y.

x � y x is much less than y.

x � y x is much greater than y.

x ∼ y x is similar to y, i.e. x and y have the same order of magnitude.

x −→ y, y ←− x Transition from x to y.

x ↔ y Interchange of x and y.

n! = 1 · 2 · · · · · (n − 1) · n : n factorial.

xy

x to the power of y. Important relations: xy+z = x

y · xz , xy =

x · x y−1, x1 = x , and x

0 = 1.

x−1 = 1/x : Reciprocal of x .

f (x)∣∣∣x=y

= f (y) : Value of function f for x = y.

|x | =√[Re(x)]2 + [Im(x)]2 : Absolute value of a (possibly com-

plex) number x . Important relation: |x+y| ≤ |x |+|y| (triangleinequality).

List of Symbols xxvii

|M| = |M tr| : Determinant of the matrix M . Cf. [20], pp. 154ff., 193ff.

‖x‖ =√∑

i

(xi )2 : Magnitude (length) of the vector x . Vec-

tors x with ‖x‖ = 1 (i.e. with magnitude 1) are callednormalized. Vectors of arbitrary magnitude fulfil the triangle

inequality ‖x + y‖ ≤ ‖x‖ + ‖y‖(

or

∥∥∥∥∥∑

i

xi

∥∥∥∥∥ ≤∑

i

‖xi‖)

and

the CAUCHY-SCHWARZ inequality∑

i

ci |xi yi | ≤√∑

i

ci (xi )2 ·√∑

i

ci (yi )2 (ci ≥ 0), in particular, |x · y| ≤ ‖x‖ · ‖y‖. Cf. [20],

pp. 133f., 288f.

|X〉 State vector of a quantum mechanical system.

|X, t〉 State vector of a quantum mechanical system at time t .

〈X | Adjoint of the state vector |X〉.〈X |Y 〉 Scalar product of the state vectors |X〉 and |Y 〉 in HILBERT space.

〈 f 〉, 〈 f (x, t)〉t =∑

x

f (x, t)P(x, t) : Mean value (expected value, average) of

the function f (x, t).

.., . represent arguments (independent variables) or subscripts/super-scripts of a function.

. . . And so on; up to.

× Indicates the continuation of a formula.

{x1, . . . , xS} Set consisting of the elements x1, . . . , xS .

[x, y] = {z with x ≤ z ≤ y} : Closed interval reaching from x to y (setof all values z with x ≤ z ≤ y).

x ∈M x is element of the set M.

∀ x For all admissible values of x .

∞ Infinity.n∏

x=1

f (x) = f (1) · . . . · f (n).

xxviii List of Symbols

∏x

Product over all admissible values of x .

n∑x=1

f (x) = f (1)+ · · · + f (n).

∑x

Sum over all admissible values of x .

∑x( �=x ′)

Sum over all admissible values of x except for x ′.

∑x �=x ′

=∑

x

∑x ′( �=x)

.

∑x1,...,xn

=∑x1

· · ·∑xn

: Sum over all n-tuples (x1, . . . , xn).

∑x1,...,xn

( f (x1,...,xn )=g(x1,...,xn ))

Sum over all n-tuples (x1, . . . , xn) which satisfy the condition

f (x1, . . . , xn) = g(x1, . . . , xn).

dn x n-dimensional differential volume element (for n = 2: Surfaceelement, n = 1: Line element).

x∫

x0

dx ′ f (x ′) = limn→∞

n∑i=1

Δx f (xi ) withΔx = x − x0

nand xi = x0+ i Δx :

Integral of the function f (x) from x0 to x .∫dn x f (x) =

∫dx1 . . .

∫dxn f (x) : n-dimensional volume integral over

f (x).∫dx · f (x) Line integral of the vector field f (x). Cf. [246], pp. 1027ff.

d

dxf (x) = lim

Δx→0

f (x +Δx)− f (x)

Δx: Derivative of the function f (x)

with respect to x . Important relations:d

dx

[f (x)g(x)

]=(

d

dxf (x)

)g(x)+ f (x)

(d

dxg(x)

)(product rule),

d

dx

(f (x)

g(x)

)=

(d

dxf (x)

)g(x)− f (x)

(d

dxg(x)

)

(g(x)

)2(quotient rule),

d

dxf(

g(x))= d

dyf (y)

∣∣∣∣y=g(x)

· d

dxg(x) (chain rule), and

List of Symbols xxix

d

dx

x∫

x0

dx ′ f (x ′) = f (x) (second fundamental theorem of cal-

culus).

∂V Surface of the volume V .

∂

∂xf (x, y) = lim

Δx→0

f (x +Δx, y)− f (x, y)

Δx: Partial derivative of

f (x, y)with respect to x .

d

dxf (x, y) = ∂

∂xf (x, y)+ ∂y

∂x

∂

∂yf (x, y) : (Total, substantial) derivative

of the function f (x, y) with respect to x .

δ

δ fF(

f (x))

Functional derivative (variational derivative) of F with

respect to the function f (x).

∇ =(∂

∂x1, . . . ,

∂

∂xn

)tr

: Nabla operator.

∇x f (x, y) =(∂ f (x, y)

∂x1, . . . ,

∂ f (x, y)

∂xn

)tr

: Gradient of the function

f (x, y) with respect to x = (x1, . . . , xn)tr.

∇ · f (x) =n∑

i=1

∂ fi (x)

∂xi: Divergence of the vector field f (x).

Chapter 1Introduction and Summary

The field of quantitative sociodynamics is still a rather young and very thrillinginterdisciplinary research area which deals with the mathematical modelling of thetemporal evolution of social systems. In view of the growing complexity of social,economic, and political developments quantitative models are becoming more andmore important—also as an aid to decision-making. From a scientific point of viewas well, a mathematical formulation of social interrelations has long been over-due. Compared with purely qualitative considerations it allows clearer definitionsof the terms used, a more concentrated reduction to the interesting interrelations,more precise and more compact descriptions of structures and relations, more reli-able conclusions, better forecasts and, thus, statements which are easier to verify[102, 231]. Apart from this it turned out that many social phenomena cannot evenapproximately be understood with static concepts. Some dynamic social processesare not comprehensible as a sequence of time-dependent equilibrium structures. Thedescription of self-organization and structure formation processes requires dynamicmathematical concepts which also can describe non-equilibrium phenomena, i.e. thetemporal evolution of systems which become unstable.

Because of the complexity of social systems a mathematical formulation of socialprocesses was, for a long time, considered to be almost impossible. Nevertheless,again and again corresponding formulations have been developed by social as wellas natural scientists. However, the proposed concepts often were too simple and didnot sufficiently do justice to those phenomena which they should describe.

Only in the recent years, powerful methods have been developed which makeit possible to describe complex systems consisting of many elements (subsystems)that interact with each other. Here pioneering were particularly the concepts of sta-tistical physics for the description of chance affected processes and discoveries inthe field of non-linear dynamics which was shaped by synergetics [106, 107], chaostheory [16, 135, 183, 254], catastrophe theory [273, 302], as well as by the theoryof phase transitions and critical phenomena [145, 177, 202, 263]. These again andagain proved their interdisciplinary explanatory power and allow the developmentof models for social processes at an actually much sounder level than it was fea-sible only a short time ago. In the meantime a remarkable list of physicists existswho already dealt with topics from the field of mathematical sociology: WEIDLICH

D. Helbing, Quantitative Sociodynamics, 2nd ed.,DOI 10.1007/978-3-642-11546-2_1, C© Springer-Verlag Berlin Heidelberg 2010

1

2 1 Introduction and Summary

[285, 286, 288–290], HAAG [103], HAKEN [107, 300], EBELING [56, 57, 59],MALCHOW [179], SCHWEITZER [257], PRIGOGINE [225, 226], ALLEN [2–6],MONTROLL [91, 185–187], MOSEKILDE [191, 192], HUBERMAN [89, 90, 147],WALLS [284], and LEWENSTEIN [168].

During the next years, in the field of the mathematical description of social phe-nomena we can surely expect a rapid scientific development and fundamentally newresults from the various disciplines involved.

1.1 Quantitative Models in the Social Sciences

Despite the difficulties with the mathematical modelling of social processes alreadyfor some time a number of formal models has been developed by social scien-tists. These include OSGOOD and TANNENBAUM’s principle of congruity [213],HEIDER’s balance theory [119, 120], and FESTINGER’s dissonance theory [74]which deal with the stability of attitude structures, to mention a few. Very interest-ing is also the model suggested by LEWIN which assumes that the behaviour of anindividual is determined by a ‘social field’ [169]. Further mathematical models weredeveloped for learning processes (cf. [231], Chap. 14) and decision processes [48,53, 174, 209, 297]. The first stochastic models for social processes go back to COLE-MAN [43] and BARTHOLOMEW [18]. In addition, numerous diffusion models [50,98, 110, 157, 178, 247] are worth mentioning. Of big importance is also the descrip-tion of the competition and cooperation of individuals within the framework of gametheory which was founded by VON NEUMANN and MORGENSTERN [196]. Last butnot least the event analysis or survival analysis received considerable attention [51,275]. That is, meanwhile a broad and substantial literature exists in the field of math-ematical sociology (cf. also [44, 187, 231]). In the following those models whichplay a particular role in the course of this book will be discussed in more detail.

1.1.1 The Logistic Model

For growth processes an exponential temporal development was assumed for a longtime. In many cases, however, this is not empirically confirmed. Rather it turnedout that growth processes with limited resources can well be characterized by thelogistic equation of PEARL and VERHULST [217, 281]. According to the logisticequation growth processes start in a roughly exponential way and then turn intoa saturation phase in which a certain system dependent maximum value is gradu-ally approached (cf. Sect. 4.5.2). Exponential growth occurs for unlimited resourcesonly.

The applications of the logistic equation reach from the description of chemicalreaction rates, the growth of animal and plant populations in the absence of enemies,and the growth of towns up to the description of the spreading of information orinnovations [18, 43, 91, 110, 130, 187, 206].