Quantitative Sociodynamics (eBook)

Preface 5
Preface of the First Edition 7
Contents 9
List of Symbols 15
Notation and Conventions 15
Frequently Occuring Symbols 17
Greek Symbols 24
Operators and Special Symbols 26
1 Introduction and Summary 30
1.1 Quantitative Models in the Social Sciences 31
1.1.1 The Logistic Model 31
1.1.2 Diffusion Models 32
1.1.3 The Gravity Model 32
1.1.4 The Game Theory 32
1.1.5 Decision Models 34
1.1.6 Final Remark 35
1.2 How to Describe Social Processes in a Mathematical Way 35
1.2.1 Statistical Physics and Stochastic Methods 36
1.2.2 Non-linear Dynamics 41
2 Dynamic Decision Behavior 45
2.1 Introduction 45
2.2 Modelling Dynamic Decision Behavior 46
2.2.1 Questioning Transitive Decisions and Homo Economicus 46
2.2.2 Probabilistic Decision Theories 48
2.2.3 Are Decisions Phase Transitions? 51
2.2.4 Fast and Slow Decisions 52
2.2.5 Complete and Incomplete Decisions 53
2.2.6 The Red-Bus-Blue-Bus Problem 54
2.2.7 The Freedom of Decision-Making 55
2.2.8 Master Equation Description of Dynamic Decision Behavior 55
2.2.9 Mean Field Approach and Boltzmann Equation 57
2.2.10 Specification of the Transition Rates of the Boltzmann Equation 58
2.3 Fields of Applications 60
2.3.1 The Logistic Equation 60
2.3.2 The Generalized Gravity Model and Its Application to Migration 60
2.3.3 Social Force Models and Opinion Formation 61
2.3.4 The Game-Dynamical Equations 63
2.3.5 Fashion Cycles and Deterministic Chaos 65
2.3.6 Polarization, Mass Psychology, and Self-Organized Behavioral Conventions 67
2.4 Summary and Outlook 69
References 69
Part I Stochastic Methods and Non-linear Dynamics 74
Overview 74
3 Master Equation in State Space 76
3.1 Introduction 76
3.2 Derivation 78
3.2.1 Derivation from the MARKOV Property 79
3.2.2 External Influences (Disturbances) 80
3.2.3 Internal Fluctuations 81
3.2.4 Derivation from Quantum Mechanics 84
3.3 Properties 91
3.3.1 Normalization 91
3.3.2 Non-negativity 92
3.3.3 The LIOUVILLE Representation 92
3.3.4 Eigenvalues 93
3.3.5 Convergence to the Stationary Solution 94
3.4 Solution Methods 95
3.4.1 Stationary Solution and Detailed Balance 95
3.4.2 Time-Dependent Solution 98
3.4.3 `Path Integral' Solution 99
3.5 Mean Value and Covariance Equations 106
4 Boltzmann-Like Equations 109
4.1 Introduction 109
4.2 Derivation 110
4.3 Subdivision into Several Types of Subsystems 113
4.4 Properties 114
4.4.1 Non-negativity and Normalization 114
4.4.2 The Gaskinetic BOLTZMANN Equation 114
4.4.3 The H-Theorem for the Gaskinetic BOLTZMANN Equation 117
4.4.4 Solution of the Gaskinetic BOLTZMANN Equation 120
4.5 Comparison of Spontaneous Transitions and Direct Interactions 121
4.5.1 Transitions Induced by Interactions 121
4.5.2 Exponential Function and Logistic Equation 122
4.5.3 Stationary and Oscillatory Solutions 123
5 Master Equation in Configuration Space 125
5.1 Introduction 125
5.2 Transitions in Configuration Space 126
5.2.1 Spontaneous Transitions 126
5.2.2 Pair Interactions 127
5.3 Mean Value and Covariance Equations 129
5.4 Corrections and Higher Order Interactions 133
5.5 Indirect Interactions and Mean Field Approaches 137
5.6 Comparison of Direct and Indirect Interactions 137
5.6.1 Differences Concerning the Covariance Equations 137
5.6.2 Differences Concerning the Mean Value Equations 138
6 The Fokker-Planck Equation 140
6.1 Introduction 140
6.2 Derivation 140
6.3 Properties 144
6.3.1 The Continuity Equation 144
6.3.2 Normalization 145
6.3.3 The LIOUVILLE Representation 146
6.3.4 Non-negativity 146
6.3.5 Eigenvalues 146
6.3.6 Convergence to the Stationary Solution 146
6.4 Solution Methods 147
6.4.1 Stationary Solution 147
6.4.2 Path Integral Solution 148
6.4.3 Interrelation with the SCHRÖDINGER Equation 149
6.5 Mean Value and Covariance Equations 150
6.5.1 Interpretation of the Jump Moments 151
6.6 Boltzmann-Fokker-Planck Equations 152
6.6.1 Self-Consistent Solution 157
7 Langevin Equations and Non-linear Dynamics 159
7.1 Introduction 159
7.2 Derivation 161
7.3 Escape Time 165
7.4 Phase Transitions, Liapunov Exponents, and Critical Phenomena 167
7.5 Routes to Chaos 169
7.5.1 RUELLE-TAKENS-NEWHOUSE Scenario and LIAPUNOV Exponents 170
7.5.2 Period Doubling Scenario and Power Spectra 171
Part II Quantitative Models of Social Processes 174
Overview 174
8 Problems and Terminology 176
8.1 Terms 176
8.1.1 System and Subsystems 176
8.1.2 State 176
8.1.3 Subpopulation 176
8.1.4 Socioconfiguration 179
8.1.5 Interaction 179
8.2 Problems with Modelling Social Processes 180
8.2.1 Complexity 180
8.2.2 Individuality 182
8.2.3 Stochasticity and Disturbances 182
8.2.4 Decisions and Freedom of Decision-Making 183
8.2.5 Experimental Problems 186
8.2.6 Measurement of Behaviours 186
8.3 Summary 187
9 Decision Theoretical Specification of the Transition Rates 189
9.1 Introduction 189
9.2 Derivation 190
9.2.1 The Multinomial Logit Model 190
9.2.2 Entropy Maximization 192
9.2.3 FECHNER'S Law 194
9.2.4 Utility and Distance Function 195
9.3 Pair Interaction Rates 198
9.3.1 Special Applications in the Social Sciences 205
9.4 Properties of the Utility Approach 207
9.4.1 Stationary Distribution 207
9.4.2 Contributions to the Utility Function 208
10 Opinion Formation Models 209
10.1 Introduction 209
10.2 Indirect Interactions 211
10.2.1 A Period Doubling Route to Chaos 213
10.2.2 A RUELLE-TAKENS-NEWHOUSE Route to Chaos 213
10.3 Direct Pair Interactions 213
10.3.1 Kinds of Pair Interactions 214
10.3.2 Oscillations 219
10.3.3 Influence of the Interaction Frequencies 223
10.3.4 Period Doubling Scenarios and Chaos 227
10.4 Generalizations 239
10.5 Spatial Spreading of Opinions 240
10.5.1 Opinion Spreading by Diffusion 240
10.5.2 Opinion Spreading by Telecommunication 242
11 Social Fields and Social Forces 246
11.1 Introduction 246
11.2 Derivation 247
11.3 The Social Force Model 250
11.3.1 Comparison with LEWIN's `Social Field Theory' 254
11.4 Computer Simulations 256
11.4.1 Imitative Processes 258
11.4.2 Avoidance Processes 265
12 Evolutionary Game Theory 267
12.1 Introduction 267
12.2 Derivation of the Game Dynamical Equations 268
12.2.1 Payoff Matrix and Expected Success 268
12.2.2 Customary Derivation 269
12.2.3 Fields of Application 269
12.2.4 Derivation from the Boltzmann-Like Equations 270
12.3 Properties of Game Dynamical Equations 273
12.3.1 Non-negativity and Normalization 273
12.3.2 Formal Solution 273
12.3.3 Increase of the Average Expected Success in Symmetrical Games 274
12.3.4 Invariant of Motion for Antisymmetrical Games 276
12.3.5 Interrelation with the Lotka-Volterra Equations 277
12.3.6 Limit Cycles and Chaos 278
12.4 Stochastic Version of the Game Dynamical Equations 280
12.4.1 Self-Organization of Behavioural Conventions for the Case of Two Equivalent Competing Strategies 283
13 Determination of the Model Parameters from Empirical Data 295
13.1 Introduction 295
13.2 The Case of Complete Data 295
13.3 The Case of Incomplete Data 298
13.3.1 Parameter Estimation 302
13.3.2 Model Reduction 308
13.4 Migration in West Germany 309
13.4.1 First Model Reduction 310
13.4.2 Second Model Reduction 312
13.4.3 Comparison of the WEIDLICH-HAAG Model and the Generalized Gravity Model 315
13.4.4 Third Model Reduction 318
13.5 Evaluation of Empirically Obtained Results 320
13.5.1 Sensitivity Analysis 320
13.5.2 Decomposition of the Utility Functions with Respect to Explanatory Variables 321
13.5.3 Prognoses 322
13.6 Examples for Decompositions of Utility Functions 322
13.6.1 Purchase Pattern 322
13.6.2 Voting Behaviour 325
13.6.3 Gaps in the Market and Foundations of New Parties 328
References 330
Index 341