Models of Society and Complex Systems

Models of Society and Complex Systems Models of Society and Complex Systems introduces readers to a variety of different mathematical tools used for modelling human behaviour and interactions, and the complex social dynamics that drive institutions, conflict, and coordination. What laws govern human affairs? How can we make sense of the complexity of societies and how do individual actions, characteristics, and beliefs interact? Social systems follow regularities which allow us to answer these questions using different mathematical approaches. This book emphasises both theory and application. It systematically introduces mathematical approaches, such as evolutionary and spatial game theory, social network analysis, agent-based modelling, and chaos theory. It provides readers with the necessary theoretical background of each toolset as well as the underlying intuition, while each chapter includes exercises and applications to real-world phenomena. By looking behind the surface of various social occurrences, the reader uncovers the reasons why social systems exhibit both cultural universals and at the same time a diversity of practices and norms to a degree that even surpasses biological variety, or why some riots turn into revolutions while others do not even make it into the news. This book is written for any scholar in the social sciences interested in studying and under- standing human behaviour, social dynamics, and the complex systems of society. It does not expect readers to have a particular background apart from some elementary knowledge and affinity for mathematics. Sebastian Ille is an Associate Professor of Economics at Northeastern University - London, and Editor-in-Chief of the International Social Science Journal. “Drawing on a rich array of historical and contemporary examples, this book provides an introduction to dynamical systems theory and how it elucidates the complex interplay of political, economic, and social factors that give rise to social norms and institutions. The exposition is exceptionally clear and tailored to different levels of mathematical preparation. It will appeal to experts as well as students across the social sciences.” — H. Peyton Young, University of Oxford & London School of Economics, UK “An interdisciplinary book on complexity, a guided tour through mathematical methods ranging from evolutionary game theory, dynamical systems, Markov chains and graph theory to bifurcations and chaos, with applications to nonlinear social sciences. A rigorous introduction delivered by an intriguing storytelling approach.” — Gian Italo Bischi, Professor of Applied Mathematics, University of Urbino, Italy Models of Society and Complex Systems Sebastian Ille Cover image: AerialPerspective Works / Getty Images First published 2023 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2023 Sebastian Ille The right of Sebastian Ille to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-367-47396-9 (hbk) ISBN: 978-0-367-47397-6 (pbk) ISBN: 978-1-003-03532-9 (ebk) DOI: 10.4324/9781003035329 Typeset in Minion Pro by codeMantra To Anastasia whose growing curiosity for the world has inspired this book – CONTENTS – List of Figures, xi List of Tables, xix Chapter 1: Introduction, 1 1.1 Aim and Scope, 3 1.2 Some Caveats, 5 1.3 For Whom Is This Book and How to Use It?, 6 1.4 Acknowledgement, 7 Chapter 2: Game Theory: Strategic Interactions, 9 2.1 Introduction, 9 2.2 Definition of a Game, 11 2.3 The Ultimatum Game, 18 2.4 Signalling, Focal Points, and Practices, 26 2.5 Strategic Complements and Substitutes, 32 2.6 Conclusion: The Limits of Game Theory, 35 Chapter 3: Evolutionary Game Theory and Dynamical Systems: Decentralised Decision-Making and Spontaneous Order, 41 3.1 Introduction, 41 3.2 The Continuous Replicator Model, 43 3.3 Regime Change, Stability and Bifurcation, 46 3.4 Multi-Population and N-Strategy Games, 52 3.5 Three Strategies with One and Two Populations, 57 3.6 Conclusion, 63 Chapter 4: Markov Chains and Stochastic Stability: Understanding Cultural Universals, 67 4.1 Introduction, 67 4.2 Markov Chains, 69 4.3 Stochastic Stability, 75 4.4 Benefit and Caveats, 79 4.5 Loss Sensitivity and Idiosyncratic Errors, 83 Contents 4.6 Conclusion, 89 vii viii Chapter 5: Individual Threshold Models and Public Signals: Fads, Riots, and Revolutions, 93 Contents 5.1 Introduction, 93 5.2 Thresholds in a Single Population, 95 5.3 Thresholds in More Than One Population, 102 5.4 Two Populations with Uniform Distributions, 103 5.5 Extensions with Individual Preferences and Choices, 109 5.6 Conclusion, 113 Chapter 6: Social Networks and Graph Theory: Small World Effects and Social Change, 117 6.1 Introduction, 117 6.2 Definitions and Elementary Measures 118 6.3 Centralities, 124 6.4 Application, 130 6.5 Preferential Attachment and the Power Law, 137 6.6 Conclusion, 142 Chapter 7: Peer Effects and Spatial Game Theory: Local and Global Efficiency, 147 7.1 Introduction, 147 7.2 Local Imitation on Regular Networks, 148 7.3 Non-Symmetric Interactions and Payoff vs. Imitation Space, 155 7.4 Other Spatial Games, 160 7.5 Extension: Co-evolving networks, 166 7.6 Conclusion, 167 Chapter 8: Agent-Based Modelling: Cascades and Self-Organised Criticality, 171 8.1 Introduction, 171 8.2 Critical Systems, 173 8.3 Complexity, Criticality, and ABM, 175 8.4 Segregation, 179 8.5 Self-Organised Criticality, 187 8.6 Conclusion, 195 Chapter 9: Chaos Theory: Non-Linear Dynamics and Social Complexity, 199 9.1 Introduction, 199 9.2 Threshold Models with Decision Reversals, 201 9.3 An Information Spin Glass, 208 9.4 Chaos in Evolutionary Games, 214 9.5 Conclusion, 218 Appendix A, 223 A.1 Elementaries, 223 A.2 Equilibrium Refinements and Discrimination Criteria, 225 A.3 Derivation of the Replicator Dynamics, 227 A.4 Extensions of the Replicator Dynamics, 228 A.5 Dynamical Systems Revisited, 232 A.6 Solving the Roots of Polynomials, 236 A.7 Deriving Modularity, 238 Bibliography, 241 Index, 251 Contents ix – FIGURES – 2.1 Benefit of footbinding with two types: the authoritarian type is shown in (a), the liberal type in (b). 10 2.2 Footbinding continued: the average parent’s actions determine the rows and benefits are shown in (a), the game between a liberal-type parent and the average parent is shown in (b). 11 2.3 Two equivalent games: The right payoff matrix has been derived from the left payoff matrix via an positive affine transformation. 14 2.4 Versions of the most common 2 × 2 Normal Form Games. 15 2.5 Payoff transformation by subtracting a constant that maintains the best responses and Nash equilibria. 18 2.6 Classic version of the Ultimatum Game reduced to two strategies: dominant strategy of responder is to accept and best response of Proposer is to offer a minimum amount. 19 2.7 Normal Form representation of the Ultimatum Game in 2.6. Reject is denoted by R and accept by A. 20 2.8 (a) shows the probability density function f (ρ) and (b) illustrates the cumulative distribution function F(ρ) for a = 20 and b = 50. 23 2.9 Probability of acceptance and expected payoff a = 20, b = 50, α = 1, β = 0.25, and σ = 100. 24 2.10 Coordination Game: Finding your friend. 27 2.11 A foot binding signalling game: women are either subservient (S) or self-willed (W) and choose whether to bind (B) their feet or to desist (D) based on their type. Men choose to marry (M) or to remain celibate (C) based on the decision of the woman. 28 2.12 Normal form of a foot binding signalling game. 30 2.13 (a) illustrates the best response correspondence of stratum 2 based on the frequency of footbinding x1 in stratum 1. (b) shows the best response correspondence of stratum 1 and stratum 2 if strategies are complementary. The intersection determines the mutual best response and the equilibrium of the interaction. Figures are drawn for α = 1, β = 5.5, and γ = 4. 34 2.14 (a) shows the best response correspondence of stratum 1 and stratum 2 if strategies are substitutes using α = 5, β = 7, and γ = −5. (b) illustrates the situation in which stratum 1’s strategy is complementary and stratum 2’s strategy is a substitute using α = 12, β = 18, γP = 6 and γG = −6. 35 Figures 2.15 Prisoner’s dilemma with pro-social preferences. 36 3.1 Expected payoffs of sharers and loners: the solid line shows πl (x) and the dashed lines show πs (x) for various parameter values. 46 xi  3.2 Plot of the state space of the one-dimensional system: the solid circles mark the xii (stable) nodes and the empty circles the repellors, the arrows indicate the direction Figures of the dynamics. In addition, the dotted line illustrates ẋ and the solid line shows the tangent at the interior fixed point. 49 3.3 Plot of the state space of the one-dimensional system with general parameter values: β = −6 + 10−4 , γ = 9: solid circles indicate nodes, empty circles indicate repellors and half-shaded circle indicate the saddle point (which is stable from the right and unstable from the left). Arrows indicate the direction of the dynamics. In addition, the dotted line illustrates ẋ = f (x). Figure 3.3d illustrates the change of x over time for the situation depicted in Figure 3.3b. 50 3.4 Examples of structurally unstable fixed points with eigenvalue λ = 0. 51 x2 −x x2 −x 3.5 (a) Saddle type surface (showing potential function V = 1 2 1 − 2 2 2 ) in a two- dimensional state space with a saddle point at (x1∗ , x2∗ ) = (0.5, 0.5), (b) The corre- sponding stable and unstable manifold in the state space. 54 3.6 Examples of fixed points with eigenvalue λ+,− = 12 Tr(J) ± iΩ. 56 3.7 Transformation of the unit simplex. 59 3.8 Unit simplices for the three Strategy Nash Demand Game with α = 0.25, β = 0.75. 59 3.9 Unit simplices for the three Strategy Nash Demand Game: coloured regions illustrate the basins of attraction of the mixed and pure node. 61 3.10 Best response areas for two sub-population interactions. 62 4.1 Accord of the sexes with common interests. 69 4.2 Probability transition matrix. 70 4.3 Probability transition matrix. 72 4.4 Probability transition matrix with state-dependent errors. 73 4.5 Probability transition matrix with state-dependent errors. 74 4.6 Agent-based simulation of stochastic play with memory and sample size 4 among two players, and state-dependent error ε = 0.00001, and λ = 0.1. Values illustrate the share of M play in the collective memory of both players, thus 0 illustrates state (CC,CC) and a value of 1 represents (MM, MM). 74 4.7 Two equivalent generic coordination games with ai = âi − di and bi = b̂i − gi . 75 4.8 Illustration of the transition probabilities. 76 4.9 Errors defined by λA = 0.001, λB = 0.001, εA = 0.01, and εB = 0.00001. Values illus- trate the share of M play in the collective memory of both players, thus 0 illustrates state C and a value of 1 represents M . 81 4.10 Simulation parameters identical to Figure 4.9, but errors are defined by λA = 0.06, λB = 0.05, εA = 0.1, and εB = 0.03. Values illustrate the share of M play in the collective memory of both players, thus 0 illustrates state C and a value of 1 represents M. 82 4.11 Illustration of the interplay of the force of selection and the force of random choice for two coordination games based on expected payoffs. 84 4.12 Error εγ for different exponents φ . 85 4.13 The histograms of strategy M players for two simulation sets, n = 21. Memory length 200, sample size 100, minimum error εmin = 0.0001. 86 4.14 The histograms of strategy M players for two simulation sets and parameters identical to Figure 4.13a at a = 10 and b ∈ (2, 23). Minimum error is εmin = 10−4 for 4.14a and 4.14b and εmin = 10−6 for 4.14c and 4.14d. 87 4.15 The minimum basin of attraction based on the sensitivity to losses φ , given a = 100 and εmin ∈ [10−10 , 10−6 ]. 88 4.16 Frequency of Musical plays based on basin of attraction of M , given by (1 − αA∗ ) α ∗ +α ∗ and the average basin of attraction across both populations, given by 1 − A 2 B . Black indicates a frequency of zero, while white indicates a relative frequency of 1. Simulations use s = 200, m = 400, φ = 2. 89 5.1 Function F(x) is shown in bold, the 45◦ line as the dotted line, and the trajectory in grey. All iterations start at x0 = 0.25 and arrows indicate the direction of movement. 98 5.2 (a) shows four different CDFs with mean µ = 0.45 and standard deviation σ = {0.1, 0.3, 0.7, 1.5}. The curve becomes flatter as σ increases. (b) has asymptotically stable fixed points at x1∗ = 0.015 and x2∗ = 0.997. The vectors illustrate the general dynamics and basins of attraction, and the grey zig-zag lines show the population paths for µ = 0.45, σ = 0.2 and initial conditions x01 = 0.35, x02 = 0.50. 99 5.3 (a) presents x∗ as a function of σ and given µ = 0.45 and (b) presents x∗ as a function of µ ∈ [0.1, 1] and σ ∈ [0.1, 3]. 100 5.4 Based on 100,000 simulations, the histograms show the frequency of the realised equilibrium values x∗ based on individual thresholds drawn randomly from a normal distribution and given various group sizes. (a) has a stable fixed point at x∗ = 0.533, (b) has a stable fixed point at x∗ = 0.901. 101 5.5 Values of a∗ are shown in blue, values of b∗ are shown in brown. Dotted and dashed lines show the corresponding value for the two unstable interior equilibria, solid lines show the values for the stable interior equilibrium: (a) shows the equilibrium quantities in relation to the population size of group B for Ra = Rb = 6 and Na = 100, (b) shows the equilibrium quantities in relation to the tolerance level of group B for Ra = 6, Na = 100, and Nb = 150. 105 5.6 The nullcline of group A for Ra = 6 and Na = 100. 106 5.7 The nullclines for Ra = Rb = 6 and Na = Nb = 100. The stable interior equilibrium lies at (a∗ , b∗ ) = (250/3, 250/3). The other two stable fixed points are at (0, Nb ) and (Na , 0). 107 5.8 The nullclines for (a) different population sizes Na = {50, 70, 90, 110} and (b) differ- ent tolerances Ra = {1, 2, 3, 4}. 108 5.9 Dynamics for two different population sizes, given Na = 100, Ra = 6, and Rb = 2. 108 5.10 F(σ ∗ )for different values of α given a normal distribution with µ = 0.6 and σ = 0.3. 110 5.11 Figure (a) shows the nullclines for x1 = F(x1 ) (solid) and x2 = F(x2 ) (dashed) and the dynamics given parameters α1 = 0.7, α2 = 0.3, µ1 = µ2 = 0.3, and σ1 = σ2 = 0.2. Figures (b) illustrates the stable equilibrium values of x1 (blue) and x2 (brown) for different α1 values and parameters as in (a). 111 xiii 6.1 Characteristic networks. 119 6.2 Network examples. 119  6.3 Adjacency Matrix of the directed and undirected network. 120 xiv 6.4 The square and cube of the adjacency matrices in Figure 6.3. 121 Figures 6.5 ¯ Shogun network showing degree centrality and modularity. 131 6.6 ¯ Shogun network showing eigenvector centrality and modularity. 132 6.7 Sample of a bipartite graph whose projection corresponds to Figure 6.2b. Set A corresponds to posts and set B to users. 134 6.8 (a) and (b) show the out-degree distribution of the bipartite graph. (c) show the degree distribution of the undirected version of the bipartite graph, and (d) shows degree distribution of the unipartite projection. 135 6.9 (a) betweenness distribution of unipartite graph, (b) distribution of geodesic distance of undirected bipartite and unipatite graphs. 137 6.10 (a)–(e) show a sample graph of 5,000 nodes generated via preferential attachment based on a different centrality measure. (f) illustrates the degree distribution for each of the sample graphs: degree - red, page-rank - blue, eigenvector - green, betweenness - brown, closeness - black. 139 6.11 (a)–(g) show the average degree over percentile range based on 100 runs per centrality for a graph of order 1, 000. (h) and (i) illustrate the degree of node with highest degree centrality and the number of nodes of degree 1, respectively. Bars indicate confidence interval at α = 5%. 140 6.12 Size distribution of 100 largest cities in 100,000s. 141 6.13 A preferential attachment graph with three identities: green, blue, red. Links have the same colour as their parent nodes. Links that connect different types are coloured in black. 142 7.1 Types of neighbourhood: neighbours of the grey cell are illustrated in black. 148 7.2 (a) flattened representation of the interaction plane, (b) plane warped into a torus. 149 7.3 Symmetric 2 × 2 game. 150 7.4 Different clusters of size r. 152 7.5 Non-symmetric 2 × 2 coordination game assuming ai , di > bi , ci . 155 7.6 Illustration of the two-player population interaction. 156 7.7 Simulation results for a = 10, b = 0, c = 4, and d = 8 on a 103 × 103 torus. Darker colours indicate higher frequency of the payoff dominant convention, lighter colours illustrate higher frequency of the risk-dominant convention. Numbers equal the average number of periods required for reaching a stable distribution. 159 7.8 The upper row shows the invasion of a single defector of a population composed of only cooperators. The lower row shows the invasion of a 3 × 3 cluster of cooperator of a population of only defectors. A black cell indicates a defector, a light grey cell indicates a cooperator. The size of the lattice is 199 × 199, parameters are a = 10, b = 0, c = 16.1, and d = 0.1. 162 7.9 (a) shows a quasi-stable distribution with mixed strategies, (b) and (c) show the invasion of a population of defectors by a 3 × 3 cluster of cooperators, given a probability that an agent updates their strategy equal to 66 percent. Parameters are a = 10, b = 0, c = 15.9, and d = 0.1. 163 7.10 3 × 3 zero sum game with no equilibrium in pure strategies and mixed equilibrium α1 = β1 = 6/11 and α2 = β2 = 3/11. 164 7.11 Different equilibrium states of the game in Figure 7.10 starting with a virtually identi- cal initial distribution at which each strategy is chosen with probability 1/3. Colour code: blue = {Comply, A}, green = {renegotiate, B}, and red = {persevere,C}, remaining colours are the combination of the primary colours: brown indicates either {comply,C} or {persevere, A}, turquoise {comply, B} or {renegotiate, A}, yellow {persevere, B} or {renegotiate,C}. 165 8.1 Number of death per 100,000 (including military and civilian) from 1400 to 1913. Data based on Roser (2020). 173 8.2 Rank-frequency distribution in log-log of war fatalities. 174 8.3 Rank-frequency distribution in log-log of migration rates. Data based on World Development Indicators (04/09/2020). 175 8.4 Analysis of systemic change based on an adapted version of Coleman’s Boat. 178 8.5 Initial setup of Schelling’s segregation model (Period 0). 180 8.6 Evolution of Schelling’s segregation model. 181 8.7 Figures show equilibrium states. Simulations initiated with an identical initial distri- bution and a similarity threshold of 40 percent and a density of 80 percent. Maximum distance and thus the size of a neighbourhood is defined by d and the number of neighbours included is given by n. 182 8.8 Simulations initiated with an identical initial distribution as in 8.7, a similarity threshold of 38 and 48 percent, respectively, and a density of 80 percent. Parameter d defines the size of the neighbourhood and n the number of neighbours. (a) and (b) show the equilibrium states, (c) and (d) the percentage share of dissatisfied agents over time. 183 8.9 Rank-frequency distribution in log-log of dissatisfied agents. 185 8.10 Multi-level co-evolutionary change: Rooftop model. 186 8.11 Different presentations of a sandpile. 188 8.12 Toppling of sand. 189 8.13 Example of a cascade. 190 8.14 Distribution in log-log of landslide lifetimes and sizes. 191 8.15 Distribution in log-log of landslide sizes in preferential networks. 192 8.16 Distribution in log-log of landslide lifetimes in preferential networks. 193 8.17 Distribution in linear-log of landslide lifetimes in preferential networks. 193 8.18 Example of a cascade following Bak et al. (1988). 198 9.1 (a) shows F L and F U in bold given α = 4 and β = 3. T he resulting net-threshold function F (xt ) is shown as dashed. (b) illustrates F = 2 x(1/α ) − xα for different values of α. 202 9.2 Figures show periods 9,950 to 10,000, given different values for α. 203 Figures 9.3 Bifurcation diagram for α ∈ (1, 2]. 203 9.4 Bifurcation diagram for α ∈ [1.96, 2.01]. 204 xv 9.5 Figures show the second iterate around the critical α value. 205  9.6 (a) Plots the fourth iterate F 4 (x) below the critical α value. (b) Plots the eighth iterate xvi F 8 (x) below the critical α value. 206 Figures 9.7 (a) shows the first four iterations starting within ±0.04 of the critical value for α = 1.987. (b) shows the critical value lines based on the x values of the first eight iterations (bold) and the following eight iterations (dashed) for a trajectory starting at x̂. 207 9.8 Plot of the sixth iteration F 6 (x) at α = 1.9915. Fixed points (x1∗ = 0.17081, x2∗ = 0.39900, x3∗ = 0.57664, x4∗ = 0.76423, x5∗ = 0.84881, x6∗ = 0.93996) are highlighted by circles. 207 9.9 Figures show different population states given ω = 1 on a 201 × 201 plane. (a) shows the initial random population state, the remaining figures show the population states after 5,000 periods for different noise parameters and thus probabilities of idiosyncratic switching. A phase transition occurs at σ ∈ (1.0, 1.3). 209 9.10 Figures plot the average number of switches in periods 500–600. (a) shows the impact of σ , given ω = 1, (b) shows the impact of ω, given σ = 0 . 210 9.11 Figures show the results for different ωs and σ = 0. 210 9.12 Figures show histograms based on the simulation with ω = 0.85 in a network of 1,000 nodes which has been shocked 10,000 times. (a) and (b) refer to the binary case, (c) and (d) show the continuous case. 212 9.13 (a) and (b) shows the share of individuals who switch their state as a consequence of a new attachment: (a) 4 runs, (b) 50 runs. (c) illustrates the network of order n = 500. The size of a node defines its eigenvector centrality, darker colours indicate that a node is more frequently discouraged from its state (i.e., obtains usually πit ≤ 0). 213 9.14 A four strategy game. 214 9.15 Figures show values of xt , yt , and zt during periods 300–500, with initial values y0 = 0.21, and z0 = 0.25. Full curves are initiated at x0 = 0.26, dotted values use x0 = 0.27. 215 9.16 Figure shows the trajectory from period 0 to 800 in the unit simplex with the same initial values as the full curves in Figure 9.15. 215 9.17 The two-population rock-paper-scissor game. 216 9.18 Figures show trajectory for periods 3,000–5,000 in unit simplex of population 1. (a) illustrates the periodic behaviour shown in the unit simplex for initial conditions (x10 = 0.5, y10 = 0.3, x20 = 0.5, y20 = 0.3), (b) for initial conditions (1/3, 0.25, 0.25, 1/3), and (c) for initial conditions (0.6, 0.3, 0.3, 0.6). 217 9.19 Figures demonstrates the super-periodic dynamics of the population frequencies of population 1 in periods 500–3,500. (a) and (b) are based on the initial values of Figure 9.18b, (c) and (d) are based on the initial values of Figure 9.18c. 217 9.20 Figure shows the chaotic dynamics for Population 1 for initial conditions (0.8, 0.1, 0.8, 0.1). 218 A.1 A game with three players. 226 A.2 Dynamics of the Predator-Prey model. 229 p A.3 Trajectories of the dynamical system defined by α = 0.7 and ẋ = αx − x2 + y2 (x + p y) and ẏ = αy − x2 + y2 (y − x). 234 Figures xvii – TABLES – 3.1 Stabilities given the characteristic value λ 48 3.2 Stabilities given the characteristic values λ1 and λ2 for f12 = 0 and f21 = 0 53 3.3 Stabilities of fixed points for the two-dimension state space 55 6.1 Network characteristics of undirected secularist networks 136 8.1 Advantages and disadvantages of ABM 177 Tables xix – 1 – Introduction M ORE than a millennium ago, the Emperor’s court was as lavish as was his harem. Yet, the poet-king Li Yu of China was confronted with a problem: his concubines and wives could substantially elevate their status by giving birth to a male heir apparent, but given their sheer numbers, the prospects of being with a child sired by the Emperor were less than slim. Meanwhile, each had potentially numerous opportunities to engage in infidelity unbeknownst to the Emperor within the palace walls. How could Li Yu ensure that the progeny was his own? The solution is shrouded in legend according to which Li Yu asked his favourite concubine Yao Niang to wrap her feet in silk to give them the shape of a lotus. After she bound her feet, her graceful dance on the tip of her toes is said to have not only mesmerised courtiers and especially the Emperor, but has inspired envy among the other ladies of the court. The latter thus fervently adopted the practice. As delightful as this account may be, it is clearly an idealised version and probably belongs to the realm of historical fiction: given Li Yu’s predicament, it seems more plausible that he recognised footbinding as a useful way to curb extramarital affairs and pregnancies.1 Footbinding was conceived as a means of control to limit the movement of women around the palace. Types of footbinding ranged from the three-inch golden lotus to the less aggressive version of the cucumber foot, but generally, the practice severely maimed the feet - it was irreversible and painful. Yet, the Chinese elite quickly embraced footbinding since binding a daughter’s feet was a hypergamous practice that opened the potential of her being accepted to the palace.2 Adopting the practice was thus of political and economic interest to the higher social strata. It was seen as a signal of status, fertility, chastity, and virtue. Literature internalised footbinding to such a degree that it became a sign of beauty; an erotic custom that served male foot fetishism and stirred men’s fantasy with a forced graceful gait. Footbinding turned into an ambiguous sign of concupiscence and purity. Probably due to a lack of historical data, it was initially assumed to be an elite practice. After all, only elite Chinese women possessed the leisure and skills to celebrate bound feet as a mark of beauty and sacrifice since they were not subject to the economic and educational constraints of female workers, farmers, or servants. Nevertheless, footbinding was soon emulated by the lower classes. While it made it impossible to work in wet rice fields, the Introduction practice was widespread in rural areas. It seemed incompatible with a society that relied heavily on labour-intensive family farming. 1 DOI: 10.4324/9781003035329-1  Several reasons have been discussed in the literature, such as the practice’s role as a marker of 2 Introduction social status even among the peasantry or of ethnic belonging used to distinguish oneself from the invading Mongols (Mackie, 1996). Indeed, the spread of footbinding as a more common practice overlaps with the adoption of the Han culture. Yet, footbinding was also adopted in culturally mixed areas and Manchu women embraced the practice quickly after an ineffectual prohibition of footbinding in 1847.3 While footbinding was prevalent, it was not universal. Turner (1997) illustrates that geographic conditions seemed to have played a major role instead as footbinding was more common in hospitable areas. Bossen and Gates (2017) provide compelling arguments for another explanation. Not only was footbinding a practice to isolate women away from the eyes of strangers but contrary to common belief, it was stimulated and not deterred by the economic constraints of the peasantry. The labour cost of domestic production was low and child labour was common. In addition, income from weaving was relatively high. Women could earn more by engaging in weaving than agricultural labour. Utilising girls for weaving already at a young age and engaging them in supporting actions, such as hemp twine and reeling silk could therefore sustain a family. Furthermore, they were employed in the production of shoes, sandals, hats, and quilts. Each of these tasks happened inside the house creating little need for long-distance travel because the produce was exchanged at local markets. Footbinding inhibited the ability to play and run and compelled a girl to focus on handwork. Additionally, it was a signal. Seen both as a sign of hand-skill and as an investment towards married life, it was a testament of a wife’s dedication - and to a certain degree, footbinding still allowed women to participate in heavier tasks, such as drying fruits or raising silkworms. The division of labour was simple: men plough and women weave. The widespread custom of footbinding was quickly abandoned in the Republican era in the years following the Nationalist Revolution in 1912. Consistent with the argument of Bossen and Gates, footbinding first disappeared in urbanised areas and among the elites after literati (such as the poet Yuan Mei in the later 18th century), intellectuals, and politicians shunned the practice in their endeavour to modernise China for global trade. The reasons were again manifold. Scholars, such as Qian Yong argued that footbinding is no longer a symbol for the gentry since it has been adopted by the lower classes. Also, footbinding was seen as damaging to China’s reputation and honour - a new perception of honour encouraged by the establishment of Christian missions after 1864.4 Especially female Protestant missionaries opposed footbinding. Thus, while it was initially seen as a symbol of family honour, the changing perception turned it into a dishonourable practice. Similarly, a young scholar by the name of Kang Youwei argued that a new status for women was necessary for China’s reformation in a 10,000 word petition to the throne in 1889.5 Already before, the Taiping Rebellion of 1850–1854 envisioned equality between men and women. Espe- cially Christian schools established after 1860 opposed the practice, and an increasing number of Chinese from the gentry and merchant classes, who studied abroad, instituted Western ideals. Kang initiated the Unbound Foot Association which counted over 10,000 members at the end of the 19th century. Yet the movement was mainly centred around elite expatriate women. Eventually, the Empress Dowager Cixi issued an anti-footbinding (but non-prohibitive) edict in 1902. Despite these efforts, it took until 1911, when Sun Yat-sen banned footbinding. The practice did no longer fit the new Chinese order. The different types of footbinding indicated class in a substantially hierarchical society. It was a symbol of the old imperial world that ended with the Emperor’s abdication in 1912 and made room for a new structure of social classes, one that had no need for localised control of women by men. Yet in rural areas, Chinese held on to footbinding for several years. Fathers started to oppose footbinding while mothers still encouraged the practice since the latter were afraid of the negative signal of a big foot to a prospective mother-in-law. It entailed foregoing a respectable marriage and condemning her daughter to hard labour on the fields. However, the introduction of global commerce increased the availability of industrially manufactured machine-made cotton. Production shifted to iron-gear looms, which were mainly operated by men, and production moved outside the house. Meanwhile, revenues from handcrafted products declined. Consequently, the opportunity costs of footbinding became too high and peasant families started abandoning the practice. The prevalence of Christian values entailed a realisation that women can be both unbound and faithful, and schools imposed restrictions on footbinding as elementary education of girls became more important. Thus, while footbinding was almost universal in Dongting, for example, among women born before 1892, it was abandoned in less than a quarter of a century (Gamble, 1954). Similarly, the region to the south of Peking, Tinghsien, completely abolished the practice during the period from 1899 to 1919 (Mackie, 1996). 1.1 Aim and Scope Several characteristics of the history of footbinding in China may pique our attention. While footbinding was initially a practice within the imperial court and the upper social stratum, its endorsement became increasingly and quickly widespread; not only among the gentry but even- tually among commoners and peasants. We have seen that for Li Yu of China and the peasantry, footbinding was a means of control, but its main advantage was that bound feet were associated with different and more opaque characteristics that probably outweighed its significant costs. Over time, footbinding co-evolved with Chinese culture and arts and became increasingly ingrained in the latter, as the practice became internalised and shaped the perception of beauty. On the other hand, despite its endurance for a millennium, footbinding was abolished within a generation. Yet, revocation was not uniform across the entire population and area of China. Intellectual centres rapidly ceased to promote footbinding, whereas more rural areas, such as Shanxi, sustained the practice significantly longer. Still, also rural areas illustrated vast differences. Footbinding is only one example of the plentiful institutions that determine the human history of habits, traditions, and in general, behaviour. Institutions constitute the recurrent behavioural rules that are shared by at least part of a population. They include practices, conven- tions, and norms to which we do not only subject our actions but which the latter reinforce.6 Institutions are then the collectively accepted and shared code of social interactions.7 In the context of institutions, our brief historical study of footbinding raises a number of broader questions that are part of the central themes with which we are concerned in this book: Introduction ˆ How are institutions adopted and when do they become prevalent? ˆ What makes an institution endure and when is it abolished by society? ˆ Under which conditions can different social practices and conventions co-exist? 3  ˆ How do certain characteristics or behaviour transform into a signal that is linked to some 4 Introduction specific qualities and how do these signals foster endorsement of a particular institution? ˆ How do individuals learn behavioural rules and which role do peer effects play in the evolution of institutions? ˆ How can we explain the co-evolutionary processes which govern institutions and which are mutually self-reinforcing? Social systems follow certain regularities which allow us to model social behaviour and dynamics on the basis of a mathematical approach. Each chapter introduces a different mathematical technique along with various models. The mathematical technique or approach describes a particular way of interpreting actions and behaviour. The models, which apply the mathematical approach, can therefore only be abstract representations of the world, but they are adequate to replicate particular regularities of society. Nevertheless, it is still important to understand each model as a reductionist explanation of a social phenomenon. The aim cannot be a realistic representation of the real world, but only an adequate. In the end, as George Box (1976, p. 792) said: “[A]ll models are wrong”.8 Nevertheless, collectively, these models offer explanations for a wide range of social phenomena, including local and global institutional change and norm evolution, the existence of consistent institutions across different regions and periods of human society – so-called cultural universals – and localised institutions that exist alongside other accepted behavioural rules. Some of these models take account of the elements of complex systems that societies are: they are adaptive, generate scale-free networks and systems, contain different levels of aggregation, and produce emergent properties - the importance and meaning of these concepts will become clearer as we proceed with our study of complex system and social dynamics. Nevertheless, the modelling of social behaviour in this abstract form comes at a cost. The models I discuss in this book present individuals in a stylised manner. Models involving a larger number of agents are built around the assumption that the relevant preferences and behavioural rules are largely identical across vast parts of the society under investigation. While this is certainly a strong assumption, members of the same social stratum who are faced with the same constraints and backgrounds are likely to have very similar motivations and options at hand. After all, share- croppers in a subsistence society are mainly concerned with feeding their family. In addition, the models I discuss here can be extended to a larger variety of preferences and types of agents in a straightforward manner but at the cost of a higher mathematical complexity. To understand social phenomena, we will see that it is frequently unnecessary to provide a detailed account of individual motivations, beliefs and depending on the model, even individual characteristics. At the same time, while social behaviour emerges from individual behaviour, society cannot be understood merely on the basis of a summation of individual actions or even less so, on the basis of a homme moyen - the average man who is representative of the mean field approximation of a distribution - as postulated by Adolphe Quetelet almost two centuries ago (see Quetelet, 1835). Societies are formed by collectives of agents and are complex systems in which different elements, entities, and dynamics interact. The resulting social behaviour often exhibits emergent properties and can therefore be rarely adequately understood from solely analysing individual actions: the aggregate does not necessarily share the same qualitative attributes as the individuals of which it is composed. The properties of a social system are critically dependent on the social connections formed by individuals and their way of interacting. When Margarete Thatcher claimed “[..] who is society? There is no such thing!” (Thatcher, 1987), she reduced society to a mere abstract concept giving all relevance to the individual, ignoring the strategic character of their actions and the complex interplay of economic, social, political, and cultural factors that motivate social phenomena. While I give credit to the methodological individualism formulated by Max Weber and the need for a proper micro-foundation to describe a social phenomenon, we will see in future chapters that a reduction of these micro-foundations to a mere study of the individual is inadequate for explaining at least some aspects of society. 1.2 Some Caveats Each of the eight mathematical approaches in this book is useful for explaining particular aspects of society. Each has not only its own potentials but also its limits, both of which I will discuss in each chapter. Again, I need to caution the reader not to over-interpret or over-generalise the results which we obtain from these models. The validity of the results in each chapter is constrained by the limiting assumptions of the underlying approach. While the approaches discussed in the early chapters of this book are more simplified and are based on stronger assumptions, the subsequent approaches are more complex and less-restrictive. Readers may fall prey to two fallacies. Some readers might be tempted to refer only to the later chapters for their work. However, abandoning restrictive assumptions does not necessarily imply a decrease in limitations. A less-restrictive approach may turn out to be unnecessarily more complex in a given context, and while being more flexible, such approaches (especially those discussed in Chapters 8 and 9 of this book) come at a cost of less control. Another fallacy is to assume that integrating more variables improves the explanatory power of a model. While this is partially true, the inclusion of each new variable negatively affects our ability to understand the internal workings of a model and the individual impact of each variable. A scholar, therefore, faces the challenging task of balancing authenticity and tractability when designing a model. The endeavour can be approached through two different methods - from the simple to the complex or from the complex to the simple.9 Either the scholar begins with a bare, reduced model and adds assumptions to the model until the latter adequately reflects the social phenomenon under investigation while keeping the model tractable and solvable. Alternatively, she starts with a model that almost fully describes the empirical data underlying the social phenomenon and gradually reduces the assumptions and thus, the complexity of the model up to the point that its effectiveness to satisfactorily describe the phenomenon is not compromised. The latter is ensured by testing the robustness of the model after each change. Both approaches should lead to essentially the same result - a balance that guarantees solvability or tractability on the one hand and avoids over-simplification on the other hand. Various approaches that we study in this book follow one of two fundamental notions of Introduction social dynamics. Evolutionary game-theoretic models are studies of dynamics and equilibria. These models understand a social system as a structure that naturally evolves towards an equilibrium over time. Consequently, an institution is a local or global attractor of the resulting dynamical system. I will discuss the technical details later, but the intuition of a local attractor is that a particular 5  environment can give rise to several potential solutions to social, economic, and political issues 6 Introduction in the form of different institutional structures. A social system then settles into an institutional structure that is not too dissimilar to the initial setup. Here, history matters to a limited degree, but as long as two societies are adequately similar (and we will see what this means later on), they must eventually establish the same set of institutions. In large societies, random variations in individual actions do not affect society as a whole. This at least holds for most situations: we will further see that some rather rare initial setups at tipping points illustrate a strong sensitivity to small random variations.10 Under these circumstances, individual actions can precipitate the evolution of one institutional solution over another or lead to a mixed state without unique institutions. In the case of a global attractor, on the other hand, history does no longer matter at all. Independent of the current characteristics of the social system, the final institutional setup is inevitable.11 The chaotic dynamics in Chapters 8 and 9 constitute the antithetic notion of the former social dynamics. Here, history does not only matter, some models push path dependency to its extreme. Two societies with minuscule variations in the initial institutional setup can diverge radically from each other over time. Individual actions then have a fundamental impact on the society as a whole. Popular science termed these evolutionary dynamics the butterfly effect. In addition, these systems are open systems. We will see that they are usually non-ergodic and cannot be studied on the basis of a Markovian system. They are path- or history-dependent and demonstrate co-evolutionary dynamics (again these concepts will become clearer subsequently).12 How then can we align these two entirely different notions of social dynamics? An easy but unsatisfactory answer would be: it depends on the system under scrutiny. I will discuss this question in Chapter 9 in greater detail, yet as we shall see, convergence and chaos are two sides of the same coin. And I can only wholeheartedly agree with Robert May’s statement: “Not only in research, but also in the everyday world of politics and economics, we would be better off if more people realised that simple non-linear systems do not necessarily possess simple dynamical properties” (May, 1976, p. 93). Last but not least, the approaches in this book further stress the importance of an interdisci- plinary perspective to competently understand and model social regularities and phenomena. It is not only the non-linearity of the social dynamics that render social systems complex, but these systems are complex because they are subject to a variety of determining factors. As our short study of footbinding in China illustrated, societies are the product of a sophisticated and compound interplay of political, social, and economic aspects that determine institutions and social dynamics. 1.3 For Whom Is This Book and How to Use It? This book is written for any scholar, across the social sciences as well as the humanities, who is interested in developing their own mathematical models. I have done my best to increase the scope of the book and present applications beyond my discipline. At the end of this book, reader will realise that footbinding and arms races, the size of cities and the influence of celebrities on Facebook, cooperation and Persian carpets, as well as drip castles and protests have more in common than what we might believe. I further hope that this book will help readers appreciate the need for more multi- and interdisciplinary perspectives when studying social dynamics and complex systems.  The book does not expect the reader to have any prior knowledge about the different approaches, and the appendix briefly reviews the most essential concepts as well as some useful methods that have proven helpful while analysing dynamical systems. A chapter should be seen as a rather cursory introduction to a particular approach. I have tried to include the most essential further readings in the conclusion to each chapter for those scholars who wish to delve deeper into a subject. However, in my experience, most of the presented approaches are already sufficiently sophisticated to deliver useful insights when applied to an empirical case or context. In addition, some of the approaches can be combined (an obvious candidate being, for example, Chapters 6 and 7). At the same time, it is important to be prudent when interpreting and generalising the results an approach delivers. Consequently, I discuss the limitations of each approach as well as the connections between different approaches in the conclusion of each chapter. To improve tractability, I print a new concept in bold whenever it is first introduced and defined. While this book has been written with a focus on self-study in mind, the content of this book can be adapted to meet the needs of a course for undergraduate and graduate students. Each chapter starts with relatively accessible examples in the earlier sections while the later sections contain more advanced material. Apart from Chapter 9, a chapter is composed of four sections in addition to an introduction and conclusion. Consequently, a judicious exposition of the first two to three sections (in addition to supplementary explanations depending on their prior background) of each chapter is suitable for undergraduate students. The full sections can be taught to graduate and postgraduate students, but it may be convenient to spread a chapter over two lectures. If a lecturer wishes to put more emphasis on evolutionary game theory, I suggest including elements from Sections A.3 and A.4 in the Appendix. 1.4 Acknowledgement Models of Society and Complex Systems retraces the various research themes that I engaged in during the past decade, but it would have been impossible without the foundational and highly inspirational work on which this book is based. Some of these works are referenced in various chapters. I am deeply grateful to these scholars and I count myself lucky that I had the opportunity to be taught by some of them. I take this opportunity to pay my special regards to Charles Anderton, Edgar Sanchez Carrera, Gian Italo Bischi, Habib Saadi, Laura Gardini, and Samuel Bowles for their valuable insights, thought-provoking suggestions, and very helpful comments which much improved this book. Last but not least, my sincere thanks goes to my wife Dina who had to suffer through the earliest versions of this book and yet remained married to me. Thank you for walking by my side and for being my shoulder to lean on. Notes Introduction 1 See also Mackie (1996). 2 For the first detailed modern study of footbinding, refer to Levy (1966). 3 See Mackie (1996) and Turner (1997). 7 8 4 In fact, intellectuals, like Linag Qichao, saw footbinding as a ridiculous custom that made China the laughingstock Introduction to foreigners, see Appiah (2010). 5 See Volz (2007) and Appiah (2010)). 6 The difference between these concepts - practices, norms, conventions - are not clearly defined in the literature and vary across disciplines. We may define a social norm as behaviour that is based on empirical and normative expectations, whereas conventions can be seen as descriptive norms that only rely on empirically learned behaviour. Practices are habitual forms of behaviour that don’t require a normative reinforcement. 7 For a broader discussion, see North (1991). 8 We may think here also of Paul Valéry’s statement: “Ce qui est simple est toujours faux. Ce qui ne l’est pas est inutilisable” (Valéry, 1942). 9 This is the principle of Occam’s razor, named after Franciscan friar William of Ockham who postulated: “[...] plurality must never be posited without necessity” (Scotus and García, 1912, p. 211). Ockham was probably the model for William of Baskervill in Umberto Eco’s The name of the Rose who was played by Sean Connery in the film adaptation. 10 Such cases of final state sensitivity have been first described in detail in Grebogi et al. (1983). 11 But again, we will see that even in these situations, the institutional setup might be defined by a recurrent periodic pattern. 12 Theoretically, a dynamical social system can even illustrate some sort of hybrid dynamics. 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