Table Of ContentPolitecnico di Milano
School of Industrial and Information Engineering
Master of Science in Mathematical Engineering
Selfish Reciprocal Altruism:
an evolutionary rule for strategy
update in social networks
Supervisor: Prof. Carlo PICCARDI
Co-supervisor: Prof. Fabio DERCOLE
Dott. Fabio DELLA ROSSA
Candidate:
Claudia RUFFONI
Personal ID 836977
Academic Year 2015-2016
Ai miei genitori,
Giovanna e Cesare
Contents
Sommario 1
Abstract 3
1 Introduction 5
1.1 The role of cooperation . . . . . . . . . . . . . . . . . . . . . . 5
1.2 About this work . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Evolutionary Dynamics 11
2.1 The Prisoner’s Dilemma game . . . . . . . . . . . . . . . . . . 11
2.2 Mechanisms to favor cooperation . . . . . . . . . . . . . . . . 13
2.3 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Network Properties . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Network Models . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Evolutionary Dynamics in Social Networks . . . . . . . . . . . 25
2.4.1 Strategy Update . . . . . . . . . . . . . . . . . . . . . 25
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ii CONTENTS
2.4.2 The PD Game in Social Networks . . . . . . . . . . . . 28
2.5 Evolution by imitation . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Evolutionary Dynamics of Social Dilemmas in Struc-
tured Heterogeneous Populations . . . . . . . . . . . . 32
3 Selfish Reciprocal Altruism 35
3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Preliminary analysis for star network . . . . . . . . . . 41
3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Explanatory example . . . . . . . . . . . . . . . . . . . 45
3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 A possible extension: Mean Field Analysis . . . . . . . . . . . 55
4 Conclusions 59
Ringraziamenti 67
List of Figures
2.1 Undirected graph . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Directed graph . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Weighted and unweighted graphs . . . . . . . . . . . . . . . . 19
2.4 Connected and disconnected graphs . . . . . . . . . . . . . . . 20
2.5 Examples of assortative and disassortative networks . . . . . . 22
2.6 Complete network . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Star network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Birth-Death process . . . . . . . . . . . . . . . . . . . . . . . . 26
2.9 Death-Birth process . . . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Link dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.11 Imitation process . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.12 Replicator Dynamics . . . . . . . . . . . . . . . . . . . . . . . 29
2.13 Heterogeneous network . . . . . . . . . . . . . . . . . . . . . . 30
2.14 Results of Santos’ model . . . . . . . . . . . . . . . . . . . . . 34
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iv LIST OF FIGURES
3.1 Initialization step . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Interaction step . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Update step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Star network with C hub . . . . . . . . . . . . . . . . . . . . . 42
3.5 Selfish reciprocal altruism: example 1 . . . . . . . . . . . . . . 46
3.6 Selfish reciprocal altruism: example 2 . . . . . . . . . . . . . . 46
3.7 Selfish reciprocal altruism: example 3 . . . . . . . . . . . . . . 47
3.8 Selfish reciprocal altruism: example 4 . . . . . . . . . . . . . . 47
3.9 Selfish reciprocal altruism: example 5 . . . . . . . . . . . . . . 47
3.10 Selfish reciprocal altruism: example 6 . . . . . . . . . . . . . . 48
3.11 Selfish reciprocal altruism: example 7 . . . . . . . . . . . . . . 48
3.12 Selfish reciprocal altruism for BA networks . . . . . . . . . . . 52
3.13 Selfish reciprocal altruism for ER networks . . . . . . . . . . . 53
3.14 Selfish reciprocal altruism for complete networks . . . . . . . . 54
3.15 Selfish reciprocal altruism for star networks . . . . . . . . . . . 55
List of Tables
2.1 Prisoner’s Dilemma example . . . . . . . . . . . . . . . . . . . 12
2.2 PD 4-parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 PD 2-parameter: T and S . . . . . . . . . . . . . . . . . . . . 12
2.4 PD 2-parameter: b and c . . . . . . . . . . . . . . . . . . . . . 13
2.5 PD 1-parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Payoff values and conditions for strategy change . . . . . . . . 41
v
Description:3.14 Selfish reciprocal altruism for complete networks 54. 3.15 Selfish reciprocal . We propose a new rule for strategy update, called selfish reciprocal altruism, where players decide to update L the number of iterations considered and m(t) the floating average at iteration t, then: m(t) = 1.
