Agent-Based Evolutionary Game Dynamics Agent-Based Evolutionary Game Dynamics A guide to implement and analyze Agent-Based Models within the framework of Evolutionary ameG Theory, using NetLogo LUIS R. IZQUIERDO, SEGISMUNDO S. IZQUIERDO, AND WILLIAM H. SANDHOLM Agent-Based Evolutionary ameG Dynamics by Luis R. Izquierdo, Segismundo S. Izquierdo & William H. Sandholm is licensed under a Creative Commons Attribution 4.0n I ternational License, except where otherwise noted. Cover picture by Dino Reichmuth. This book was produced with Pressbooks (https://pressbooks.com) and rendered with Prince. Contents Preface vii 0. Introduction 0.1. Introduction to ve olutionary game theory 2 0.2. Introduction to agent-based modeling 14 0.3. Introduction to eN tLogo 21 0.4. The fundamentals of NetLogo 26 1. Our first agent-based evolutionary model 1.0. Our very first model 44 1.1. Extension to any number of strategies 56 1.2. Noise and initial onditionsc 67 1.3. Interactivity and fficiencye 77 1.4. Analysis of these models 90 2. Spatial interactions on a grid 2.0. Spatial chaos in the risoner'P s Dilemma 117 2.1. Robustness and fragility 132 2.2. Extension to any number of strategies 145 2.3. Other types of neighborhoods and other revision protocols 165 3. Games on networks 4. Revision protocols and general payoff functions 5. Endogenous networks 6. Multipopulation games 7. Solving the mean dynamics at runtime Models implemented in this book 188 References 190 Preface 1. What is this book about? The objective of this book is to help you learn to implement and analyze evolutionary models fo social interactions in finite populations. heT following paragraphs explain why these two skills –i.e. model implementation and analysis– are key for scientific modeling. Model implementation To use a scientific model rigorously, it is important to be fully aware of all the assumptions embedded in it, and also of the various alternative assumptions that could have been chosen. If we don’t understand all the details of a model, we run the risk of over-extrapolating its scope and of drawing unsound conclusions. A great way to understand a model in depth is to implement it in computer code following an agent-based approach. We believe this is true regardless of whether the model is currently expressed in natural language (and may even exist only in your mind) or, alternatively, it is written down in mathematical language (e.g. using equations). Coding a model expressed in natural language is very useful because it ensures that the model is both consistent and completely specified. A computer implementation fo a model is necessarily consistent because the language used to code it (i.e. the programming language) is formal, so it does not allow ambiguities to infiltrate; symbols and instructions used in programming languages have always the same meaning regardless of context. A computer implementation fo a model must also be completely specified before it can be run, since computers do not make assumptions yb themselves. Thus, to execute a model in a computer, there cannot be any loose ends in the description fo the model. This contrasts with models expressed in natural language, where it is easy to leave aspects of the model partially unspecified ften–o unintentionally–, since our brains are particularly oodg at using context to unconsciously fill the details. This implies that the audience may be understanding something slightly different from what is meant to be communicated, and results may be driven by assumptions that are not made explicitly. By contrast, computers need all assumptions to be spelt out, and this requirement makes the process of scientific modelling more sound and rigorous. If a model is written in the language of mathematics, the problems outlined in the paragraph above are no longer an issue. Thus, is it really worth implementing a mathematical model in computer code following an agent-based approach? We believe that, in many cases, it certainly is. The reason is that many mathematical models onc tain assumptions that are desirable for analytical tractability, but which also weaken the link between the model and the real world. These assumptions made orf analytical onc venience tend to elevate the mathematical model to a higher level of abstraction and aggregation. yB contrast, the agent-based approach has the advantage of forcing the programmer to implement the microfoundations fo a model explicitly, considering each individual agent as a separate Agent-Based Evolutionary Game Dynamics | vii entity. This requirement helps the modeler be aware of all the assumptions that are made in the mathematical model, and it also allows for an assessment of their significance. Model analysis Once the model is implemented, the way to fully understand it is to analyze it. In this book, we will see several techniques that are useful to analyze finite-population ve olutionary models, including arkM ov chain analyses, Monte Carlo simulations, mean dynamics, stochastic stability analyses and diffusion approximations. orF each of these techniques, we give a brief introduction, illustrate its usefulness with concrete examples, and provide references for the interested reader to learn more about it. 2. How to use this book This book has been written and ormaf tted ollof wing a hands-on approach. To make the most of it, we encourage you to have NetLogo open, and to code the models as you read the book. We are hopeful and confident that if you go through the whole text, implement the proposed models, and try to do some of the exercises included at the end of most sections, ouy will master the art of implementing and analyzing agent-based evolutionary dynamics. Nonetheless, we also have in mind two other types of less committed eaders:r • If you are interested in learning to analyze finite-population ve olutionary models, and in their relation with thero models in Evolutionary ameG Theory, but you do not wish to program, then you should skip all the sections preceded by the label CODE . • If you want to become proficient in coding agent-based models, but you are not interested in learning how to analyze these models, please do reconsider your preference. If your preference persists after careful reflection,ou y may want to skip the section titledAnaly “ sis of these models”, which you will find ta the end of most chapters. Our hope is that, regardless of the discipline you are coming from, your background and your preferences, this book will help you learn new and exciting aw ys of understanding evolutionary systems. viii | Luis R. Izquierdo, Segismundo S. Izquierdo, and William H. Sandholm 0. INTRODUCTION Agent-Based Evolutionary Game Dynamics | 1 0.1. Introduction to ve olutionary game theory Evolutionary ameG Theory (EGT) is a branch of a more general discipline called game theory. Therefore, to understand what EGT is about, we believe it is useful to get familiar with the basic ideas underlying game theory first. 1. What is game theory? Game theory is a discipline devoted to studying social interactions where individuals’ decisions are interdependent, i.e. situations where the outcome of the interaction orf any individual generally depends not only on her own choices, but also on the choices made by every other individual. Thus, several scholars have pointed out that game theory could well be defined as ‘interactive decision theory’ (Myerson, 1997, p. 1). Some examples of interactive decisions for which game theory is useful are: 1. Choosing the side of the road on which to drive. 2. Choosing WhatsApp or Facebook messenger as your default text messaging app. 3. Choosing which restaurant to go in the following situation: Imagine that over breakfast your partner and you decided to have lunch together, but you did not specify where. Right before lunch time ouy discover that you left oury cellphone at home, and there is no other way to contact your partner quickly. Fortunately, you are not completely lost: there are only two restaurants that you both love in town. Which one should you go to? Interactive social interactions like the ones outlined above are modeled in game theory as games. A game is an abstract representation fo a social interaction which is meant to capture its most basic properties. In particular, a game typically comprises: • the set of individuals who interact (called players), • the different choices available to each of the individuals when they are called upon to act (named actions or pure strategies), • the information individuals have at the time fo making their decisions, • and a payoff function that assigns a value to each individual for each possible combination fo choices made by every individual (Fig. 1). In most cases, payoffs eprr esent the preferences of 1 each individual over each possible outcome of the social interaction, though there are some 1. A common misconception about game theory relates to the roots of players’ preferences. There is no assumption in game theory that dictates that players’ preferences are formed in complete disregard of each other’s interests. On the contrary, preferences in game theory are assumed to account for anything, i.e. they may include altruistic 2 | Agent-Based Evolutionary Game Dynamics evolutionary models where payoffs eprr esent Darwinian fitness. Player 2 Player 2 chooses A Player 2 chooses B Player 1 chooses A 1 , 1 0 , 0 Player 1 Player 1 chooses B 0 , 0 2 , 2 Figure 1. Payoff matrix of a 2-player 2-strategy game. For each possible combination of pure strategies there is a corresponding pair of numbers (x , y) in the matrix whose first element x represents the payoff for player 1, and whose second element y represents the payoff for player 2.