Classification by Decomposition A Partitioning of the Space of 2 × 2 Symmetric Games Department of Mathematics, Linköping University Mikael Böörs, Tobias Wängberg LiTH-MAT-EX–2017/10—SE Credits: 16 hp Level: G2 Supervisor: Tom Everitt, Marcus Hutter, Computer Science Department, Australian National University Examiner: Martin Singull, Department of Mathematics, Linköping University Linköping: May 2017 Abstract Game theory is the study of strategic interaction between rational agents. The need for understanding interaction arises in many different fields, such as: eco- nomics, psychology, philosophy, computer science and biology. The purpose of game theory is to analyse the outcomes and strategies of these interactions, in mathematical models called games. Some of these games have stood out from the rest, e.g. Prisoner’s Dilemma, Chicken and Stag Hunt. These games, com- monly referred to as the standard games, have attracted interest from many fields of research. In order to understand why these games are interesting and how they differ from each other and other games, many have attempted to sort games into interestingly different classes. In this thesis some already existing classifications are reviewed based on their mathematical structure and how well justified they are. Emphasis is put on mathematical simplicity because it makes the classification more generalisable to larger game spaces. From this review we conclude that none of the classifications captures both of these aspects. We therefore propose a classification of symmetric 2×2 games based on decomposi- tion. We show that our proposed method captures everything that the previous classifications capture. Our method arguably explains the interesting differ- ences between the games, and we justify this claim by computer experiments. Moreover it has a simple mathematical structure. We also provide some results concerning the size of different game spaces. Keywords: Game theory, Classification, 2 × 2 Games, Symmetric games, Decomposi- tion, Partition, Number of Games URL for electronic version: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-137991 Böörs, Wängberg, 2017. iii Acknowledgements First and foremost we would like to express our deep gratitude to our supervi- sors Tom Everitt and Dr. Marcus Hutter at the Australian National University for inviting us to write this thesis under their superior guidance and for their enthusiasm about this project. We would also like to thank our examiner Dr. Martin Singull at Linköping University for taking the time to read and examine this thesis. Johan Persson deserves a special thanks both for opposing our thesis and for his personal friendship. Also our fantastic classmates Erik Landstedt and André Malm deserve a special thanks for their highly valued friendship and for their companionship through many late nights of studying. My personal thanks goes to my family, my close friends and everyone else who have supported me and brightened my days throughout the years - you know who you are. -Mikael Böörs Apart from those mentioned earlier, I wish to also express my profound gratitude towards my family and friends. You have always been there to support me throughout the years. No accomplishment would have been possible without you. Thank you. -Tobias Wängberg Böörs, Wängberg, 2017. v Nomenclature The notation used in the thesis. Symbol Explanation R; N; ::: set of real,natural numbers k; l 2 N indices for the natural numbers i; j 2 N indices for players ui payoff function for player i U set of payoff functions Ai set of actions available to player i a action profile si strategy of player i Si set of strategies of player i s strategy profile S set of strategy profiles 2S Powerset of a set S Γ(G) The mixed extension of a game G Σi set of probability distributions over Si Σ set of mixed strategy profiles σi mixed strategy of player i σ mixed strategy profile Pi payoff matrix of player i G game end of proof P probability ~v vector ∼ strategical equivalence Böörs, Wängberg, 2017. vii viii NE Nash equilibrium HO Huertas optimality PO Pareto optimality zk Conflict parameter ck Common interest parameter Contents 1 Introduction 1 2 Background 3 2.1 Game Theory . .3 2.1.1 Introduction to Game Theory . .3 2.1.2 Strategic Games and Equilibria . .5 2.1.3 Zero-Sum Game . .8 2.1.4 Mixed Strategies . 10 2.1.5 Pareto Optimality . 13 2.1.6 2 × 2 Games . 14 2.1.7 Standard 2 × 2 Games . 18 2.2 Classification of Games . 24 2.2.1 Classification of all ordinal 2 × 2 games by Rapoport, Guyer, and Gordon . 25 2.2.2 A Geometric Classification System for 2 × 2 Inverval- Symmetric Games, (Harris, 1969) . 29 2.2.3 A Classification of 2 × 2 Bimatrix Games (Borm, 1987) . 32 2.2.4 A Cartography for 2 × 2 Symmetric Games (Huertas- Rosero, 2003) . 41 2.2.5 A Topologically-Based Classification of the 2 × 2 Ordinal Games (Robinson and Goforth, 2003) . 49 2.2.6 Analysis . 53 3 Results 55 3.1 Classification by Decomposition . 55 3.1.1 Decomposition of Symmetric 2 × 2 Games . 55 3.1.2 Stereographic Projection . 59 3.2 Nash and Huertas Optimality . 62 3.3 Deterministic and Non-deterministic Classes . 69 Böörs, Wängberg, 2017. ix x Contents 3.4 Analysis of the Regions . 74 3.4.1 Stronger Zero-Sum . 78 3.4.2 Weaker Zero-Sum . 83 3.4.3 Mixed Strength . 87 3.4.4 Comparison Between the Standard Game Regions . 92 3.4.5 Conclusions . 93 3.5 Experimental Analysis . 93 3.5.1 Experiment Design . 93 3.5.2 Analysis of Action Frequencies . 97 3.5.3 Analysis of Strategy Frequencies . 105 3.6 Analysis of the Standard Game Regions . 107 3.6.1 Analysis of the Prisoner’s Dilemma Regions . 108 3.6.2 Analysis of the Stag Hunt Regions . 111 3.6.3 Analysis of the Chicken Regions . 112 3.6.4 Analysis of the Leader and Hero Regions . 113 3.7 Comparison with Reviewed Classifications . 114 3.8 Number of m × n Games . 117 3.8.1 Distinct players . 117 3.8.2 Non-distinct players . 119 4 Discussion 123 4.1 Outlook on Further Topics . 126 A Detailed Class Description 133 B Computer Experiment Plots 139 C Some Results Regarding the Number of Non-Strictly Ordinal Games With Distinct Players 143 Chapter 1 Introduction Game theory is described as the study of strategic interaction between ratio- nal agents. The first underlying assumption that the interactions are strategic means that the choices of the agents involved in the interaction affects each other and that they are aware of this. They therefore have to reason strategi- cally about not only their own action, but also about the actions of others. The second assumption states that the agent is rational. An agent is rational if it always takes an action which maximises its expected utility, according to the agents subjective preferences and information. The purpose of game theory is to model complex real world situations as abstract models with strategic inter- action between rational agents. These models are referred to as games, and are used to understand how agents will interact in various situations. In order to fully understand games, they need to be classified based on prop- erties that capture the essence of these games. A classification is a systematic way to investigate the properties of different games. If a game is to be useful, we need to understand what kind of real world situation it can represent. To achieve this we make up stories that enables us to relate these games to different real world scenarios, e.g. the story of the prisoners in the Prisoner’s Dilemma. A lot of different stories could, however, describe the same game. Therefore the essential properties of the game needs to be identified, upon which a classifi- cation is built. There is a need to understand what games are essentially the same and what games are different, and why. A classification of games is one approach to achieving this goal. An interesting question in game theory is what makes a game interesting and how the properties that define an interesting game can be formalised. When Böörs, Wängberg, 2017. 1 2 Chapter 1. Introduction defining a classification of games the goal is to divide games with different kinds of interesting properties into different classes. Therefore, before one defines a classification, the natural question to answer is what properties are interesting? and perhaps even more important, why do we find these properties interesting? We suggest that for a classification to be truly useful, it should have: 1. simple mathematical structure, 2. a priori justified conditions, 3. a posteriori justified conditions and 4. no ad hoc conditions. We motivate this as follows. We consider how well they justify their conditions a priori, that is how well the authors of the classification motivate their choice of conditions used to classify. There should also be some justification of whether the resulting regions captures all aspects which are considered interesting about the games. We call this a posteriori justification. Furthermore, most classifica- tions classify the smallest form of non-trivial games, that is the 2 × 2 games. These are games with only two players with two actions each. These games do, however, only make up a small fraction of the total amount of games. Hence, it is important that the classification can be generalised by adding, for example, more players. We therefore advocate that a classification should have a simple mathematical structure. A lot of research has already been done on the topic of classifying games. When reviewing previous research we argue that the previous classifications fail, on at least some account, to fulfil the conditions stated above.