Coordinate Systems for Games: Simplifying the "me" and "we" Interactions (Static & Dynamic Game Theory: Foundations & Applications)

Description

This monograph develops a method of creating convenient coordinate systems for game theory that will allow readers to more easily understand, analyze, and create games at various levels of complexity. By identifying the unique characterization of games that separates the individual’s strategic interests from the group’s collective behavior, the authors construct a single analytical methodology that readers will be able to apply to a wide variety of games. With its emphasis on practicality and approachability, readers will find this book an invaluable tool, and a viable alternative to the ad hoc analytical approach that has become customary for researchers utilizing game theory. The introductory chapters serve two important purposes: they review several games of fundamental importance, and also introduce a dynamic that is inherent in games, but has gone unexplored until now. After this has been established, readers will advance from simple 2 x 2 games to games with more player strategies and dynamics. For interested readers, a rigorous treatment of the underlying mathematics is conveniently gathered at the end of the book. Additional topics of interest, such as extensive form and coalitional games, are presented to help readers visualize more complex settings that will be vital in aiding the understanding of advanced topics, such as coalition-free Nash points, multi-player repeated games, and more. Coordinate Systems for Games is ideal for a wide variety of researchers interested in game theory, including social scientists, economists, mathematicians, computer scientists, and more. The authors' approachable style also makes this accessible to an audience at any scale of experience, from beginning non-specialists to more practiced researchers. Preface Book Outline Contents 1 Introduction 1.1 Review 1.2 An Inherent Dynamic 1.2.1 Nash Dynamics 1.2.2 Structure of 2times2 Games 1.3 Standard Games 1.3.1 Games with One Nash Cell 1.3.2 Games with Two Nash Cells 1.4 Mixed Strategies 1.4.1 Games Without Nash Cells 1.4.2 Finding the Probabilities 1.4.3 Matching Pennies 1.4.4 Another Source of Mixed Strategies 1.5 Nash Theorem and Defining Generic and Degenerate Settings 2 Two-Player, Two-Strategy Games 2.1 Preview 2.2 Introduction 2.3 The Nash Decomposition 2.3.1 The Nash, or ``me'' Component 2.3.2 The Behavioral ``we'' Factor, and Kernel Terms 2.3.3 More Examples 2.4 Gaining Intuition; Comparing Examples 2.4.1 The Nash Component and Game Dynamic 2.4.2 Behavior Component and Cooperation 2.4.3 What Does All of This Mean? 2.4.4 Creating Exotic Examples 2.4.5 Does Size Matter? 2.5 Structure of Standard Games 2.5.1 Prisoner's Dilemma 2.5.2 Collective Action 2.5.3 Cooperation 2.5.4 Hawk–Dove 2.5.5 Battle of the Sexes 2.5.6 Stag Hunt 2.6 Risk Versus Payoff Dominance 2.6.1 Payoff Dominance 2.6.2 Risk Dominance 2.6.3 Examples 2.6.4 A 50–50 Choice 2.7 Recap 3 Consequences 3.1 Introduction 3.2 Symmetric Games 3.3 Zero-Sum and Identical Play 3.3.1 CoCo Solutions 3.3.2 Zero-Sum Games 3.3.3 Identical Play Games 3.4 Using Vectors to Analyze and Design Games 3.4.1 Review of Vector Analysis 3.4.2 Intuition and a Vector Basis 3.4.3 Designing Games 3.4.4 Utility Functions 3.4.5 Congestion and Potential Games 3.4.6 Zero-Sum, Identical Play Games 3.5 Missing Terms When Analyzing Beliefs 3.6 Roadmap of 2times2 Games 3.6.1 The Nash Portion Is a Square; Ok, A Torus 3.6.2 Likelihood of Different Strategic Structures 3.6.3 Extending a Monderer–Shapley Observation 3.6.4 Why Not Symmetric, Antisymmetric? 3.7 The Behavioral Portion and a Circle 4 Two-Person Multi-strategy Games 4.1 Introduction 4.2 Decomposition 4.2.1 More on Risk and Payoff Dominance 4.2.2 More on Zero-Sum and Identical Play Games 4.3 Examples 4.3.1 Ultimatum Game 4.3.2 Legislator Pay Raise Game 4.4 Modeling and Creating Games 4.4.1 Differences in the mathcalGB Structure 4.4.2 Dimension Counting 4.5 Structure of Two-Person Games 4.5.1 Where's Waldo? 4.5.2 Next Response Dynamic 4.6 Structure of the Next Response Dynamic 4.6.1 Number of Hyperbolic Cells 4.6.2 The Next Response Dynamic 4.7 Some Games 4.7.1 Colonel Blotto 4.7.2 Rock-Paper-Scissors 4.7.3 Those Repulsive Cells 4.8 Computing Mixed Equilibria 4.8.1 Help from the Decomposition 4.8.2 Moving to 3times3 Blocks 5 Extensive Form Games 5.1 What Is with What? 5.2 Centipede Game 5.3 Equilibria and Subgame Equilibrium 5.4 Nature's Involvement in a 2times2 Game 5.5 From a Decomposition to Coordinates 5.5.1 Nash Component for the General Game 5.5.2 Changing Probabilities 5.5.3 The Behavioral Terms for the General Game 5.6 Paradise for Designers! 5.7 Details of the Decomposition 5.7.1 Subgames and Decisions 5.7.2 Player 2 5.7.3 Player 1 5.7.4 Alternative Basis for Behavioral Coordinates 6 Multiplayer Games 6.1 Hawk–Dove–Dove 6.1.1 Analyzing Hawk–Dove–Dove 6.1.2 Designing a Hawk–Dove–Dove Game 6.2 Nash Dynamic Structure 6.2.1 Adding Complexity 6.2.2 Those Retaliatory Strategies 6.3 New Symmetries 6.3.1 A Wedding Between Voting and Game Theory 6.3.2 Another Designer's Delight! Coalition-Free Nash Points? 6.3.3 More Strategies, More Cycles 6.4 Comments 7 The Underlying Mathematics 7.1 Decompositions 7.2 Definitions 7.3 Symmetry 7.4 Algebraic Structure of Games 7.4.1 Symmetries of the Space of Games 7.4.2 Homomorphisms 7.4.3 Module 7.5 Best Response Algebraic Structures 7.6 Decompositions of Games 7.7 Finding the Structure 7.7.1 Conjugacy Classes 7.7.2 Nash Decomposition 7.7.3 Row-and-Column Decomposition–Potential Games 7.7.4 Pareto Decomposition–CoCo Solutions 7.8 Conclusion 8 Summary Appendix References Index