The topology of the 2x2 games: a new periodic table / David Robinson, David Goforth

Game theory has implications for all the social sciences and beyond. It now provides the theoretical basis for almost all teaching in economics, and 2x2 games provide the very basis of game theory. Here, Goforth and Robinson here have delivered a well-written and knowledgeable, 'periodic table' of the most common games including: \* the prisoner's dilemma \* coordination games \* chicken \* the battle of the sexes. This book will provide a valuable reference for students of microeconomics and business mathematics. A New Foundation -- 1 2x2 Games And The Strategic Form -- 1.1 Form And Solution -- Form -- Solution -- 1.2 2x2 Games In Strategic Form -- Equivalent Games -- Representative Games -- 1.3 Conventions For Payoff Matrices -- Four Matrices Per Game -- 1.3.1 Summary -- 2 144 Games -- 2.1 Introduction -- 2.2 Payoff Space -- 2.2.1 The Inducement Correspondence -- 2.2.2 Analysis In Payoff Space -- Nash Equilibrium -- Best Response Analysis -- 2.3 Order Graphs -- Order Graphs And Matrices -- 2.4 Counting The 2x2 Games -- 2.4.1 Using Order Graphs To Count The 2x2 Games -- 2.4.2 Numbering The 2x2 Games -- 2.5 All 144 Games -- 2.5.1 Types Of Order Graphs -- Wirings -- Assignment -- Iv Contents -- Rotations And Reflections -- 2.5.2 Quasi-symmetric Games -- 2.5.3 Assignment And Reflection -- 2.6 Summing Up -- 2.7 Appendix: Payoff Patterns And Indexing -- Row, Column And Stack -- 3 Elementary Topology Of The 2x2 Games -- 3.1 About Topologies -- 3.2 What Is A Neighbour? -- 3.2.1 Talking About The Neighbours -- Bad Neigbourhood? -- Strange Neighbours -- Symmetric Neighbours -- 3.3 Groups -- 3.4 Constructing The Graph Of 2x2 Games. 3.4.1 The Subgraph Of A Single Swap: Z2 -- 3.4.2 Non-overlapping Swaps: Z2xz2 -- 3.4.3 Overlapping Operations: P6 -- 3.4.4 Slices: P24 -- 3.4.5 Structure Of A Stack -- 3.4.6 Layers: P6xp6 -- The No Conflict Layer -- 3.4.7 Topology Of A Layer -- 3.4.8 The Euler: Poincaŕe Characteristic -- 3.4.9 The Four-layered Torus -- 3.4.10 Tiling The Layers -- 3.4.11 Pipes And Hotspots -- 3.5 Structure And Content -- 4 Symmetric Games -- 4.1 The Seven Most Studied 2x2 Games -- 4.2 The Nature Of A Symmetric Game -- 4.3 Counting The Symmetric Games -- 4.3.1 Identifying The Symmetric Games -- Example -- 4.4 The Space Of Symmetric Games -- 4.5 A Map Of The Symmetric Games -- 4.5.1 Types Of Symmetric Games -- 4.5.2 A Flying Octahedron -- 4.6 Do The Symmetric Games Matter? -- 4.7 Appendix: Other Subspaces Under Symmetric Operations.70 -- Six-game Subspaces -- Twelve-game Subspaces -- Twenty-four-game Subspaces -- 5 A Family For The Pd. 5.1 The Most Famous Game -- 5.2 The Nature Of The Prisoner's Dilemma -- Three Descriptions Of The Pd -- 5.3 Overlapping Neighbourhoods -- The Dominant Strategy Layout -- Symmetry -- Connection Between Layers -- Restitching The Dominant Strategy Layout -- 5.3.1 Intersecting Regions -- 5.4 Games Of The Prisoner's Dilemma Family -- Social Dilemmas -- 5.5 An Alibi For A Prisoner -- The Prisoner's Dilemma, Classic -- An Alibi Game -- 5.6 The Asymmetry Of The Alibi Games -- 5.6.1 Evolution With Pdf Games -- 5.6.2 Bargaining In Alibi Games -- Solutions -- 5.7 Rank-sum Inefficiency -- 5.8 Conclusions -- 6 Connecting The Layers -- 6.1 Importance Of Tiles -- 6.2 Instability Zone And X12 Swaps -- 6.3 Pipes At Last -- 6.3.1 The Prisoner's Dilemma Pipe: A Microcosm -- 6.3.2 Pipes And Layers -- 6.4 Four Kinds Of Pipes -- 7 37 Holes -- 7.1 Location And Structure Of Hotspots -- 7.2 How Many Holes? -- 7.3 Hotspots And Their Games -- Vi Contents -- 7.3.1 Two Equilibria -- Battles Of The Sexes -- Coordination Games -- 7.3.2 No-equilibrium Hotspot -- 7.3.3 The Other Hotspots -- 7.4 Geography Of The Social Dilemmas -- 8 Classifying Conflict. 8.1 Conflict, No Conflict, Common And Mixed Interests -- 8.2 Terminology -- 8.3 A Single Surface Map -- 8.3.1 Linking The Layers -- 8.3.2 Villarceau Circles On The 144-game Torus -- 8.4 Contours Of Conflict And Cooperation -- 8.5 Giver And Taker: The Type Games -- 8.6 Counting Conflict Correctly -- 8.7 Structure Of Conflict -- 9 A Periodic Table For The 2x2 Games -- 9.1 The Periodic Table Of The 2x2 Games -- On The Choice Of The Numbering System -- 9.2 Axes Of Symmetry -- 9.3 Conflict And Common Interest -- 9.4 Pipes And Tiles -- 9.5 Two, One, Or No Dominant Strategies -- 9.5.1 Two, One, Or No Nash Equilibria -- 9.5.2 Pareto Optimality -- 9.5.3 Dominant Strategies And Unmixed Interests -- 9.6 Social Dilemmas -- 9.7 Cross-classifications -- 10 Real Payoffs -- 10.1 A Real-valued Version Of The Model -- 10.2 An Evolutionary Investigation -- A Very Wide View -- A Closer View -- 10.2.1 The Ecology Of Errors -- Results -- Chaos Crossing The Pd Region -- 10.3 Implications. David Robinson, David Goforth. Includes Bibliographical References (p. 169-172) And Index.