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El. knyga: Topology of 2x2 Games

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(Laurentian University, Canada), (Laurentian University, Canada)
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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 xv
1 2 x 2 games and the strategic form 1(8)
1.1 Form and solution
2(2)
1.2 2 x 2 games in strategic form
4(2)
1.3 Conventions for payoff matrices
6(2)
1.4 Summary
8(1)
2 144 games 9(24)
2.1 Introduction
9(1)
2.2 The strategic form in payoff space
10(5)
2.2.1 The inducement correspondence
11(1)
2.2.2 Using payoff-space representations to analyse games
12(3)
2.3 Order graphs
15(1)
2.4 Counting the 2 x 2 games
16(4)
2.4.1 Using order graphs to count the 2 x 2 games
17(2)
2.4.2 Numbering the 2 x 2 games
19(1)
2.5 All 144 games
20(5)
2.5.1 Types of order graphs
21(3)
2.5.2 Quasi-symmetric games
24(1)
2.5.3 Assignment and reflection
25(1)
2.6 Summing up
25(2)
2.7 Appendix: Relating payoff patterns to the indexing system
27(6)
3 Elementary topology of 2 x 2 games 33(24)
3.1 About topologies
35(1)
3.2 What is a neighbour?
36(4)
3.2.1 Talking about the neighbours
37(3)
3.3 Groups
40(3)
3.4 Constructing the graph of 2 x 2 games
43(13)
3.4.1 Subgraph/subspace/subgroup generated by a single swap: Z2
44(1)
3.4.2 Two non-overlapping operations: Z2 x Z2
45(1)
3.4.3 Overlapping operations: P6
46(2)
3.4.4 Slices: P24
48(2)
3.4.5 Structure of a stack
50(1)
3.4.6 Layers: P6 x P6
50(2)
3.4.7 Topology of a layer
52(1)
3.4.8 The Euler - Poincare characteristic
52(1)
3.4.9 The four-layered torus
53(1)
3.4.10 Tiling the layers
54(1)
3.4.11 Pipes and hotspots
54(2)
3.5 Structure and content
56(1)
4 Symmetric games 57(16)
4.1 The seven most studied 2 x 2 games
57(1)
4.2 The nature of a symmetric game
58(1)
4.3 Counting the symmetric games
59(3)
4.3.1 Identifying the symmetric games
60(2)
4.4 The space of symmetric games
62(3)
4.5 A map of the symmetric games
65(3)
4.5.1 Types of symmetric games
65(2)
4.5.2 The world of the symmetric games: a flying octahedron
67(1)
4.6 Do the symmetric games matter'?
68(2)
4.7 Appendix: Other subspaces under the symmetric operations
70(3)
5 A Family for the PD 73(20)
5.1 The most famous game
73(1)
5.2 The nature of the Prisoner's Dilemma
74(2)
5.3 Overlapping neighbourhoods
76(5)
5.3.1 Conditions defining the PD as intersecting regions
80(1)
5.4 The Prisoner's Dilemma Family
81(2)
5.5 An alibi for one prisoner
83(2)
5.6 The asymmetry of the Alibi games
85(4)
5.6.1 Evolution with PDF games
85(1)
5.6.2 Bargaining in Alibi games
86(3)
5.7 Rank-sum inefficiency in the PDF
89(1)
5.8 Concluding remarks
90(3)
6 Connecting the layers 93(12)
6.1 Least among equals: the X12 swaps
93(2)
6.2 Instability zone - X12 swaps matter
95(2)
6.3 Pipes at last
97(5)
6.3.1 The pipe with the PD, microcosm of the 2 x 2 games
98(2)
6.3.2 Pipes and layers
100(2)
6.4 Four kinds of pipes
102(3)
7 37 Holes, Coordination games, Battles of the Sexes and the hotspots 105(10)
7.1 Location and structure of hotspots
106(2)
7.2 How many holes? Thirty-seven
108(1)
7.3 Hotspots and their games
109(4)
7.3.1 The two-equilibrium hotspot
109(3)
7.3.2 The no-equilibrium hotspot
112(1)
7.3.3 The other hotspots
113(1)
7.4 Geography of the social dilemmas
113(2)
8 Classifying conflict 115(16)
8.1 Conflict, no conflict, mixed interests
115(3)
8.2 Describing conflict using inducement correspondences
118(1)
8.3 A single-surface map of the 144 games
119(2)
8.4 The pure cases
121(3)
8.5 Giver and taker: the Type games
124(1)
8.6 Mixed motive games
125(1)
8.7 Extending Schelling
126(1)
8.8 Completing the classification
126(3)
8.9 Structure of conflict
129(2)
9 A Periodic Table for the 2 x 2 Games 131(16)
9.1 The Periodic Table of the 2 x 2 games: indexing
132(3)
9.2 Axes of symmetry
135(1)
9.3 Conflict and common interest
136(1)
9.4 Pipes and tiles
137(1)
9.5 Two, one or no dominant strategies
138(2)
9.5.1 Two, one or no Nash equilibria
139(1)
9.5.2 Dominant strategies and unmixed interests
139(1)
9.6 Social dilemmas
140(1)
9.7 Previous typologies
141(2)
9.8 A summary
143(4)
10 The real world 147(12)
10.1 A real-valued version of the model
148(2)
10.2 An evolutionary investigation
150(6)
10.2.1 The ecology of errors
153(3)
10.3 In conclusion: ordinal boundaries and real behaviour
156(3)
Glossary 159(10)
Bibliography 169(4)
Index 173
Goforth, David; Robinson, David
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