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The Topology of the 2×2 Games: a New Periodic Table

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The Topology of the 2×2 Games: a New Periodic Table The Topology of the 2×2 Games The Topology of 2×2 Games provides the reader with a unique new guide to this basic tenet of microeconomics. David Robinson and David Goforth examine the structure of players’ preferences and the resulting topology to create a greater awareness of these games and the relationships that exist between them. This systematic treatment of 2×2 games – presented in tabular format – will provide researchers and teachers in game theory and microeconomics with an indispensable tool. David Robinson is Associate Professor of Economics, and David Goforth is Assistant Professor of Computer Science. They are both at Laurentian University, Canada. Routledge Advances in Game Theory Edited by Christian Schmidt Volume 1 Game Theory and Economic Analysis A Quiet Revolution in Economics Christian Schmidt Volume 2 Negotiation Games Applying Game Theory to Bargaining and Arbitration Steven J. Brams Volume 3 The Topology of the 2×2 Games A New Periodic Table David Robinson and David Goforth The Topology of the 2×2 Games A New Periodic Table David Robinson and David Goforth First published 2005 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Ave, New York, NY 10016 Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” © 2005 David Robinson and David Goforth All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catolog record for this title has been requested ISBN 0-203-34027-2 Master e-book ISBN ISBN 0–415–33609–0 (hbk) The TOPOLOGY of the 2 × 2 games: a new periodic table David Robinson David Goforth June 1, 2005 iv Contents A New Foundation xv 12×2 games and the strategic form 1 1.1Formandsolution...................... 2 1.2 2×2gamesinstrategicform................ 4 1.3Conventionsforpayoffmatrices.............. 6 1.4Summary.......................... 8 2 144 games 9 2.1 Introduction ......................... 9 2.2Thestrategicforminpayoffspace............. 10 2.2.1 The inducement correspondence .......... 11 2.2.2 Using payoff-space representations to analyse games 12 2.3Ordergraphs........................ 15 2.4 Counting the 2 × 2games................. 16 2.4.1 Using order graphs to count the 2 × 2 games . 17 2.4.2 Numbering the 2×2games............. 19 2.5All144games........................ 20 2.5.1 Typesofordergraphs................ 21 2.5.2 Quasi-symmetricgames.............. 24 2.5.3 Assignment and reflection............. 25 2.6Summingup......................... 25 2.7 Appendix: Relating payoff patterns to the indexing system 27 3 Elementary topology of 2 × 2 games 33 3.1 About topologies . ..................... 35 3.2 What is a neighbour? .................... 36 3.2.1 Talking about the neighbours . .......... 37 3.3 Groups . ........................... 40 v vi CONTENTS 3.4 Constructing the graph of 2 × 2games........... 43 3.4.1 Subgraph/subspace/subgroup generated by a single swap: Z2 ...................... 44 3.4.2 Two non-overlapping operations: Z2 × Z2 ..... 45 3.4.3 Overlapping operations: P6 ............. 46 3.4.4 Slices: P24 ..................... 48 3.4.5 Structureofastack................. 50 3.4.6 Layers: P6 × P6 ................... 50 3.4.7 Topology of a layer . ................ 52 3.4.8 The Euler – Poincar´echaracteristic........ 52 3.4.9 Thefour-layeredtorus............... 53 3.4.10 Tiling the layers . ................ 54 3.4.11Pipesandhotspots................. 54 3.5Structureandcontent.................... 56 4 Symmetric games 57 4.1 The seven most studied 2 × 2games............ 57 4.2Thenatureofasymmetricgame.............. 58 4.3Countingthesymmetricgames............... 59 4.3.1 Identifyingthesymmetricgames.......... 60 4.4Thespaceofsymmetricgames............... 62 4.5Amapofthesymmetricgames............... 65 4.5.1 Typesofsymmetricgames............. 65 4.5.2 The world of the symmetric games: a flying octa- hedron....................... 67 4.6Dothesymmetricgamesmatter?.............. 68 4.7 Appendix: Other subspaces under the symmetric opera- tions............................. 70 5 A Family for the PD 73 5.1Themostfamousgame................... 73 5.2ThenatureofthePrisoner’sDilemma........... 74 5.3 Overlapping neighbourhoods ................ 76 5.3.1 Conditions defining the PD as intersecting regions 80 5.4ThePrisoner’sDilemmaFamily.............. 81 5.5Analibiforoneprisoner.................. 83 5.6TheasymmetryoftheAlibigames............. 85 5.6.1 EvolutionwithPDFgames............. 85 5.6.2 BargaininginAlibigames............. 86 5.7 Rank-sum inefficiencyinthePDF............. 89 CONTENTS vii 5.8Concludingremarks..................... 90 6 Connecting the layers 93 6.1 Least among equals: the X12 swaps............. 93 6.2 Instability zone – X12 swapsmatter............. 95 6.3Pipesatlast......................... 97 6.3.1 The pipe with the PD, microcosm of the 2×2 games 98 6.3.2 Pipesandlayers................... 100 6.4Fourkindsofpipes..................... 102 7 37 Holes, Coordination games, Battles of the Sexes and the hotspots 105 7.1Locationandstructureofhotspots............. 106 7.2Howmanyholes?Thirty-seven............... 108 7.3Hotspotsandtheirgames.................. 109 7.3.1 The two-equilibrium hotspot . .......... 109 7.3.2 The no-equilibrium hotspot . .......... 112 7.3.3 Theotherhotspots................. 113 7.4Geographyofthesocialdilemmas............. 113 8 Classifying conflict 115 8.1 Conflict, no conflict,mixedinterests............ 115 8.2 Describing conflict using inducement correspondences . 118 8.3Asingle-surfacemapofthe144games........... 119 8.4Thepurecases........................ 121 8.5Giverandtaker:theTypegames.............. 124 8.6Mixedmotivegames.................... 125 8.7 Extending Schelling ..................... 126 8.8 Completing the classification................ 126 8.9 Structure of conflict..................... 129 9 A Periodic Table for the 2×2 Games 131 9.1 The Periodic Table of the 2 × 2games:indexing...... 132 9.2Axesofsymmetry...................... 135 9.3 Conflictandcommoninterest................ 136 9.4 Pipes and tiles . ..................... 137 9.5Two,oneornodominantstrategies............. 138 9.5.1 Two, one or no Nash equilibria . .......... 139 9.5.2 Dominantstrategiesandunmixedinterests.... 139 9.6Socialdilemmas....................... 140 viii CONTENTS 9.7 Previous typologies ..................... 141 9.8Asummary......................... 143 10 The real world 147 10.1Areal-valuedversionofthemodel............. 148 10.2Anevolutionaryinvestigation................ 150 10.2.1Theecologyoferrors................ 153 10.3 In conclusion: ordinal boundaries and real behaviour . 156 Glossary 159 Bibliography 169 Index 173 List of Figures 1.1Elementsofagamemodel................. 3 2.1PayoffcombinationsforthePrisoner’sDilemma...... 10 2.2 Inducement correspondences for the Prisoner’s Dilemma . 11 2.3 Best responses and Nash equilibrium for PD . ..... 13 2.4OrdergraphforthePrisoner’sDilemma.......... 15 2.5 Ordinal and real-valued representations of the strategic form inpayoffspaceandstrategyspace............. 16 2.6Orientingthepayoffmatrix................. 17 2.7 An approach to counting the 2 × 2games......... 18 2.8 Arrangement of the 2 × 2 games by indices, grc ...... 20 2.9 Elementary payoff patterns, wirings, and numbers of isome- tries............................. 21 2.10 Transformations of g315 ................... 23 2.11Somequasi-symmetricgames............... 24 2.12 Reflections (R) and reassignments (A) .......... 26 2.13 Layer 2:Columngenerallydoeswell........... 30 2.14 Layer 1:WithPD,ChickenandBoS............ 30 2.15 Layer 3:Bestoutcomefeasibleforboth.......... 31 2.16 Layer 4:Rowgenerallydoeswell............. 31 3.1 Changing the ordinal values for the two lowest-ranked out- comesfortherowplayer.................. 36 3.2 The neighbourhood of g412 ................. 38 3.3 Games generated by C12 and C23 from g111 ......... 47 3.4Slice............................. 49 3.5 The stack and the position of highest payoffs . ..... 52 3.6Torus............................ 53 3.7TilesonandlinksfromLayer1............... 54 ix x LIST OF FIGURES 4.1Symmetricgames...................... 59 4.2 Counting the symmetric games geometrically: steps 1 and 2 60 4.3 Identifiersforthesymmetricgames............ 61 4.4ConstructingthePDfromtwonumbers.......... 62 4.5TurningthePDintoChicken................ 62 4.6Thelinked-loopmodelofthesymmetricgames...... 63 4.7Symmetricgamesasverticesofapolyhedron....... 64 4.8Thespaceofthe12symmetricgames........... 66 4.9 Evolution of the flyingoctahedron............. 68 4.10Symmetricstructures.................... 71 4.11Wingedoctahedron..................... 72 5.1 The dominant strategy layout for Layer 1, showing position of g111,Prisoner’sDilemma................. 76 5.2 Four layers in dominant strategy layout – symmetric games on the diagonal . ..................... 77 5.3 Corner of 4 layers with some X34 links – PD and PDF . 78 5.4 Restitched dominant
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