Econophysics of the Kolkata Restaurant Problem and Related Games: Classical and Quantum Strategies for Multi-agent, Multi-choice Repetitive Games

212 Pages·2017·4.97 MB·English

by  Bikas K. Chakrabarti

New Economic Windows
Bikas K. Chakrabarti
Arnab Chatterjee
Asim Ghosh
Sudip Mukherjee
Boaz Tamir
Econophysics of the
Kolkata Restaurant
Problem and
Related Games
Classical and Quantum Strategies for
Multi-agent, Multi-choice Repetitive
Games
Econophysics of the Kolkata Restaurant Problem
and Related Games
New Economic Windows
Series editors
MARISA FAGGINI, MAURO GALLEGATI, ALAN P. KIRMAN, THOMAS LUX
Series Editorial Board
Jaime Gil Aluja
Departament d’Economia i Organització d’Empreses, Universitat de Barcelona,
Barcelona, Spain
Fortunato Arecchi
Dipartimento di Fisica, Università degli Studi di Firenze and INOA, Florence, Italy
David Colander
Department of Economics, Middlebury College, Middlebury, VT, USA
Richard H. Day
Department of Economics, University of Southern California, Los Angeles, USA
Steve Keen
School of Economics and Finance, University of Western Sydney, Penrith, Australia
Marji Lines
Dipartimento di Scienze Statistiche, Università degli Studi di Udine, Udine, Italy
Alfredo Medio
Dipartimento di Scienze Statistiche, Università degli Studi di Udine, Udine, Italy
Paul Ormerod
Directors of Environment Business-Volterra Consulting, London, UK
Peter Richmond
School of Physics, Trinity College, Dublin 2, Ireland
J. Barkley Rosser
Department of Economics, James Madison University, Harrisonburg, VA, USA
Sorin Solomon Racah
Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel
Pietro Terna
Dipartimento di Scienze Economiche e Finanziarie, Università degli Studi di Torino,
Torino, Italy
Kumaraswamy (Vela) Velupillai
Department of Economics, National University of Ireland, Galway, Ireland
Nicolas Vriend
Department of Economics, Queen Mary University of London, London, UK
Lofti Zadeh
Computer Science Division, University of California Berkeley, Berkeley, CA, USA
More information about this series at http://www.springer.com/series/6901
Bikas K. Chakrabarti Arnab Chatterjee
(cid:129)
Asim Ghosh Sudip Mukherjee
(cid:129)
Boaz Tamir
Econophysics of the Kolkata
Restaurant Problem
and Related Games
Classical and Quantum Strategies
for Multi-agent, Multi-choice Repetitive
Games
123
BikasK.Chakrabarti and
CondensedMatterPhysicsDivision
SahaInstituteofNuclearPhysics RaghunathpurCollege
Kolkata Purulia
India India
and SudipMukherjee
DepartmentofPhysics
EconomicResearchUnit BarasatGovernmentCollege
IndianStatisticalInstitute Kolkata
Kolkata India
India
and
ArnabChatterjee
CondensedMatterPhysicsDivision CondensedMatterPhysicsDivision
SahaInstituteofNuclearPhysics SahaInstituteofNuclearPhysics
Kolkata Kolkata
India India
and BoazTamir
DepartmentofScienceTechnologyandSociety,
TCSInnovationLabs FacultyofInterdisciplinaryStudies
Delhi Bar-IlanUniversity
India Ramat-Gan
Israel
AsimGhosh
DepartmentofComputerScience and
AaltoUniversitySchoolofScience
Espoo Iyar,TheIsraeliInstituteforAdvancedResearch
Finland ZikhronYaakov
Israel
ISSN 2039-411X ISSN 2039-4128 (electronic)
NewEconomic Windows
ISBN978-3-319-61351-2 ISBN978-3-319-61352-9 (eBook)
DOI 10.1007/978-3-319-61352-9
LibraryofCongressControlNumber:2017945223
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
TheKolkata Restaurant ProblemormorespecificallytheKolkata PaiseRestaurant
(KPR)problemisarepetitivemany-choiceandmany-agentgamewheretheplayers
collectively learn from past experience, and the identification of successful strate-
gieshelptoavoidthecrowdinordertograbthebest(minoritychoice)opportunity
available. Although the binary-choice many-agent games, called minority games,
havebeenstudiedearlier,thesemany-choicegameshavemanydimensions,andthe
successful strategies here can have intriguing features. Detailed studies on several
stochasticstrategies,whichensuremaximumutilizationoftheresources,havebeen
made. These are mostly classical (using statistical physics tricks). Quantum
strategies are also being formulated with interesting features. This book intends to
give an introduction on this recent development.
Along with the main text on these interdisciplinary developments, we have
included six appendices for the benefit of the readers. Five of them are invited
contributions from our colleagues: Statistical Physics: A Brief Introduction by
Purusattam Ray (Institute of Mathematical Sciences, Chennai), Quantum
Mechanics: A Brief Introduction by Parthasarathi Mitra (Saha Institute of Nuclear
Physics, Kolkata), Game Theory (Classical): A Brief Introduction by Priyodorshi
Banerjee (Indian Statistical Institute, Kolkata), Manipushpak Mitra (Indian
StatisticalInstitute,Kolkata)andConanMukherjee(LundUniversity,Lund,Indian
Institute of Technology Bombay, Mumbai), Minority Game: An Overview and
Recent Results by V. Sasidevan (Frankfurt Institute for Advanced Studies,
Frankfurt),andExtendingKPRProblemtoDynamicMatchinginMobilityMarkets
byLaylaMartinandPaulKaraenke(TechnicalUniversityofMunich,Munich).We
areextremely thankfulfor their contributions tothis volume. Another appendix on
Econophysics: A Brief Introduction has also been added by us. Each chapter,
including of course the appendices, contains sufficient introductory materials, with
referencestootherchaptersorappendicesfordetails,sothatthereadercanchoose
each chapter almost independently.
We are grateful to our collaborators Soumyajyoti Biswas, Anindya S.
Chakrabarti, Anirban Chakraborti, Damien Challet, Daniele De Martino, Deepak
Dhar, Matteo Marsili, Tapan Naskar, V. Sasidevan, and Yi-Cheng Zhang for their
v
vi Preface
collaborations in the development of this and related game models. We are also
gratefultoMauroGallegatiandothereditorsoftheNewEconomicWindowsseries
for their encouragement to publish this book in their esteemed series.
This book is addressed to students and researchers in economics (game theory
andmodelgames),physics(statisticalphysics,quantummechanics),andcomputer
science (job scheduling, etc). We hope they will find material here sufficiently
exciting and useful for inspiring researches in these directions.
Kolkata, India Bikas K. Chakrabarti
Delhi, India Arnab Chatterjee
Espoo, Finland Asim Ghosh
Kolkata, India Sudip Mukherjee
Ramat-Gan, Israel Boaz Tamir
March 2017
Acknowledgements
We thank P. Banerjee, L. Martin, P. Karaenke, M. Mitra, P. Mitra, C. Mukherjee,
P. Ray, and V. Sasidevan for contributing appendices to this book. We would like
tothankourcollaboratorsS.Biswas,A.S.Chakrabarti,A.Chakraborti,D.Challet,
D.DeMartino,D.Dhar,M.Marsili,T.Naskar,V.Sasidevan,andY.-C.Zhangfor
collaborationsanddiscussionsatvariousstages.WearealsogratefultoS.Biswas,
A.S.Chakrabarti,D.Ghosh,andL.Martinfortheirhelpbyprovidinguswiththeir
yet unpublished results of investigations regarding extended Kolkata Restaurant
problems in various contexts, through preprints and thesis copy (private commu-
nication) and checking our summary of their results, given in Chap. 7.
vii
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Background .... .... ..... .... .... .... .... .... ..... .... 1
1.1.1 Minority Game ..... .... .... .... .... .... ..... .... 2
1.1.2 Kolkata Restaurant Problem ... .... .... .... ..... .... 2
1.2 Motivation of the Book .... .... .... .... .... .... ..... .... 3
1.3 Plan of the Book .... ..... .... .... .... .... .... ..... .... 5
2 Kolkata Paise Restaurant Problem .. .... .... .... .... ..... .... 7
2.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 7
2.2 Stochastic Learning Strategies ... .... .... .... .... ..... .... 9
2.2.1 Random Choice Strategies .... .... .... .... ..... .... 9
2.2.2 Rank Dependent Strategies.... .... .... .... ..... .... 10
2.2.3 Strict Crowd-Avoiding Case... .... .... .... ..... .... 12
2.2.4 Stochastic Crowd Avoiding Case ... .... .... ..... .... 12
2.3 Convergence to a Fair Social Norm with Deterministic
Strategies .. .... .... ..... .... .... .... .... .... ..... .... 13
2.3.1 A ‘Fair’ Strategy.... .... .... .... .... .... ..... .... 14
2.3.2 Asymptotically Fair Strategy... .... .... .... ..... .... 14
2.4 Summary and Discussion... .... .... .... .... .... ..... .... 14
3 Phase Transition in the Kolkata Paise Restaurant Problem.... .... 17
3.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 17
3.2 The Models .... .... ..... .... .... .... .... .... ..... .... 18
3.3 Results from Numerical Simulations... .... .... .... ..... .... 21
3.3.1 Model A. .... ..... .... .... .... .... .... ..... .... 21
3.3.2 Model B . .... ..... .... .... .... .... .... ..... .... 22
3.4 Analytical Treatment of the Models in Mean Field Case .... .... 23
3.4.1 Approximate Analysis of the Critical Point
and Faster-Is-Slower Effect.... .... .... .... ..... .... 26
3.4.2 Analysis of the Finite Size Effects on the Time
to Reach the Absorbing State .. .... .... .... ..... .... 28
3.5 Summary and Discussions .. .... .... .... .... .... ..... .... 29
ix
x Contents
4 Zipf’s Law from Kolkata Paise Restaurant Problem.... ..... .... 31
4.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 31
4.2 Model. .... .... .... ..... .... .... .... .... .... ..... .... 33
4.3 Results .... .... .... ..... .... .... .... .... .... ..... .... 34
4.3.1 Distribution of Sizes. .... .... .... .... .... ..... .... 34
4.3.2 Utilization.... ..... .... .... .... .... .... ..... .... 37
4.3.3 Evolution with Fitness ... .... .... .... .... ..... .... 39
4.4 Empirical Evidences.. ..... .... .... .... .... .... ..... .... 40
4.5 Summary and Discussions .. .... .... .... .... .... ..... .... 41
5 Minority Game and Kolkata Paise Restaurant Problem . ..... .... 43
5.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 43
5.2 Strategy of the Agents ..... .... .... .... .... .... ..... .... 44
5.2.1 Uniform Approximation in Guessing the Excess
Crowd... .... ..... .... .... .... .... .... ..... .... 45
5.2.2 Nonuniform Guessing of the Excess Crowd... ..... .... 48
5.2.3 Following an Annealing Schedule... .... .... ..... .... 50
5.3 Effect of Random Traders... .... .... .... .... .... ..... .... 52
5.4 Summary and Discussions .. .... .... .... .... .... ..... .... 54
6 From Classical Games, the Kokata Paise Restuarant
Game, to Quantum Games..... .... .... .... .... .... ..... .... 55
6.1 A Short Introduction to Classical Games ... .... .... ..... .... 55
6.1.1 Definitions and Preliminaries... .... .... .... ..... .... 56
6.1.2 Repeated Games.... .... .... .... .... .... ..... .... 62
6.1.3 Games and Evolution Theory .. .... .... .... ..... .... 66
6.2 KPR.. .... .... .... ..... .... .... .... .... .... ..... .... 69
6.2.1 Some Simple KPR Results .... .... .... .... ..... .... 70
6.2.2 Phase Transition .... .... .... .... .... .... ..... .... 72
6.2.3 Minority Games .... .... .... .... .... .... ..... .... 74
6.2.4 KPR Non-stochastic . .... .... .... .... .... ..... .... 79
6.3 Quantum Games. .... ..... .... .... .... .... .... ..... .... 79
6.3.1 Quantum Strategies.. .... .... .... .... .... ..... .... 84
6.3.2 Nash Equilibrium in Quantum Games.... .... ..... .... 84
6.3.3 Quantum Coin Tossing and Bit Commitment.. ..... .... 89
6.3.4 Strong and Weak Coin Tossing. .... .... .... ..... .... 91
6.3.5 Quantum Games and Semidefinite Programming .... .... 93
6.4 Quantum KPR .. .... ..... .... .... .... .... .... ..... .... 100
6.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 102
7 Some Recent Developments: A Brief Discussion.... .... ..... .... 105
7.1 KPR Under Dynamic Setting .... .... .... .... .... ..... .... 105
7.2 Reinforcement Learning.... .... .... .... .... .... ..... .... 106
7.3 KPR and Wealth Distribution in Society ... .... .... ..... .... 110
7.4 Summary and Discussions .. .... .... .... .... .... ..... .... 111