Smoothed Analysis in Nash Equilibria and the Price of Anarchy

87 Pages·2012·1.6 MB·English

UNIVERSIDADEFEDERALDORIOGRANDEDOSUL
INSTITUTODEINFORMÁTICA
PROGRAMADEPÓS-GRADUAÇÃOEMCOMPUTAÇÃO
FÉLIXCARVALHORODRIGUES
Smoothed Analysis in Nash Equilibria and
the Price of Anarchy
Thesispresentedinpartialfulfillment
oftherequirementsforthedegreeof
MasterofComputerScience
Profa.Dra.LucianaSaleteBuriol
Advisor
Prof.Dr.MarcusRitt
Coadvisor
PortoAlegre,December2011
CIP–CATALOGING-IN-PUBLICATION
Rodrigues,FélixCarvalho
Smoothed Analysis in Nash Equilibria and the Price of
Anarchy / Félix Carvalho Rodrigues. – Porto Alegre: PPGC
daUFRGS,2011.
87f.: il.
Thesis(Master)–UniversidadeFederaldoRioGrandedoSul.
ProgramadePós-GraduaçãoemComputação,PortoAlegre,BR–
RS, 2011. Advisor: Luciana Salete Buriol; Coadvisor: Marcus
Ritt.
1.Algorithmicgametheory. 2.Smoothedanalysis. 3.Lemke-
Howsonalgorithm. 4.Bimatrixgames. 5.Frank-Wolfealgorithm.
6. Network games. 7. Traffic assignment problem. 8. Price of
anarchy. I.Buriol,LucianaSalete. II.Ritt,Marcus. III.Título.
UNIVERSIDADEFEDERALDORIOGRANDEDOSUL
Reitor: Prof. CarlosAlexandreNetto
Vice-Reitor: Prof. RuiVicenteOppermann
Pró-ReitordePós-Graduação: Prof. AldoBoltenLucion
DiretordoInstitutodeInformática: Prof. LuísdaCunhaLamb
CoordenadordoPPCG:ÁlvaroFreitasMoreira
Bibliotecária-ChefedoInstitutodeInformática: BeatrizReginaBastosHaro
“A common mistake that people make when trying to design something
completely foolproof is to underestimate the ingenuity of complete fools.”
— DOUGLAS ADAMS
(THE HITCHHIKER’S GUIDE TO THE GALAXY)

ACKNOWLEDGMENTS
My most profound gratitude goes to Prof. Guido Schäfer, who has made this thesis
possible and whom without it would not be possible for me to go into the game theory
field and leave with a sane mind. For all his ideas, corrections, patience and specially
guidance,whichIcarrynotonlyforthisthesisbutformylife.
I thank my advisors, Prof. Luciana Buriol and Prof. Marcus Ritt, both which had
the patience and willingness to explore a different field and provided invaluable support
throughthewholetimeIwaswritingthisthesis.
I thank everybody at CWI (Centrum Wiskunde & Informatica) in Amsterdam, spe-
cially Bart de Keijzer, Tony Huynh and again Guido Schäfer for making me feel at home
while so far away. I am specially grateful to Bart and his family, for letting me stay in
theirhomeandofferingsuchwarmhospitalitythewholeperiodIwasthere.
I would also like to thank the “European South American Network for Combinato-
rialOptimizationunderUncertainty”project,programmeacronym/referenceFP7-People/
247574, for the financial support while I was at Amsterdam, which made this thesis pos-
sible.
I thank Kao Félix and Letícia Nunes for the friendship and the needed moments of
normallifewhiledoingthisMaster’sprogram.
Finally, I would like to give a special thank you to my father Osvaldir, my mother
HelenaandmysisterPriscila. WithouttheirsupportIwouldneverbeallowedtospendso
muchtime onthe academic worldand onthisthesis, withso little worryon otheraspects
oflife.

CONTENTS
LIST OF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . 9
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
RESUMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 StructureofthisThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 GAME THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 NashEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.1 PPADComplexityClass . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 NetworkGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 BimatrixGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 TheLemke-HowsonAlgorithm . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 PriceofAnarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 THE TRAFFIC ASSIGNMENT PROBLEM . . . . . . . . . . . . . . . . 31
3.1 TheFrank-WolfeAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 TrafficAssignmentProbleminGameTheory . . . . . . . . . . . . . . . 34
3.2.1 PriceofAnarchyintheTrafficAssignmentProblem . . . . . . . . . . . . 34
4 SMOOTHED ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1 PerturbationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 SmoothedPriceofAnarchy . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 SMOOTHED COMPLEXITY OF BIMATRIX GAMES . . . . . . . . . . 41
5.1 KnownWorst-CaseInstances . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.1 AnExactImplementationoftheLemke-HowsonAlgorithm . . . . . . . . 43
5.2.2 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 SMOOTHED PRICE OF ANARCHY IN THE TRAFFIC ASSIGNMENT
PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1 LowerBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1.1 LinearLatencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1.2 PolynomialLatencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 ExperimentalResultsinBenchmarkInstances . . . . . . . . . . . . . . 63
6.2.1 ExperimentalSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 DiscussionandFutureWork . . . . . . . . . . . . . . . . . . . . . . . . 70
7.1.1 Hypercubebasedinstances . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.1.2 SmoothedPriceofAnarchyindifferentcontexts . . . . . . . . . . . . . . 72
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
APPENDIX A ANÁLISE SUAVISADA EM EQUILíBRIOS NASH E NO
PREÇO DA ANARQUIA . . . . . . . . . . . . . . . . . . . 77
A.1 TeoriadosJogos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2 JogosBimatrizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.2.1 OalgoritmodeLemke-Howson . . . . . . . . . . . . . . . . . . . . . . . 78
A.3 PreçodaAnarquia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
A.4 ProblemadeAtribuiçãodeTráfego . . . . . . . . . . . . . . . . . . . . . 79
A.4.1 PreçodaAnarquiaparaoProblemadeAtribuiçãodeTráfego . . . . . . . 80
A.5 AnáliseSuavizada(SmoothedAnalysis) . . . . . . . . . . . . . . . . . . . 81
A.6 ComplexidadeSuavizadadeJogosBimatrizes . . . . . . . . . . . . . . . 81
A.6.1 ResultadosExperimentaisparaJogosBimatrizes . . . . . . . . . . . . . . 82
A.7 PreçodaAnarquiaSuavizado(SmoothedPriceofAnarchy) . . . . . . . 83
A.8 PreçodaAnarquiaSuavizadoparaoProblemadeAtribuiçãodeTráfego 84
A.8.1 ResultadosExperimentaisemInstânciasdeBenchmark . . . . . . . . . . 86
A.9 Conclusões . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
LIST OF ABBREVIATIONS AND ACRONYMS
FW Frank-Wolfealgorithm
LH Lemke-Howsonalgorithm
NE NashEquilibrium
PoA PriceofAnarchy
PoS PriceofStability
TAP TrafficAssignmentProblem
SPoA SmoothedPriceofAnarchy