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An Introduction to Dynamic Games PDF

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An Introduction to Dynamic Games A. Haurie J. Krawczyk March 28, 2000 2 Contents 1 Foreword 9 1.1 WhatareDynamicGames? . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 OriginsoftheseLectureNotes . . . . . . . . . . . . . . . . . . . . . 9 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 I Elements of Classical Game Theory 13 2 DecisionAnalysiswithManyAgents 15 2.1 TheBasicConceptsofGameTheory . . . . . . . . . . . . . . . . . . 15 2.2 GamesinExtensiveForm . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Descriptionofmoves,informationandrandomness . . . . . . 16 2.2.2 ComparingRandomPerspectives . . . . . . . . . . . . . . . 18 2.3 Additionalconceptsaboutinformation . . . . . . . . . . . . . . . . . 20 2.3.1 Completeandperfectinformation . . . . . . . . . . . . . . . 20 2.3.2 Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Bindingagreement . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 GamesinNormalForm . . . . . . . . . . . . . . . . . . . . . . . . 21 3 4 CONTENTS 2.4.1 Playinggamesthroughstrategies . . . . . . . . . . . . . . . . 21 2.4.2 Fromtheextensiveformtothestrategicornormalform . . . 22 2.4.3 MixedandBehaviorStrategies . . . . . . . . . . . . . . . . . 24 3 Solutionconceptsfornoncooperativegames 27 3.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 MatrixGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.1 Saddle-Points . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Mixedstrategies . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.3 AlgorithmsfortheComputationofSaddle-Points . . . . . . . 34 3.3 BimatrixGames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 NashEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.2 ShortcommingsoftheNashequilibriumconcept . . . . . . . 38 3.3.3 Algorithms for the Computation of Nash Equilibria in Bima- trixGames . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Concavem-PersonGames . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.1 ExistenceofCoupledEquilibria . . . . . . . . . . . . . . . . 45 3.4.2 NormalizedEquilibria . . . . . . . . . . . . . . . . . . . . . 47 3.4.3 UniquenessofEquilibrium . . . . . . . . . . . . . . . . . . . 48 3.4.4 Anumericaltechnique . . . . . . . . . . . . . . . . . . . . . 50 3.4.5 Avariationalinequalityformulation . . . . . . . . . . . . . . 50 3.5 Cournotequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.1 ThestaticCournotmodel . . . . . . . . . . . . . . . . . . . . 51 CONTENTS 5 3.5.2 Formulation of a Cournot equilibrium as a nonlinear comple- mentarityproblem . . . . . . . . . . . . . . . . . . . . . . . 52 3.5.3 ComputingthesolutionofaclassicalCournotmodel . . . . . 55 3.6 Correlatedequilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.1 Exampleofagamewithcorrelatedequlibria . . . . . . . . . 56 3.6.2 Ageneraldefinitionofcorrelatedequilibria . . . . . . . . . . 59 3.7 Bayesianequilibriumwithincompleteinformation . . . . . . . . . . 60 3.7.1 Exampleofagamewithunknowntypeforaplayer . . . . . . 60 3.7.2 Reformulationasagamewithimperfectinformation . . . . . 61 3.7.3 AgeneraldefinitionofBayesianequilibria . . . . . . . . . . 63 3.8 AppendixonKakutaniFixed-pointtheorem . . . . . . . . . . . . . . 64 3.9 exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 II Repeated and sequential Games 67 4 Repeatedgamesandmemorystrategies 69 4.1 Repeatingagameinnormalform . . . . . . . . . . . . . . . . . . . 70 4.1.1 Repeatedbimatrixgames . . . . . . . . . . . . . . . . . . . . 70 4.1.2 Repeatedconcavegames . . . . . . . . . . . . . . . . . . . . 71 4.2 Folktheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Repeatedgamesplayedbyautomata . . . . . . . . . . . . . . 74 4.2.2 Minimaxpoint . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.3 Setofoutcomesdominatingtheminimaxpoint . . . . . . . . 76 4.3 CollusiveequilibriuminarepeatedCournotgame . . . . . . . . . . . 77 6 CONTENTS 4.3.1 Finitevsinfinitehorizon . . . . . . . . . . . . . . . . . . . . 79 4.3.2 A repeated stochastic Cournot game with discounting and im- perfectinformation . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Shapley’sZeroSumMarkovGame 83 5.1 Processandrewardsdynamics . . . . . . . . . . . . . . . . . . . . . 83 5.2 Informationstructureandstrategies . . . . . . . . . . . . . . . . . . 84 5.2.1 Theextensiveformofthegame . . . . . . . . . . . . . . . . 84 5.2.2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Shapley’s-Denardooperatorformalism . . . . . . . . . . . . . . . . . 86 5.3.1 Dynamicprogrammingoperators . . . . . . . . . . . . . . . 86 5.3.2 Existenceofsequentialsaddlepoints . . . . . . . . . . . . . 87 6 Nonzero-sumMarkovandSequentialgames 89 6.1 SequentialGamewithDiscretestateandactionsets . . . . . . . . . . 89 6.1.1 Markovgamedynamics . . . . . . . . . . . . . . . . . . . . 89 6.1.2 Markovstrategies . . . . . . . . . . . . . . . . . . . . . . . . 90 6.1.3 Feedback-Nashequilibrium . . . . . . . . . . . . . . . . . . 90 6.1.4 Sobel-Whittoperatorformalism . . . . . . . . . . . . . . . . 90 6.1.5 ExistenceofNash-equilibria . . . . . . . . . . . . . . . . . . 91 6.2 SequentialGamesonBorelSpaces . . . . . . . . . . . . . . . . . . . 92 6.2.1 Descriptionofthegame . . . . . . . . . . . . . . . . . . . . 92 6.2.2 Dynamicprogrammingformalism . . . . . . . . . . . . . . . 92 CONTENTS 7 6.3 ApplicationtoaStochasticDuopoloyModel . . . . . . . . . . . . . . 93 6.3.1 Astochasticrepeatedduopoly . . . . . . . . . . . . . . . . . 93 6.3.2 Aclassoftriggerstrategiesbasedonamonitoringdevice . . . 94 6.3.3 Interpretationasacommunicationdevice . . . . . . . . . . . 97 III Differential games 99 7 Controlleddynamicalsystems 101 7.1 Acapitalaccumulationprocess . . . . . . . . . . . . . . . . . . . . . 101 7.2 Stateequationsforcontrolleddynamicalsystems . . . . . . . . . . . 102 7.2.1 Regularityconditions . . . . . . . . . . . . . . . . . . . . . . 102 7.2.2 Thecaseofstationarysystems . . . . . . . . . . . . . . . . . 102 7.2.3 Thecaseoflinearsystems . . . . . . . . . . . . . . . . . . . 103 7.3 Feedbackcontrolandthestabilityissue . . . . . . . . . . . . . . . . 103 7.3.1 Feedbackcontrolofstationarylinearsystems . . . . . . . . . 104 7.3.2 stabilizingalinearsystemwithafeedbackcontrol . . . . . . 104 7.4 Optimalcontrolproblems . . . . . . . . . . . . . . . . . . . . . . . . 104 7.5 Amodelofoptimalcapitalaccumulation . . . . . . . . . . . . . . . . 104 7.6 Theoptimalcontrolparadigm . . . . . . . . . . . . . . . . . . . . . 105 7.7 TheEulerequationsandtheMaximumprinciple . . . . . . . . . . . . 106 7.8 AneconomicinterpretationoftheMaximumPrinciple . . . . . . . . 108 7.9 Synthesisoftheoptimalcontrol . . . . . . . . . . . . . . . . . . . . 109 7.10 Dynamicprogrammingandtheoptimalfeedbackcontrol . . . . . . . 109 8 CONTENTS 7.11 Competitivedynamicalsystems . . . . . . . . . . . . . . . . . . . . 110 7.12 Competitionthroughcapitalaccumulation . . . . . . . . . . . . . . . 110 7.13 Open-loopdifferentialgames . . . . . . . . . . . . . . . . . . . . . . 110 7.13.1 Open-loopinformationstructure . . . . . . . . . . . . . . . . 110 7.13.2 Anequilibriumprinciple . . . . . . . . . . . . . . . . . . . . 110 7.14 Feedbackdifferentialgames . . . . . . . . . . . . . . . . . . . . . . 111 7.14.1 Feedbackinformationstructure . . . . . . . . . . . . . . . . 111 7.14.2 Averificationtheorem . . . . . . . . . . . . . . . . . . . . . 111 7.15 WhyarefeedbackNashequilibriaoutcomesdifferentfromOpen-loop Nashoutcomes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.16 Thesubgameperfectnessissue . . . . . . . . . . . . . . . . . . . . . 111 7.17 Memorydifferentialgames . . . . . . . . . . . . . . . . . . . . . . . 111 7.18 Characterizingallthepossibleequilibria . . . . . . . . . . . . . . . . 111 IV A Differential Game Model 113 7.19 AGameofR&DInvestment . . . . . . . . . . . . . . . . . . . . . . 115 7.19.1 DynamicsofR&D competition . . . . . . . . . . . . . . . . 115 7.19.2 ProductDifferentiation . . . . . . . . . . . . . . . . . . . . . 116 7.19.3 Economicsofinnovation . . . . . . . . . . . . . . . . . . . . 117 7.20 Informationstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.20.1 Statevariables . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.20.2 Piecewiseopen-loopgame. . . . . . . . . . . . . . . . . . . . 118 7.20.3 ASequentialGameReformulation . . . . . . . . . . . . . . . 118 Chapter 1 Foreword 1.1 What are Dynamic Games? DynamicGamesaremathematicalmodelsoftheinteractionbetweendifferentagents who are controlling a dynamical system. Such situations occur in many instances like armed conflicts (e.g. duel between a bomber and a jet fighter), economic competition (e.g. investments in R&D for computer companies), parlor games (Chess, Bridge). Theseexamplesconcerndynamicalsystemssincetheactionsoftheagents(alsocalled players)influencetheevolutionovertimeofthestateofasystem(positionandvelocity of aircraft, capital of know-how for Hi-Tech firms, positions of remaining pieces on a chess board, etc). The difficulty in deciding what should be the behavior of these agentsstemsfromthefactthateachactionanagenttakesatagiventimewillinfluence the reaction of the opponent(s) at later time. These notes are intended to present the basic concepts and models which have been proposed in the burgeoning literature on gametheoryforarepresentationofthesedynamicinteractions. 1.2 Origins of these Lecture Notes These notes are based on several courses on Dynamic Games taught by the authors, in different universities or summer schools, to a variety of students in engineering, economics and management science. The notes use also some documents prepared in cooperationwithotherauthors,inparticularB.Tolwinski[Tolwinski,1988]. Thesenotesarewrittenforcontrolengineers,economistsormanagementscien- tists interested in the analysis of multi-agent optimization problems, with a particular 9 10 CHAPTER1. FOREWORD emphasis on the modeling of conflict situations. This means that the level of mathe- maticsinvolvedinthepresentationwillnotgobeyondwhatisexpectedtobeknownby a student specializing in control engineering, quantitative economics or management science. Thesenotesareaimedatlast-yearundergraduate,firstyeargraduatestudents. The Control engineers will certainly observe that we present dynamic games as an extensionofoptimalcontrolwhereaseconomistswillseealsothatdynamicgamesare only a particular aspect of the classical theory of games which is considered to have been launched in [VonNeumann&Morgenstern1944]. Economic models of imper- fectcompetition,presentedasvariationsonthe”classic”Cournotmodel[Cournot,1838], will serve recurrently as an illustration of the concepts introduced and of the theories developed. Aninterestingdomainofapplicationofdynamicgames,whichisdescribed in these notes, relates to environmental management. The conflict situations occur- ringinfisheriesexploitationbymultipleagentsorinpolicycoordinationforachieving global environmental control (e.g. in the control of a possible global warming effect) arewellcapturedintherealmofthistheory. The objects studied in this book will be dynamic. The term dynamic comes from Greek dynasthai (which means to be able) and refers to phenomena which undergo a time-evolution. In these notes, most of the dynamic models will be discrete time. This implies that, for the mathematical description of the dynamics, difference (rather than differential) equations will be used. That, in turn, should make a great part of the notesaccessible,andattractive,tostudentswhohavenotdoneadvancedmathematics. However,therewillstillbesomedevelopmentsinvolvingacontinuoustimedescription ofthedynamicsandwhichhavebeenwrittenforreaderswithastrongermathematical background. 1.3 Motivation There is no doubt that a course on dynamic games suitable for both control engineer- ing students and economics or management science students requires a specialized textbook. Sinceweemphasizethedetaileddescriptionofthedynamicsofsomespecificsys- tems controlled by the players we have to present rather sophisticated mathematical notions, related to control theory. This presentation of the dynamics must be accom- panied by an introduction to the specific mathematical concepts of game theory. The originality of our approach is in the mixing of these two branches of applied mathe- matics. There are many good books on classical game theory. A nonexhaustive list in-

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