Essays on Pareto Optimality in Cooperative Games

169 Pages·2022·2.846 MB·English

by  Yaning Lin, Weihai Zhang

Yaning Lin
Weihai Zhang
Essays on Pareto
Optimality in
Cooperative
Games
Essays on Pareto Optimality in Cooperative Games
·
Yaning Lin Weihai Zhang
Essays on Pareto Optimality
in Cooperative Games
YaningLin WeihaiZhang
SchoolofMathematicsandStatistics CollegeofElectricalEngineering
ShandongUniversityofTechnology andAutomation
Zibo,Shandong,China ShandongUniversityofScience
andTechnology
Qingdao,Shandong,China
ISBN 978-981-19-5048-3 ISBN 978-981-19-5049-0 (eBook)
https://doi.org/10.1007/978-981-19-5049-0
©SpringerNatureSingaporePteLtd.2022
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Preface
Thisbookcontainsadiscussionofsomestochasticdifferentialgamesanddifference
games.Onlyonemodeofgameisconsidered,thatis,Paretooptimalityforcoopera-
tion.Thereasonforustoconfinetheresearchinteresttocooperativegamesisnotthat
noncooperative games are less interesting, but that cooperative game theory is far
lessdevelopedthannoncooperativegametheory.Inthisbook,necessary/sufficient
conditionsfortheexistenceofParetosolutionsarederived,thecharacterizationof
Pareto solutions is discussed, and many practical applications in economics and
managementscienceareprovided.
Tomakethebookself-contained,Chap.1givesabriefintroductiontothehistor-
ical development of game theory and provides some basics of Pareto optimality.
Chapter 2 discusses Pareto optimality in the finite horizon stochastic differential
games. Chapter 3 reveals the essential differences from Chap. 2 and extends the
obtained results in Chap. 2 from the finite horizon case to the infinite horizon
situation. By introducing a new generalized differential Riccati equation (GDRE),
Chap. 4 puts forward a sufficient condition for the solvability of the finite horizon
stochastic singular linear quadratic (LQ) optimal control problem. Then, for the
finitehorizonLQParetogameofthestochasticsingularsystems,Chap.4provides
a sufficient condition to derive all Pareto-efficient strategies by the weighted sum
minimization method. Employing the equivalent transformation method, Chap. 5
studiestheLQoptimalcontrolofthestochasticsingularsystemsininfinitehorizon
and gives a necessary and sufficient condition for the solvability. Further, for the
infinite horizon stochastic singular LQ Pareto game, under the solvability of the
corresponding stochastic algebraic Riccati equation (SARE), all Pareto solutions
arederivedbasedonthesolutionsofanintroducedgeneralizedalgebraicLyapunov
equation(GALE).Chapter6isconcernedwithPareto-basedguaranteedcostcontrol
(GCC) for the uncertain mean-field stochastic systems. By the discussion of the
convexityofthecostfunctionals,Pareto-basedGCCproblemissettledbytheGCC
oftheweightedsumcostfunctional,andnecessaryconditionsfortheexistenceof
Pareto-basedguaranteedcostcontrollersarederived.Chapter7investigatesexistence
conditionsofParetosolutionsinthefinitehorizoncooperativedifferencegames.The
v
vi Preface
infinitehorizoncounterpartissolvedinChap.8,inwhichsomeessentialdifferences
areelaborated.
This book is particularly suitable for the readers interested in multi-objective
optimization and can benefit researchers, engineers and graduate students in the
fields of control theory and control engineering, economics, management science
andoperationsresearch.
Zibo,China YaningLin
Qingdao,China WeihaiZhang
June2022
Acknowledgements
WewouldliketothankthestaffofSpringer,inparticularKowsalyaRaghunathan,
AugustusVinothandJasmineDou,fortheirhelpandsupport.YaningLinwouldlike
tothankherrespectivePh.D.advisorProfessorWeihaiZhangforhispatientguidance
andconstantencouragement.YaningLinacknowledges supportfromtheNational
NaturalScienceFoundationofChina(No.61903234)andtheNaturalScienceFoun-
dation of Shandong Province (No. ZR2019MF008). Weihai Zhang acknowledges
supportfromtheNationalNaturalScienceFoundationofChina(No.61973198),the
ResearchFundfortheTaishanScholarProjectofShandongProvinceofChinaand
SDUSTResearchFund(No.2015TDJH105).
Finally, both authors would like to thank their families for their long-lasting
understandingandsupport.
vii
Contents
1 Introduction ................................................... 1
1.1 Background ............................................... 1
1.2 ParetoOptimality ........................................... 2
1.3 ResearchStatus ............................................ 4
1.3.1 ResearchStatusofParetoOptimalityinCooperative
DifferentialGames ................................... 4
1.3.2 ResearchStatusofDifferenceGames ................... 5
References ..................................................... 5
2 ExistenceConditionsofParetoSolutionsinFiniteHorizon
StochasticDifferentialGames ................................... 7
2.1 Introduction ............................................... 7
2.2 NonlinearCase ............................................. 8
2.2.1 NecessaryConditions ................................. 10
2.2.2 SufficientConditions ................................. 12
2.3 LQCase .................................................. 15
2.3.1 Open-LoopSolution .................................. 16
2.3.2 Closed-LoopSolution ................................ 19
2.4 Examples ................................................. 24
2.5 Conclusions ............................................... 30
References ..................................................... 30
3 ExistenceConditionsofParetoSolutionsinInfiniteHorizon
StochasticDifferentialGames ................................... 31
3.1 Introduction ............................................... 31
3.2 NonlinearCase ............................................. 31
3.2.1 NecessaryConditions ................................. 33
3.2.2 SufficientConditions ................................. 34
3.3 LQCase .................................................. 36
3.3.1 Open-LoopSolution .................................. 36
3.3.2 Closed-LoopSolution ................................ 41
3.4 Examples ................................................. 45
ix
x Contents
3.5 Conclusions ............................................... 47
References ..................................................... 48
4 LQParetoGameoftheStochasticSingularSystemsinFinite
Horizon ....................................................... 49
4.1 Introduction ............................................... 49
4.2 FiniteHorizonLQOptimalControloftheStochasticSingular
Systems ................................................... 50
4.3 FiniteHorizonStochasticSingularLQParetoGame ............. 58
4.4 Examples ................................................. 62
4.5 Conclusions ............................................... 66
References ..................................................... 66
5 LQParetoGameoftheStochasticSingularSystemsinInfinite
Horizon ....................................................... 69
5.1 Introduction ............................................... 69
5.2 Indefinite Infinite Horizon LQ Optimal Control
oftheStochasticSingularSystems ............................ 69
5.3 InfiniteHorizonStochasticSingularLQParetoGame ............ 75
5.4 Examples ................................................. 80
5.5 Conclusions ............................................... 83
References ..................................................... 83
6 Pareto-Based Guaranteed Cost Control of the Uncertain
Mean-FieldStochasticSystems .................................. 85
6.1 Introduction ............................................... 85
6.2 Preliminaries .............................................. 86
6.3 ParetoGameoftheNominalMean-FieldStochasticSystems ..... 93
6.4 Pareto-Based Guaranteed Cost Control of the Uncertain
Mean-FieldStochasticSystems ............................... 100
6.5 LMIApproach ............................................. 105
6.6 Examples ................................................. 108
6.7 Conclusions ............................................... 111
References ..................................................... 112
7 ParetoOptimalityinFiniteHorizonCooperativeDifference
Games ........................................................ 115
7.1 Introduction ............................................... 115
7.2 NonlinearCase ............................................. 115
7.2.1 NecessaryConditions ................................. 118
7.2.2 SufficientConditions ................................. 121
7.3 LQCase .................................................. 123
7.3.1 Open-LoopSolution .................................. 124
7.3.2 Closed-LoopSolution ................................ 128
7.4 Examples ................................................. 132
7.5 Conclusions ............................................... 138
References ..................................................... 138
Contents xi
8 ParetoOptimalityinInfiniteHorizonCooperativeDifference
Games ........................................................ 139
8.1 Introduction ............................................... 139
8.2 NonlinearCase ............................................. 139
8.2.1 NecessaryConditions ................................. 142
8.2.2 SufficientConditions ................................. 145
8.3 LQCase .................................................. 146
8.3.1 Open-LoopSolution .................................. 147
8.3.2 Closed-LoopSolution ................................ 150
8.4 Examples ................................................. 152
8.5 Conclusions ............................................... 157
References ..................................................... 157
Index ............................................................. 159