Newton’s Method, Bellman Recursion and Differential Dynamic Programming for Unconstrained Nonlinear Dynamic Games

Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense, energy systems and etc. However, compared to single-agent control problems, the computational methods for dynamic games are relatively limited...

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Veröffentlicht in:	Dynamic games and applications Jg. 12; H. 2; S. 394 - 442
Hauptverfasser:	Di, Bolei, Lamperski, Andrew
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2022 Springer Nature B.V
Schlagworte:	Algorithms Approximation Communications Engineering Computer Systems Organization and Communication Networks Dynamic programming Dynamical systems Economic models Economic Theory/Quantitative Economics/Mathematical Methods Economics Game Theory Games Management Science Mathematics Mathematics and Statistics Multiagent systems Networks Nonlinear dynamics Operations Research Optimal control Social and Behav. Sciences Noncooperative dynamic games Feedback Nash equilibrium Newton’s method Differential dynamic programming Open-loop Nash equilibrium
ISSN:	2153-0785, 2153-0793
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Abstract	Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense, energy systems and etc. However, compared to single-agent control problems, the computational methods for dynamic games are relatively limited. As in the single-agent case, only specific dynamic games can be solved exactly, so approximation algorithms are required. In this paper, we show how to extend the Newton step algorithm, the Bellman recursion and the popular differential dynamic programming (DDP) for single-agent optimal control to the case of full-information nonzero sum dynamic games. We show that the Newton’s step can be solved in a computationally efficient manner and inherits its original quadratic convergence rate to open-loop Nash equilibria, and that the approximated Bellman recursion and DDP methods are very similar and can be used to find local feedback O ( ε 2 ) -Nash equilibria. Numerical examples are provided.
AbstractList	Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense, energy systems and etc. However, compared to single-agent control problems, the computational methods for dynamic games are relatively limited. As in the single-agent case, only specific dynamic games can be solved exactly, so approximation algorithms are required. In this paper, we show how to extend the Newton step algorithm, the Bellman recursion and the popular differential dynamic programming (DDP) for single-agent optimal control to the case of full-information nonzero sum dynamic games. We show that the Newton’s step can be solved in a computationally efficient manner and inherits its original quadratic convergence rate to open-loop Nash equilibria, and that the approximated Bellman recursion and DDP methods are very similar and can be used to find local feedback O ( ε 2 ) -Nash equilibria. Numerical examples are provided. Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense, energy systems and etc. However, compared to single-agent control problems, the computational methods for dynamic games are relatively limited. As in the single-agent case, only specific dynamic games can be solved exactly, so approximation algorithms are required. In this paper, we show how to extend the Newton step algorithm, the Bellman recursion and the popular differential dynamic programming (DDP) for single-agent optimal control to the case of full-information nonzero sum dynamic games. We show that the Newton’s step can be solved in a computationally efficient manner and inherits its original quadratic convergence rate to open-loop Nash equilibria, and that the approximated Bellman recursion and DDP methods are very similar and can be used to find local feedback O(ε2)-Nash equilibria. Numerical examples are provided.
Author	Lamperski, Andrew Di, Bolei
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Cites_doi	10.1007/s10288-007-0054-4 10.1007/978-3-319-27335-8_34-1 10.1016/j.orl.2006.03.004 10.1007/s13235-014-0105-3 10.1007/s00032-011-0163-6 10.1007/978-3-319-44374-4 10.1512/iumj.1984.33.33040 10.1007/s13235-010-0005-0 10.1007/978-3-319-98237-3_13 10.1109/TAC.2004.838489 10.1016/S0005-1098(99)00119-3 10.1007/s13235-016-0201-7 10.1007/BF00940728 10.1016/j.reseneeco.2004.08.001 10.1137/S0363012901389457 10.1109/TCST.2012.2231960 10.1007/978-0-8176-8355-9_9 10.1137/19M1288802 10.1007/s13235-018-0268-4 10.1137/1.9781611974287 10.1109/CDC.2015.7402805 10.1016/j.ejcon.2013.05.015 10.1007/978-3-319-01059-5 10.1007/s10107-007-0160-2 10.1007/s10957-020-01742-6 10.1109/TSP.2016.2551693 10.1007/s11425-016-0264-6 10.1109/ICASSP.2015.7178335 10.1007/BF01753310 10.1109/TAC.1971.1099685 10.1109/TPWRS.2003.820692 10.1007/s10287-006-0033-9 10.1109/9.86943 10.1007/978-3-030-00205-3 10.1016/j.automatica.2012.01.004 10.1109/CDC.2016.7799217 10.1007/s13235-016-0206-2 10.1134/S0005117917080136 10.1007/978-3-319-44374-4_5 10.1023/A:1019097208499 10.1016/j.jcp.2020.109907 10.1142/8442 10.1109/TAC.2011.2173412 10.1007/s40687-020-00215-6 10.1007/BF00934463 10.1007/s12532-018-0139-4 10.2139/ssrn.66448 10.1007/978-3-319-44374-4_3 10.1109/ACC.2015.7172215 10.1109/TAC.2017.2719158
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References	MazalovVVRettievaANAvrachenkovKELinear-quadratic discrete-time dynamic potential gamesAutom Remote Control20177881537154437025661405.9149310.1134/S0005117917080136 Pan Y, Ozguner U (2004) Sliding mode extremum seeking control for linear quadratic dynamic game. In: Proceedings of the 2004 American Control Conference, vol 1. IEEE, pp 614–619 FacchineiFFischerAPiccialliVGeneralized Nash equilibrium problems and Newton methodsMath Program20091171–216319424213041166.9001510.1007/s10107-007-0160-2 FacchineiFFischerAPiccialliVOn generalized Nash games and variational inequalitiesOper Res Lett200735215916423113951303.9102010.1016/j.orl.2006.03.004 Basar T, Haurie A, Zaccour G (2018) Nonzero-sum differential games. In Basar T, Zaccour G (eds) Handbook of dynamic game theory. Springer, pp 61–110 Dechert WD (1997) Non cooperative dynamic games: a control theoretic approach. Unpublished, available on request to the author Dutang C (2013) A survey of GNE computation methods: theory and algorithms. hal-00813531. https://hal.archives-ouvertes.fr/hal-00813531 EngwerdaJFeedback Nash equilibria in the scalar infinite horizon LQ-gameAutomatica200036113513918336810951.9101410.1016/S0005-1098(99)00119-3 ZazoSMacuaSVSánchez-FernándezMZazoJDynamic potential games with constraints: fundamentals and applications in communicationsIEEE Trans Signal Process201664143806382135157181414.9472410.1109/TSP.2016.2551693 EvansLCSouganidisPEDifferential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equationsIndiana Univ Math J19843357737977561581169.9131710.1512/iumj.1984.33.33040 BausoDGame theory with engineering applications2016PhiladelphiaSIAM1337.9100210.1137/1.9781611974287 Murray D, Yakowitz S (1984) Differential dynamic programming and Newton’s method for discrete optimal control problems. J Optim Theory Appl 43(3):395–414 KushnerHJNumerical approximations for stochastic differential gamesSIAM J Control Optim200241245748619202671014.6003510.1137/S0363012901389457 Carlson D, Haurie A, Zaccour G (2016) Infinite horizon concave games with coupled constraints. In: Handbook of dynamic game theory, pp 1–44 Frihauf P, Krstic M, Başar T (2013) Nash equilibrium seeking for dynamic systems with non-quadratic payoffs. In: Advances in dynamic games. Springer, pp 179–198 ContrerasJKluschMKrawczykJBNumerical solutions to Nash–Cournot equilibria in coupled constraint electricity marketsIEEE Trans Power Syst200419119520610.1109/TPWRS.2003.820692 BasarTOlsderGJDynamic noncooperative game theory1999PhiladelphiaSIAM0946.91001 KrawczykJBUryasevSRelaxation algorithms to find Nash equilibria with economic applicationsEnviron Model Assess200051637310.1023/A:1019097208499 DiehlMBjornbergJRobust dynamic programming for min-max model predictive control of constrained uncertain systemsIEEE Trans Autom Control200449122253225721067551365.9313110.1109/TAC.2004.838489 KrikelisNRekasiusZOn the solution of the optimal linear control problems under conflict of interestIEEE Trans Autom Control197116214014729081010.1109/TAC.1971.1099685 FrihaufPKrsticMBaşarTFinite-horizon LQ control for unknown discrete-time linear systems via extremum seekingEur J Control201319539940733001591293.4908010.1016/j.ejcon.2013.05.015 Zazo S, Macua SV, Sánchez-Fernández M, Zazo J (2015) Dynamic potential games in communications: fundamentals and applications. arXiv preprint arXiv:1509.01313 Andersson JAE, Gillis J, Horn G, Rawlings JB, Diehl M (2018) CasADi—a software framework for nonlinear optimization and optimal control. Math Program Comput (in press) Sun W, Theodorou EA, Tsiotras P (2015) Game theoretic continuous time differential dynamic programming. In: American Control Conference (ACC), 2015. IEEE, pp 5593–5598 Dunn JC, Bertsekas DP (1989) Efficient dynamic programming implementations of Newton’s method for unconstrained optimal control problems. J Optim Theory Appl 63(1):23–38 Paszke A, Gross S, Chintala S, Chanan G, Yang E, DeVito Z, Lin Z, Desmaison A, Antiga L, Lerer A (2017) Automatic differentiation in pytorch. In: NIPS-W LiaoLZShoemakerCAConvergence in unconstrained discrete-time differential dynamic programmingIEEE Trans Autom Control199136669270611196610760.4901610.1109/9.86943 DarbonJLangloisGPMengTOvercoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architecturesRes Math Sci20207315041233951445.3511910.1007/s40687-020-00215-6 HaurieAKrawczykJBZaccourGGames and dynamic games2012SingaporeWorld Scientific Publishing Company1396.9100310.1142/8442 FacchineiFPangJSFinite-dimensional variational inequalities and complementarity problems2007BerlinSpringer1062.90001 Bigi G, Castellani M, Pappalardo M, Passacantando M (2018) Nonlinear programming techniques for equilibria. EURO advanced tutorials on operational research. Springer. https://doi.org/10.1007/978-3-030-00205-3 ExarchosITheodorouETsiotrasPStochastic differential games: a sampling approach via fbsdesDyn Games Appl2018948650539465541429.9103510.1007/s13235-018-0268-4 Cacace S, Cristiani E, Falcone M (2011) Numerical approximation of Nash equilibria for a class of non-cooperative differential games. arXiv preprint arXiv:1109.3569 Tassa Y (2011) Theory and implementation of biomimetic motor controllers. Hebrew University of Jerusalem KebriaeiHIannelliLDiscrete-time robust hierarchical linear-quadratic dynamic gamesIEEE Trans Autom Control201863390290937725191390.9106410.1109/TAC.2017.2719158 BressanANoncooperative differential gamesMilan J Math201179235742728620231238.4905510.1007/s00032-011-0163-6 IsaacsRDifferential games: a mathematical theory with applications to warfare and pursuit, control and optimization1999ChelmsfordCourier Corporation1233.91001 González-SánchezDHernández-LermaOA survey of static and dynamic potential gamesSci China Math201659112075210235618291371.9100910.1007/s11425-016-0264-6 O’Donoghue B, Stathopoulos G, Boyd S (2013) A splitting method for optimal control. IEEE Trans Control Syst Technol 21(6):2432–2442 DarbonJMengTOn some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equationsJ Comput Phys2021425109907416606210.1016/j.jcp.2020.109907 BuckdahnRCardaliaguetPQuincampoixMSome recent aspects of differential game theoryDyn Games Appl2011117411428007861214.9101310.1007/s13235-010-0005-0 Sethi SP (2019) Differential games. In: Optimal control theory. Springer, pp 385–407 KrawczykJBCoupled constraint Nash equilibria in environmental gamesResour Energy Econ200527215718110.1016/j.reseneeco.2004.08.001 BressanANguyenKTStability of feedback solutions for infinite horizon noncooperative differential gamesDyn Games Appl201881427837646901390.9106010.1007/s13235-016-0206-2 FrihaufPKrsticMBasarTNash equilibrium seeking in noncooperative gamesIEEE Trans Autom Control20125751192120729238791369.9101010.1109/TAC.2011.2173412 EngwerdaJFeedback Nash equilibria for linear quadratic descriptor differential gamesAutomatica201248462563129015711238.4905610.1016/j.automatica.2012.01.004 González-SánchezDHernández-LermaODynamic potential games: the discrete-time stochastic caseDyn Games Appl20144330932832460321302.9101910.1007/s13235-014-0105-3 Krawczyk JB, Petkov V (2018) Multistage games. In: Handbook of dynamic game theory, pp 157–213 NocedalJWrightSJNumerical optimization20062BerlinSpringer1104.65059 FacchineiFKanzowCGeneralized Nash equilibrium problems4OR20075317321023534151211.9116210.1007/s10288-007-0054-4 Berridge S, Krawczyk JB (1997) Relaxation algorithms in finding Nash equilibria. Available at SSRN 66448 Nakamura-ZimmererTGongQKangWAdaptive deep learning for high-dimensional Hamilton–Jacobi–Bellman equationsSIAM J Sci Comput2021432A1221A124742411901467.4902810.1137/19M1288802 Zazo S, Valcarcel S, Matilde S, Zazo J et al (2015) A new framework for solving dynamic scheduling games. In: 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, pp 2071–2075 BasarTOn the uniqueness of the Nash solution in linear-quadratic differential gamesInt J Game Theory197652–365904413770354.9010110.1007/BF01753310 Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, Corrado GS, Davis A, Dean J, Devin M, Ghemawat S, Goodfellow I, Harp A, Irving G, Isard M, Jia Y, Jozefowicz R, Kaiser L, Kudlur M, Levenberg J, Mané D, Monga R, Moore S, Murray D, Olah C, Schuster M, Shlens J, Steiner B, Sutskever I, Talwar K, Tucker P, Vanhoucke V, Vasudevan V, Viégas F, Vinyals O, Warden P, Wattenberg M, Wicke M, Yu Y, Zheng X (2015) TensorFlow: large-scale machine learning on heterogeneous systems. https://www.tensorflow.org Lin W (2013) Differential games for multi-agent systems under distributed information. Ph.D. thesis, University of Central Florida information Duncan TE, Pasik-Duncan B (2015) Some stochastic differential games with state dependent noise. In: 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, pp 3773–3777 EngwerdaJA numerical algorithm to calculate the unique feedback Nash equilibrium in a large scalar LQ differential gameDyn Games Appl20177463565636984451391.9104810.1007/s13235-016-0201-7 Basar T, Zaccour G (2018) Handbook of Dynamic Game Theory. Springer González-SánchezDHernández-LermaODiscrete-time stochastic control and dynamic potential games: the Euler-equation approach2013BerlinSpringer1344.9300110.1007/978-3-319-01059-5 KrawczykJNumerical solutions to coupled-constraint (or generalised Nash) equilibrium problemsCMS20074218320423006361134.9130310.1007/s10287-006-0033-9 Plaksin A (2020) Viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations for time-delay systems. arXiv e-prints arXiv:2001.07905 Sun W, Theodorou EA, Tsiotras P (2016) Stochastic game theoretic trajectory optimization in continuous time. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp 6167–6172 F Facchinei (399_CR29) 2009; 117 399_CR34 399_CR2 399_CR1 399_CR4 F Facchinei (399_CR28) 2007; 35 399_CR6 I Exarchos (399_CR27) 2018; 9 399_CR8 399_CR9 J Darbon (399_CR16) 2020; 7 VV Mazalov (399_CR49) 2017; 78 R Buckdahn (399_CR12) 2011; 1 M Diehl (399_CR19) 2004; 49 399_CR20 399_CR63 399_CR22 J Darbon (399_CR17) 2021; 425 399_CR21 J Nocedal (399_CR52) 2006 399_CR60 HJ Kushner (399_CR46) 2002; 41 P Frihauf (399_CR32) 2012; 57 399_CR61 J Engwerda (399_CR23) 2000; 36 399_CR18 F Facchinei (399_CR30) 2007; 5 399_CR13 399_CR57 399_CR56 399_CR59 399_CR14 399_CR58 399_CR53 399_CR55 D González-Sánchez (399_CR35) 2013 399_CR54 T Basar (399_CR5) 1999 399_CR50 JB Krawczyk (399_CR44) 2000; 5 S Zazo (399_CR62) 2016; 64 T Basar (399_CR3) 1976; 5 R Isaacs (399_CR39) 1999 J Krawczyk (399_CR41) 2007; 4 D Bauso (399_CR7) 2016 A Haurie (399_CR38) 2012 T Nakamura-Zimmerer (399_CR51) 2021; 43 D González-Sánchez (399_CR36) 2014; 4 J Contreras (399_CR15) 2004; 19 J Engwerda (399_CR25) 2012; 48 399_CR48 LC Evans (399_CR26) 1984; 33 A Bressan (399_CR11) 2018; 8 399_CR43 JB Krawczyk (399_CR42) 2005; 27 LZ Liao (399_CR47) 1991; 36 A Bressan (399_CR10) 2011; 79 P Frihauf (399_CR33) 2013; 19 H Kebriaei (399_CR40) 2018; 63 F Facchinei (399_CR31) 2007 J Engwerda (399_CR24) 2017; 7 D González-Sánchez (399_CR37) 2016; 59 N Krikelis (399_CR45) 1971; 16
References_xml	– reference: Cacace S, Cristiani E, Falcone M (2011) Numerical approximation of Nash equilibria for a class of non-cooperative differential games. arXiv preprint arXiv:1109.3569 – reference: Andersson JAE, Gillis J, Horn G, Rawlings JB, Diehl M (2018) CasADi—a software framework for nonlinear optimization and optimal control. Math Program Comput (in press) – reference: O’Donoghue B, Stathopoulos G, Boyd S (2013) A splitting method for optimal control. IEEE Trans Control Syst Technol 21(6):2432–2442 – reference: KebriaeiHIannelliLDiscrete-time robust hierarchical linear-quadratic dynamic gamesIEEE Trans Autom Control201863390290937725191390.9106410.1109/TAC.2017.2719158 – reference: Carlson D, Haurie A, Zaccour G (2016) Infinite horizon concave games with coupled constraints. In: Handbook of dynamic game theory, pp 1–44 – reference: EvansLCSouganidisPEDifferential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equationsIndiana Univ Math J19843357737977561581169.9131710.1512/iumj.1984.33.33040 – reference: Krawczyk JB, Petkov V (2018) Multistage games. In: Handbook of dynamic game theory, pp 157–213 – reference: FacchineiFPangJSFinite-dimensional variational inequalities and complementarity problems2007BerlinSpringer1062.90001 – reference: Dutang C (2013) A survey of GNE computation methods: theory and algorithms. hal-00813531. https://hal.archives-ouvertes.fr/hal-00813531 – reference: FacchineiFFischerAPiccialliVGeneralized Nash equilibrium problems and Newton methodsMath Program20091171–216319424213041166.9001510.1007/s10107-007-0160-2 – reference: KrawczykJNumerical solutions to coupled-constraint (or generalised Nash) equilibrium problemsCMS20074218320423006361134.9130310.1007/s10287-006-0033-9 – reference: BressanANguyenKTStability of feedback solutions for infinite horizon noncooperative differential gamesDyn Games Appl201881427837646901390.9106010.1007/s13235-016-0206-2 – reference: ContrerasJKluschMKrawczykJBNumerical solutions to Nash–Cournot equilibria in coupled constraint electricity marketsIEEE Trans Power Syst200419119520610.1109/TPWRS.2003.820692 – reference: DarbonJLangloisGPMengTOvercoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architecturesRes Math Sci20207315041233951445.3511910.1007/s40687-020-00215-6 – reference: FrihaufPKrsticMBasarTNash equilibrium seeking in noncooperative gamesIEEE Trans Autom Control20125751192120729238791369.9101010.1109/TAC.2011.2173412 – reference: ExarchosITheodorouETsiotrasPStochastic differential games: a sampling approach via fbsdesDyn Games Appl2018948650539465541429.9103510.1007/s13235-018-0268-4 – reference: Plaksin A (2020) Viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations for time-delay systems. arXiv e-prints arXiv:2001.07905 – reference: Bigi G, Castellani M, Pappalardo M, Passacantando M (2018) Nonlinear programming techniques for equilibria. EURO advanced tutorials on operational research. Springer. https://doi.org/10.1007/978-3-030-00205-3 – reference: HaurieAKrawczykJBZaccourGGames and dynamic games2012SingaporeWorld Scientific Publishing Company1396.9100310.1142/8442 – reference: Paszke A, Gross S, Chintala S, Chanan G, Yang E, DeVito Z, Lin Z, Desmaison A, Antiga L, Lerer A (2017) Automatic differentiation in pytorch. In: NIPS-W – reference: Duncan TE, Pasik-Duncan B (2015) Some stochastic differential games with state dependent noise. In: 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, pp 3773–3777 – reference: Murray D, Yakowitz S (1984) Differential dynamic programming and Newton’s method for discrete optimal control problems. J Optim Theory Appl 43(3):395–414 – reference: DarbonJMengTOn some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equationsJ Comput Phys2021425109907416606210.1016/j.jcp.2020.109907 – reference: EngwerdaJFeedback Nash equilibria in the scalar infinite horizon LQ-gameAutomatica200036113513918336810951.9101410.1016/S0005-1098(99)00119-3 – reference: Berridge S, Krawczyk JB (1997) Relaxation algorithms in finding Nash equilibria. Available at SSRN 66448 – reference: IsaacsRDifferential games: a mathematical theory with applications to warfare and pursuit, control and optimization1999ChelmsfordCourier Corporation1233.91001 – reference: Zazo S, Valcarcel S, Matilde S, Zazo J et al (2015) A new framework for solving dynamic scheduling games. In: 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, pp 2071–2075 – reference: EngwerdaJFeedback Nash equilibria for linear quadratic descriptor differential gamesAutomatica201248462563129015711238.4905610.1016/j.automatica.2012.01.004 – reference: Frihauf P, Krstic M, Başar T (2013) Nash equilibrium seeking for dynamic systems with non-quadratic payoffs. In: Advances in dynamic games. Springer, pp 179–198 – reference: FacchineiFKanzowCGeneralized Nash equilibrium problems4OR20075317321023534151211.9116210.1007/s10288-007-0054-4 – reference: Basar T, Haurie A, Zaccour G (2018) Nonzero-sum differential games. In Basar T, Zaccour G (eds) Handbook of dynamic game theory. Springer, pp 61–110 – reference: Dunn JC, Bertsekas DP (1989) Efficient dynamic programming implementations of Newton’s method for unconstrained optimal control problems. J Optim Theory Appl 63(1):23–38 – reference: Zazo S, Macua SV, Sánchez-Fernández M, Zazo J (2015) Dynamic potential games in communications: fundamentals and applications. arXiv preprint arXiv:1509.01313 – reference: Sethi SP (2019) Differential games. In: Optimal control theory. Springer, pp 385–407 – reference: González-SánchezDHernández-LermaOA survey of static and dynamic potential gamesSci China Math201659112075210235618291371.9100910.1007/s11425-016-0264-6 – reference: BasarTOn the uniqueness of the Nash solution in linear-quadratic differential gamesInt J Game Theory197652–365904413770354.9010110.1007/BF01753310 – reference: NocedalJWrightSJNumerical optimization20062BerlinSpringer1104.65059 – reference: BuckdahnRCardaliaguetPQuincampoixMSome recent aspects of differential game theoryDyn Games Appl2011117411428007861214.9101310.1007/s13235-010-0005-0 – reference: FacchineiFFischerAPiccialliVOn generalized Nash games and variational inequalitiesOper Res Lett200735215916423113951303.9102010.1016/j.orl.2006.03.004 – reference: ZazoSMacuaSVSánchez-FernándezMZazoJDynamic potential games with constraints: fundamentals and applications in communicationsIEEE Trans Signal Process201664143806382135157181414.9472410.1109/TSP.2016.2551693 – reference: BressanANoncooperative differential gamesMilan J Math201179235742728620231238.4905510.1007/s00032-011-0163-6 – reference: DiehlMBjornbergJRobust dynamic programming for min-max model predictive control of constrained uncertain systemsIEEE Trans Autom Control200449122253225721067551365.9313110.1109/TAC.2004.838489 – reference: Sun W, Theodorou EA, Tsiotras P (2015) Game theoretic continuous time differential dynamic programming. In: American Control Conference (ACC), 2015. IEEE, pp 5593–5598 – reference: Lin W (2013) Differential games for multi-agent systems under distributed information. Ph.D. thesis, University of Central Florida information – reference: KrikelisNRekasiusZOn the solution of the optimal linear control problems under conflict of interestIEEE Trans Autom Control197116214014729081010.1109/TAC.1971.1099685 – reference: KushnerHJNumerical approximations for stochastic differential gamesSIAM J Control Optim200241245748619202671014.6003510.1137/S0363012901389457 – reference: EngwerdaJA numerical algorithm to calculate the unique feedback Nash equilibrium in a large scalar LQ differential gameDyn Games Appl20177463565636984451391.9104810.1007/s13235-016-0201-7 – reference: MazalovVVRettievaANAvrachenkovKELinear-quadratic discrete-time dynamic potential gamesAutom Remote Control20177881537154437025661405.9149310.1134/S0005117917080136 – reference: Nakamura-ZimmererTGongQKangWAdaptive deep learning for high-dimensional Hamilton–Jacobi–Bellman equationsSIAM J Sci Comput2021432A1221A124742411901467.4902810.1137/19M1288802 – reference: González-SánchezDHernández-LermaODiscrete-time stochastic control and dynamic potential games: the Euler-equation approach2013BerlinSpringer1344.9300110.1007/978-3-319-01059-5 – reference: KrawczykJBCoupled constraint Nash equilibria in environmental gamesResour Energy Econ200527215718110.1016/j.reseneeco.2004.08.001 – reference: LiaoLZShoemakerCAConvergence in unconstrained discrete-time differential dynamic programmingIEEE Trans Autom Control199136669270611196610760.4901610.1109/9.86943 – reference: Tassa Y (2011) Theory and implementation of biomimetic motor controllers. Hebrew University of Jerusalem – reference: Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, Corrado GS, Davis A, Dean J, Devin M, Ghemawat S, Goodfellow I, Harp A, Irving G, Isard M, Jia Y, Jozefowicz R, Kaiser L, Kudlur M, Levenberg J, Mané D, Monga R, Moore S, Murray D, Olah C, Schuster M, Shlens J, Steiner B, Sutskever I, Talwar K, Tucker P, Vanhoucke V, Vasudevan V, Viégas F, Vinyals O, Warden P, Wattenberg M, Wicke M, Yu Y, Zheng X (2015) TensorFlow: large-scale machine learning on heterogeneous systems. https://www.tensorflow.org/ – reference: Pan Y, Ozguner U (2004) Sliding mode extremum seeking control for linear quadratic dynamic game. In: Proceedings of the 2004 American Control Conference, vol 1. IEEE, pp 614–619 – reference: BasarTOlsderGJDynamic noncooperative game theory1999PhiladelphiaSIAM0946.91001 – reference: Dechert WD (1997) Non cooperative dynamic games: a control theoretic approach. Unpublished, available on request to the author – reference: FrihaufPKrsticMBaşarTFinite-horizon LQ control for unknown discrete-time linear systems via extremum seekingEur J Control201319539940733001591293.4908010.1016/j.ejcon.2013.05.015 – reference: Sun W, Theodorou EA, Tsiotras P (2016) Stochastic game theoretic trajectory optimization in continuous time. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp 6167–6172 – reference: González-SánchezDHernández-LermaODynamic potential games: the discrete-time stochastic caseDyn Games Appl20144330932832460321302.9101910.1007/s13235-014-0105-3 – reference: KrawczykJBUryasevSRelaxation algorithms to find Nash equilibria with economic applicationsEnviron Model Assess200051637310.1023/A:1019097208499 – reference: BausoDGame theory with engineering applications2016PhiladelphiaSIAM1337.9100210.1137/1.9781611974287 – reference: Basar T, Zaccour G (2018) Handbook of Dynamic Game Theory. Springer – volume: 5 start-page: 173 issue: 3 year: 2007 ident: 399_CR30 publication-title: 4OR doi: 10.1007/s10288-007-0054-4 – ident: 399_CR14 doi: 10.1007/978-3-319-27335-8_34-1 – volume: 35 start-page: 159 issue: 2 year: 2007 ident: 399_CR28 publication-title: Oper Res Lett doi: 10.1016/j.orl.2006.03.004 – volume: 4 start-page: 309 issue: 3 year: 2014 ident: 399_CR36 publication-title: Dyn Games Appl doi: 10.1007/s13235-014-0105-3 – volume: 79 start-page: 357 issue: 2 year: 2011 ident: 399_CR10 publication-title: Milan J Math doi: 10.1007/s00032-011-0163-6 – ident: 399_CR6 doi: 10.1007/978-3-319-44374-4 – volume: 33 start-page: 773 issue: 5 year: 1984 ident: 399_CR26 publication-title: Indiana Univ Math J doi: 10.1512/iumj.1984.33.33040 – volume: 1 start-page: 74 issue: 1 year: 2011 ident: 399_CR12 publication-title: Dyn Games Appl doi: 10.1007/s13235-010-0005-0 – ident: 399_CR57 doi: 10.1007/978-3-319-98237-3_13 – ident: 399_CR18 – volume: 49 start-page: 2253 issue: 12 year: 2004 ident: 399_CR19 publication-title: IEEE Trans Autom Control doi: 10.1109/TAC.2004.838489 – volume: 36 start-page: 135 issue: 1 year: 2000 ident: 399_CR23 publication-title: Automatica doi: 10.1016/S0005-1098(99)00119-3 – volume: 7 start-page: 635 issue: 4 year: 2017 ident: 399_CR24 publication-title: Dyn Games Appl doi: 10.1007/s13235-016-0201-7 – ident: 399_CR21 doi: 10.1007/BF00940728 – volume: 27 start-page: 157 issue: 2 year: 2005 ident: 399_CR42 publication-title: Resour Energy Econ doi: 10.1016/j.reseneeco.2004.08.001 – volume: 41 start-page: 457 issue: 2 year: 2002 ident: 399_CR46 publication-title: SIAM J Control Optim doi: 10.1137/S0363012901389457 – ident: 399_CR53 doi: 10.1109/TCST.2012.2231960 – ident: 399_CR34 doi: 10.1007/978-0-8176-8355-9_9 – volume: 43 start-page: A1221 issue: 2 year: 2021 ident: 399_CR51 publication-title: SIAM J Sci Comput doi: 10.1137/19M1288802 – volume: 9 start-page: 486 year: 2018 ident: 399_CR27 publication-title: Dyn Games Appl doi: 10.1007/s13235-018-0268-4 – volume-title: Game theory with engineering applications year: 2016 ident: 399_CR7 doi: 10.1137/1.9781611974287 – ident: 399_CR20 doi: 10.1109/CDC.2015.7402805 – volume-title: Finite-dimensional variational inequalities and complementarity problems year: 2007 ident: 399_CR31 – volume: 19 start-page: 399 issue: 5 year: 2013 ident: 399_CR33 publication-title: Eur J Control doi: 10.1016/j.ejcon.2013.05.015 – volume-title: Discrete-time stochastic control and dynamic potential games: the Euler-equation approach year: 2013 ident: 399_CR35 doi: 10.1007/978-3-319-01059-5 – ident: 399_CR22 – volume: 117 start-page: 163 issue: 1–2 year: 2009 ident: 399_CR29 publication-title: Math Program doi: 10.1007/s10107-007-0160-2 – ident: 399_CR54 – ident: 399_CR56 doi: 10.1007/s10957-020-01742-6 – ident: 399_CR61 doi: 10.1109/TSP.2016.2551693 – volume-title: Numerical optimization year: 2006 ident: 399_CR52 – volume: 59 start-page: 2075 issue: 11 year: 2016 ident: 399_CR37 publication-title: Sci China Math doi: 10.1007/s11425-016-0264-6 – ident: 399_CR63 doi: 10.1109/ICASSP.2015.7178335 – volume: 5 start-page: 65 issue: 2–3 year: 1976 ident: 399_CR3 publication-title: Int J Game Theory doi: 10.1007/BF01753310 – ident: 399_CR60 – volume: 16 start-page: 140 issue: 2 year: 1971 ident: 399_CR45 publication-title: IEEE Trans Autom Control doi: 10.1109/TAC.1971.1099685 – ident: 399_CR48 – volume: 19 start-page: 195 issue: 1 year: 2004 ident: 399_CR15 publication-title: IEEE Trans Power Syst doi: 10.1109/TPWRS.2003.820692 – volume: 4 start-page: 183 issue: 2 year: 2007 ident: 399_CR41 publication-title: CMS doi: 10.1007/s10287-006-0033-9 – volume: 36 start-page: 692 issue: 6 year: 1991 ident: 399_CR47 publication-title: IEEE Trans Autom Control doi: 10.1109/9.86943 – ident: 399_CR9 doi: 10.1007/978-3-030-00205-3 – volume: 48 start-page: 625 issue: 4 year: 2012 ident: 399_CR25 publication-title: Automatica doi: 10.1016/j.automatica.2012.01.004 – ident: 399_CR59 doi: 10.1109/CDC.2016.7799217 – volume: 8 start-page: 42 issue: 1 year: 2018 ident: 399_CR11 publication-title: Dyn Games Appl doi: 10.1007/s13235-016-0206-2 – volume: 78 start-page: 1537 issue: 8 year: 2017 ident: 399_CR49 publication-title: Autom Remote Control doi: 10.1134/S0005117917080136 – ident: 399_CR55 – volume-title: Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization year: 1999 ident: 399_CR39 – ident: 399_CR4 doi: 10.1007/978-3-319-44374-4_5 – volume: 5 start-page: 63 issue: 1 year: 2000 ident: 399_CR44 publication-title: Environ Model Assess doi: 10.1023/A:1019097208499 – ident: 399_CR13 – volume: 425 start-page: 109907 year: 2021 ident: 399_CR17 publication-title: J Comput Phys doi: 10.1016/j.jcp.2020.109907 – volume-title: Games and dynamic games year: 2012 ident: 399_CR38 doi: 10.1142/8442 – volume: 57 start-page: 1192 issue: 5 year: 2012 ident: 399_CR32 publication-title: IEEE Trans Autom Control doi: 10.1109/TAC.2011.2173412 – volume: 64 start-page: 3806 issue: 14 year: 2016 ident: 399_CR62 publication-title: IEEE Trans Signal Process doi: 10.1109/TSP.2016.2551693 – volume-title: Dynamic noncooperative game theory year: 1999 ident: 399_CR5 – volume: 7 start-page: 1 issue: 3 year: 2020 ident: 399_CR16 publication-title: Res Math Sci doi: 10.1007/s40687-020-00215-6 – ident: 399_CR50 doi: 10.1007/BF00934463 – ident: 399_CR2 doi: 10.1007/s12532-018-0139-4 – ident: 399_CR8 doi: 10.2139/ssrn.66448 – ident: 399_CR1 – ident: 399_CR43 doi: 10.1007/978-3-319-44374-4_3 – ident: 399_CR58 doi: 10.1109/ACC.2015.7172215 – volume: 63 start-page: 902 issue: 3 year: 2018 ident: 399_CR40 publication-title: IEEE Trans Autom Control doi: 10.1109/TAC.2017.2719158
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Snippet	Dynamic games arise when multiple agents with differing objectives control a dynamic system. They model a wide variety of applications in economics, defense,...
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SubjectTerms	Algorithms Approximation Communications Engineering Computer Systems Organization and Communication Networks Dynamic programming Dynamical systems Economic models Economic Theory/Quantitative Economics/Mathematical Methods Economics Game Theory Games Management Science Mathematics Mathematics and Statistics Multiagent systems Networks Nonlinear dynamics Operations Research Optimal control Social and Behav. Sciences
Title	Newton’s Method, Bellman Recursion and Differential Dynamic Programming for Unconstrained Nonlinear Dynamic Games
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