Necessary conditions of overtaking equilibrium for infinite horizon differential games

Dmitry V. Khlopin

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 5 (2013) no. 2, pp. 105-136.

Résumé

This paper is devoted to refining the necessary conditions of Nash equilibrium for infinite horizon games. The right-hand endpoint of the trajectories is free; the search for equilibrium is conducted in the class of open-loop strategies. For adjoint variables of each player, the necessary asymptotic condition of overtaking Nash equilibrium is obtained. For certain cases (in particular, when the 'strictly optimal solution' criterium is employed), it allows to explicitly specify the expression for the adjoint variable. S. M. Aseev, A. V. Kryazhimskii, and V. M. Veliov used such an expression as a necessary condition of optimality for certain infinite horizon control problems. In this paper we also specify an example of linear game with the opposite objective functionals. In this game, there exist multiple Nash equilibria that depend on the choice of the sequence along which the players optimize their actions. These equilibria produce absolutely different results. Also Sorger version of the 'Lanchester-type' differential game is considered.

Détail

Keywords: dynamic optimization, differential games, infinite horizon problem, transversality condition for infinity, necessary conditions of equilibrium, overtaking equilibrium, unbounded cost.

Bibliographie

[1] Aseev S. M., Kryazhimskii A. V., “Printsip maksimuma Pontryagina i zadachi optimalnogo ekonomicheskogo rosta”, Trudy MI RAN, 257, 2007, 3–271 | MR | Zbl

[2] Aseev S. M., Kryazhimskii A. V., Besov K. O., “Zadachi optimalnogo upravleniya na beskonechnom promezhutke vremeni v ekonomike”, UMN, 67:2(404) (2012), 3–64 | DOI | MR | Zbl

[3] Brykalov S. A., Golovina O. N., Kryazhimskii A. V., “Ravnovesie po Neshu v igrakh mnogikh lits s vyborom momentov vremeni i integralnymi funktsionalami platy”, Sovremennaya matematika i ee prilozheniya, 26 (2005), 42–53 | MR

[4] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977 | MR

[5] Gamkrelidze R. V., Osnovy optimalnogo upravleniya, Izd-vo Tbil. universiteta, Tbilisi, 1977 | MR

[6] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961

[7] Fedorchuk V. V., Filippov V. V., Obschaya topologiya. Osnovnye konstruktsii, Fizmatlit, M., 2006

[8] Khlopin D. V., “O rasshirenii konfliktno-upravlyaemykh zadach na beskonechnom promezhutke”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 1, 105–112

[9] Khlopin D. V., “O neobkhodimykh kraevykh usloviyakh dlya silno optimalnogo upravleniya v zadachakh na beskonechnom promezhutke”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2013, no. 1, 49–58

[10] Khlopin D. V., “O $\tau$-ischezayuschei sopryazhennoi peremennoi v zadachakh upravleniya na beskonechnom promezhutke”, Tezisy konferentsii po differentsialnym uravneniyam i dinamicheskim sistemam (Vladimir, 29 iyunya – 4 iyulya 2012 goda), 173–174

[11] Chentsov A. G., Konechno-additivnye mery i relaksatsii ekstremalnykh zadach, UIF “Nauka”, Ekaterinburg, 1993

[12] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR

[13] Yang L., Lektsii po variatsionnomu upravleniyu i teorii optimalnogo upravleniya, Mir, M., 1974

[14] Aseev S. M., Kryazhimskii A. V., “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons”, SIAM J. Control Optim., 43 (2004), 1094–1119 | DOI | MR | Zbl

[15] Aseev S. M., Veliov V. M., Needle Variations in Infinite-Horizon Optimal Control, Interim Rept., IR-2012-04, IIASA, 2012

[16] Aseev S. M., Veliov V. M., “Maximum Principle for infinite-horizon optimal control problems with dominating discount”, Dynamics of Continuous, Discrete and Impulsive Systems. Series B, 19:1–2 (2012), 43–63 | MR | Zbl

[17] Aubin J., Clarke F., “Shadow Prices and Duality for a Class of Optimal Control Problems”, SIAM J. Control Optim., 17 (1979), 567–586 | DOI | MR | Zbl

[18] Bagh A., “Variational convergence: Approximation and existence of equilibria in discontinuous games”, J. of Econ. Theory, 145 (2010), 1244–1268 | DOI | MR | Zbl

[19] Başar T., Olsder G. J., Dynamic Noncooperative Game Theory, SIAM, 1998 | MR

[20] Bogusz D., “On the existence of a classical optimal solution and of an almost strongly optimal solution for an infinite-horizon control problem”, J. Optim. Theory Appl., 156:2 (2013), 650–682 | DOI | MR | Zbl

[21] Carlson D., “Uniformly overtaking and weakly overtaking optimal solutions in infinite-horizon optimal control: when optimal solutions are agreeable”, J. Optim. Theory Appl., 64:1 (1990), 55–69 | DOI | MR | Zbl

[22] Carlson D., “Nonconvex and Relaxed Infinite-Horizon Optimal Control Problems”, J. Optim. Theory Appl., 78:3 (1993), 465–491 | DOI | MR | Zbl

[23] Carlson D., Haurie A., “A turnpike theory for infinite horizon open-loop differential games with decoupled dynamics”, New Trends in Dynamic Games and Applications, Annals of the Society of Dynamic Games, 3, ed. G. J. Olsder, 1995, 353–376 | MR | Zbl

[24] Carlson D., Haurie A., “Infinite horizon dynamic games with coupled state constraints”, Advances in Dynamic Games. Annals of the International Society of Dynamic Games, 5 (2000), 195–212 | DOI | MR | Zbl

[25] Carlson D., Haurie A., Moresino F., Pourtallier O., “Equilibria in Infinite Horizon Dynamic Games”, Troisième Cycle, Romand de Recherche Opèrationnelle, Zinal, March, 1999

[26] Carlson D., Haurie A., “Infinite horizon control and dynamic games”, Encyclopedia of Optimization, eds. Floudas C. A., Pardalos P. M., Springer, N.Y., 2009, 1598–1603 | DOI

[27] Carlson D., Haurie A., Leizarowitz A., Infinite Horizon Optimal Control. Deterministic and Stochastic Systems, Springer, Berlin, 1991 | MR | Zbl

[28] Chentsov A. G., Asymptotic attainability, Kluwer, Dordrecht, 1997 | MR | Zbl

[29] Clarke F., Optimization and Nonsmooth Analysis, Wiley, N.-Y., 1983 | MR | Zbl

[30] Daniel H. W., “Survey of measurable selection theorems: an update”, Lect. Notes Math., 794, Springer, N.-Y., 1980, 176–219 | DOI | MR

[31] Davidson R., Harris R., “Nonconvexities in Continuous-Time Investment Theory”, Review of Economic Studies, 43 (1981), 235–253 | DOI | MR

[32] Glicksberg I. L., “A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points”, Proc. of Amer. Math. Soc., 3 (1952), 170–174 | MR | Zbl

[33] Halkin H., “Necessary Conditions for Optimal Control Problems with Infinite Horizons”, Econometrica, 42 (1974), 267–272 | DOI | MR | Zbl

[34] Hammond P. J., Kennan J., “Uniformly Optimal Infinite Horizon Plans”, Intern. Econ. Review, 20:2 (1979), 283–296 | DOI | MR | Zbl

[35] Khlopin D. V., Necessity of vanishing shadow price in infinite horizon control problems, 33 pp., arXiv: 1207.5358

[36] Michel P., “On the transversality condition in infinite horizon optimal problems”, Econometrica, 50 (1982), 975–984 | DOI | MR

[37] Oliu-Barton M., Vigeral G., “A uniform Tauberian theorem in optimal control”, Advances in Dynamic Games. Annals of the International Society of Dynamic Games, 12 (2012), 199–215 | MR

[38] Seierstad A., “Necessary conditions for nonsmooth, infinite-horizon optimal control problems”, J. Optim. Theory Appl., 103:1 (1999), 201–230 | DOI | MR

[39] Seierstad A., “Fields of extremals and infinite horizon optimal control problems”, Optim. Control Appl. Meth., 19 (1998), 377–392 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[40] Seierstad A., Sydsæter K., Optimal control theory with economic applications, North-Holland, Amsterdam, 1987 | MR | Zbl

[41] Sorger G., “Competitive dynamic advertising: A modification of the Case game”, J. Economic Dynamics and Control, 13 (1989), 55–80 | DOI | MR | Zbl

[42] Stern L., “Criteria of optimality in the infinite-time optimal control problem”, J. Optim. Theory Appl., 44:3 (1984), 497–508 | DOI | MR | Zbl

[43] Tolstonogov A., Differential inclusions in a Banach space, Kluwer, Dordrecht, 2000 | MR | Zbl

[44] Vives X., “Nash equilibrium with strategic complementarities”, Journal of Mathematical Economics, 19 (1990), 305–321 | DOI | MR | Zbl

[45] Wachs A. O., Schochetman I. E., Smith R. L., “Average Optimality in Nonhomogeneous Infinite Horizon Markov Decision Processes”, Math. Oper. Res., 36:1 (2011), 147–164 | DOI | MR | Zbl

[46] Zaslavski A., Khan M., “On locally optimal programs in the Robinson-Solow-Srinivasan model”, J. Econ., 99 (2010), 65–92 | DOI | Zbl