Решение по Штакельбергу игры среднего поля с ведущим игроком
Ключевые слова:
игры среднего поля, решение по Штакельбергу, игры бесконечного числа лиц
Аннотация
Статья посвящена исследованию системы большого числа однотипных игроков, взаимодействующих с внешним окружением. Мы моделируем это окружение как ведущего (внешнего) игрока. Основное предположение нашей модели состоит в том, что малые игроки могут влиять друг на друга и на ведущего игрока лишь через те или иные усредненные характеристики. Подобные модели носят название игр среднего поля с ведущим игроком. Мы предполагаем, что время непрерывно и динамика ведущего и малых игроков описывается обыкновенными дифференциальными уравнениями. Мы рассматриваем решение по Штакельбергу с ведущим игроком-лидером, то есть предполагается, что ведущий игрок объявляет заранее свое управление малым игрокам. Основной результат статьи состоит в доказательстве существования решения по Штакельбергу игры среднего поля с ведущим игроком в классе обобщенных программных управлений.
Литература
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25. Lasry J.-M., Lions P.-L. Mean field games // Japanese Journal of Mathematics. 2007. Vol. 2. P. 229-260. DOI: 10.1007/s11537-007-0657-8
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28. Nguyen S.L., Huang M. Mean field LQG games with a major player: Continuum parameters for minor players // 2011 50th IEEE Conference on Decision and Control and European Control Conference. IEEE, 2011. DOI: 10.1109/cdc.2011.6160790
29. Nguyen S.L., Huang M. Mean field LQG games with mass behavior responsive to a major player // 2012 IEEE 51st IEEE Conference on Decision and Control (CDC). IEEE, 2012. P. 5772-5797. DOI: 10.1109/CDC.2012.6425984
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2. Ambrosio L., Gigli N., Savaré G. Gradient flows: in metric spaces and in the space of probability measures. Lectures in Mathematics. ETH Zurich, Basel: Birkhäuser, 2005.
3. Averboukh Yu.V. Deterministic limit of mean field games associated with nonlinear Markov processes // Applied Mathematics & Optimization. 2018. DOI: 10.1007/s00245-018-9486-9
4. Bensoussan A., Chau M.H.M., Lai Y., Yam S.C.P. Linear-quadratic mean field Stackelberg games with state and control delays // SIAM Journal on Control and Optimization. 2017. Vol. 55. No. 4. P. 2748-2781. DOI: 10.1137/15M1052937
5. Bensoussan A., Chau M.H.M., Yam S.C.P. Mean field Stackelberg games: Aggregation of delayed instructions // SIAM Journal on Control and Optimization. 2015. Vol. 53. No. 4. P. 2237-2266. DOI: 10.1137/140993399
6. Bensoussan A., Frehse J., Yam P. Mean field games and mean field type control theory. Springer Briefs in Mathematics. New York: Springer, 2013.
7. Bogachev V.I. Measure theory. Volume 2. Berlin: Springer, 2007.
8. Cardaliaguet P., Delarue F., Lasry J.-M., Lions P.-L. The master equation and the convergence problem in mean field games // arXiv: 1509.02505 [math.AP]. 2015. http://arxiv.org/pdf/1509.02505.pdf
9. Carmona R., Delarue F. The master equation for large population equilibriums // Stochastic Analysis and Applications 2014. Berlin: Springer, 2014. P. 77-128. DOI: 10.1007/978-3-319-11292-3_4
10. Carmona R., Delarue F. Probabilistic theory of mean field games with applications. I. New York: Springer, 2018. DOI: 10.1007/978-3-319-58920-6
11. Carmona R., Delarue F. Probabilistic theory of mean field games with applications. II. New York: Springer, 2018. DOI: 10.1007/978-3-319-56436-4
12. Carmona R., Delarue F., Lachapelle A. Control of McKean-Vlasov dynamics versus mean field games // Mathematics and Financial Economics. 2013. Vol. 7. No. 2. P. 131-166. DOI: 10.1007/s11579-012-0089-y
13. Carmona R., Wang P. An alternative approach to mean field game with major and minor players, and applications to herders impacts // Applied Mathematics & Optimization. 2017. Vol. 76. No. 1. P. 5-27. DOI: 10.1007/s00245-017-9430-4
14. Carmona R., Zhu X. A probabilistic approach to mean field games with major and minor players // The Annals of Applied Probability. 2016. Vol. 26. No. 3. P. 1535-1580. DOI: 10.1214/15-AAP1125
15. Fischer M. On the connection between symmetric $n$-player games and mean field games // The Annals of Applied Probability. 2017. Vol. 27. No. 2. P. 757-810. DOI: 10.1214/16-aap1215
16. Huang J., Wang S., Wu Z. Mean field linear-quadratic-gaussian (LQG) games: major and minor players // arXiv: 1403.3999 [math.OC]. 2014. http://arxiv.org/pdf/1403.3999.pdf
17. Huang M., Caines P.E., Malhamé R.P. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $\varepsilon$-Nash equilibria // IEEE Transactions on Automatic Control. 2007. Vol. 52. Issue 9. P. 1560-1571. DOI: 10.1109/TAC.2007.904450
18. Caines P.E., Huang M., Malhamé R.P. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle // Communications in Information and Systems. 2006. Vol. 6. No. 3. P. 221-251. DOI: 10.4310/CIS.2006.v6.n3.a5
19. Kolokoltsov V.N., Troeva M. On the mean field games with common noise and the McKean-Vlasov SPDEs // arXiv: 1506.04594 [math.PR]. 2015. http://arxiv.org/pdf/1506.04594.pdf
20. Lacker D. Mean field games via controlled martingale problems: existence of Markovian equilibria // Stochastic Processes and their Applications. 2015. Vol. 125. No. 7. P. 2856-2894. DOI: 10.1016/j.spa.2015.02.006
21. Lacker D. A general characterization of the mean field limit for stochastic differential games // Probability Theory and Related Fields. 2016. Vol. 165. P. 581-648. DOI: 10.1007/s00440-015-0641-9
22. Lacker D. On the convergence of closed-loop Nash equilibria to the mean field game limit // arXiv: 1808.02745 [math.PR]. 2018. http://arxiv.org/pdf/1808.02745.pdf
23. Lasry J.-M., Lions P.-L. Jeux à champ moyen. I - Le cas stationnaire // Comptes Rendus Mathematique. 2006. Vol. 343. P. 619-625. DOI: 10.1016/j.crma.2006.09.019
24. Lasry J.-M., Lions P.-L. Jeux à champ moyen. II - Horizon fini et contrôle optimal // Comptes Rendus Mathematique. 2006. Vol. 343. P. 679-684. DOI: 10.1016/j.crma.2006.09.018
25. Lasry J.-M., Lions P.-L. Mean field games // Japanese Journal of Mathematics. 2007. Vol. 2. P. 229-260. DOI: 10.1007/s11537-007-0657-8
26. Lasry J.-M., Lions P.-L. Mean-field games with a major player // Comptes Rendus Mathematique. 2018. Vol. 356. P. 886-890. DOI: 10.1016/j.crma.2018.06.001
27. Lions P.-L. College de France course on mean-field games. College de France, 2007-2011.
28. Nguyen S.L., Huang M. Mean field LQG games with a major player: Continuum parameters for minor players // 2011 50th IEEE Conference on Decision and Control and European Control Conference. IEEE, 2011. DOI: 10.1109/cdc.2011.6160790
29. Nguyen S.L., Huang M. Mean field LQG games with mass behavior responsive to a major player // 2012 IEEE 51st IEEE Conference on Decision and Control (CDC). IEEE, 2012. P. 5772-5797. DOI: 10.1109/CDC.2012.6425984
30. Shi J., Wang G., Xiong J. Stochastic linear quadratic Stackelberg differential game with overlapping information // arXiv:1804.07466 [math.OC]. 2010. http://arxiv.org/pdf/1804.07466.pdf
Поступила в редакцию
2018-10-15
Опубликована 2018-11-20
Опубликована 2018-11-20