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Linear-Quadratic Mixed Stackelberg–Nash Stochastic Differential Game with Major–Minor Agents

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Abstract

In this paper, we study a controlled linear-quadratic-Gaussian large population system combining three types of interactive agents mixed, which are respectively, major leader, minor leaders, and minor followers. In reality, they may represent three typical types of participants involved in market price formation: major supplier, minor suppliers and minor producers. The Stackelberg–Nash–Cournot (SNC) approximate equilibrium is derived from the combination of a major–minor mean-field game (MFG) and a leader–follower Stackelberg game. Although all agents are of forward states in that only initial conditions are specified in their dynamics, our SNC analysis provides an MFG framework that is naturally in a forward–backward state in that both initial and terminal conditions are specified. This result differs from those reported in the literature on standard MFG frameworks, mainly as a result of the adoption of a Stackelberg structure. Through variational analysis, the consistency condition system can be represented by some fully-coupled forward–backward-stochastic-differential-equations with a high-dimensional block structure in an open-loop case. To sufficiently address the related solvability, we also derive the feedback form of the SNC approximate. equilibrium strategy via some coupled Riccati equations. Our study includes various mean-field game models as its special cases.

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Notes

  1. Hadamard product (also known as the Schur product or the entrywise product) is a binary operation that takes two matrices of the same dimensions, and produces another matrix where each element (ij) is the product of elements (ij) of the original two matrices.

References

  1. Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63(3), 341–356 (2011)

    Article  MathSciNet  Google Scholar 

  2. Aurell, A., Djehiche, B.: Modeling tagged pedestrian motion: a mean-field type game approach. Transp. Res. Part B 121, 168–183 (2019)

    Article  Google Scholar 

  3. Bardi, M.: Explicit solutions of some linear-quadratic mean field games. Netw. Heterog. Media 7(2), 243–261 (2012)

    Article  MathSciNet  Google Scholar 

  4. Basar, T.: Stochastic Stagewise Stackelberg Strategies for Linear Quadratic Systems. Stochastic Control Theory and Stochastic Differential Systems, pp. 264–276. Springer, Berlin (1979)

    Google Scholar 

  5. Bauso, D., Tembine, H., Basar, T.: Opinion dynamics in social networks through mean-field games. SIAM J. Control Optim. 54(6), 3225–3257 (2016)

    Article  MathSciNet  Google Scholar 

  6. Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springerbriefs in Mathematics, vol. 101. Springer, New York (2013)

    Book  Google Scholar 

  7. Bensoussan, A., Chau, M., Yam, S.: Mean field stackelberg games: aggregation of delayed instructions. SIAM J. Control Optim. 53(4), 2237–2266 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bensoussan, A., Chen, S., Sethi, S.: The maximum principle for global solutions of stochastic stackelberg differential games. SIAM J. Control Optim. 53(4), 1956–1981 (2015)

    Article  MathSciNet  Google Scholar 

  9. Bensoussan, A., Chau, M., Yam, S.: Mean field games with a dominating player. Appl. Math. Optim. 74(1), 91–128 (2016)

    Article  MathSciNet  Google Scholar 

  10. Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51(4), 2705–2734 (2013)

    Article  MathSciNet  Google Scholar 

  11. Como, G., Fagnani, F.: Scaling limits for continuous opinion dynamics systems. In: 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1562–1566. IEEE (2009)

  12. Du, H., Huang, J., Qin, Y.: A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications. IEEE Trans. Autom. Control 58, 3212–3217 (2013)

    Article  MathSciNet  Google Scholar 

  13. Freiling, G., Jank, G., Abou-Kandil, H.: On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games. IEEE Trans. Autom. Control 41(2), 264–269 (1996)

    Article  MathSciNet  Google Scholar 

  14. Garnier, J., Papanicolaou, G., Yang, T.: Large deviations for a mean field model of systemic risk. SIAM J. Finan. Math. 4(1), 151–184 (2013)

    Article  MathSciNet  Google Scholar 

  15. Guéant, O., Lasry, J., Lions, P.: Mean Field Games and Applications. Paris-Princeton Lectures on Mathematical Finance 2010, pp. 205–266. Springer, Berlin (2011)

  16. Hu, Y., Huang, J., Nie, T.: Linear-quadratic-gaussian mixed mean-field games with heterogeneous input constraints. SIAM J. Control Optim. 56(4), 2835–2877 (2018)

    Article  MathSciNet  Google Scholar 

  17. Huang, M.: Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM J. Control Optim. 48(5), 3318–3353 (2010)

    Article  MathSciNet  Google Scholar 

  18. Huang, M., Caines, P., Malhamé, R.: Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. In: Proceedings of the 42nd IEEE Conference on Decision and Control, IEEE, vol. 1, pp. 98–103 (2003)

  19. Huang, M., Malhamé, R., Caines, P.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)

    Article  MathSciNet  Google Scholar 

  20. Huang, J., Wang, S., Wu, Z.: Backward-forward linear-quadratic mean-field games with major and minor agents. Probab. Uncertain. Quant. Risk 1(8), 1–27 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Huang, J., Wang, S., Wu, Z.: Backward mean-field linear-quadratic-gaussian (LQG) games: full and partial information. IEEE Trans. Autom. Control 61(12), 3784–3796 (2016)

    Article  MathSciNet  Google Scholar 

  22. Jourdain, B., Méléard, S., Woyczynski, W.: Nonlinear stochastic differential equations driven by lévy processes and related differential equations. AlEA-Latin-Am. J. Probab. Math. Stat. 4, 31–46 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Juang, J.: Global existence and stability of solutions of matrix Riccati equations. J. Math. Anal. Appl. 258(1), 1–12 (2001)

    Article  MathSciNet  Google Scholar 

  24. Kohlmann, M., Tang, S.: Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging. Stoch. Process. Appl. 97(2), 255–288 (2002)

    Article  MathSciNet  Google Scholar 

  25. Lasry, J., Lions, P.: Jeux à champ moyen. I—Le cas stationnaire. Comptes Rendus Mathématique 343(9), 619–625 (2006)

    Article  MathSciNet  Google Scholar 

  26. Lasry, J., Lions, P.: Jeux à champ moyen. II—Horizon fini et contrôle optimal. Comptes Rendus Mathématique 343(10), 679–684 (2006)

    Article  MathSciNet  Google Scholar 

  27. Lasry, J., Lions, P.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)

    Article  MathSciNet  Google Scholar 

  28. Ma, J., Ren, H.: The impact of variable cost on a dynamic Cournot-Stackelberg game with two decision-making stages. Commun. Nonlinear Sci. Numer. Simul. 62(SEP.), 184–201 (2018)

    Article  MathSciNet  Google Scholar 

  29. Ma, J., Yong, J.: Forward-Backward Stochastic Differential Equations and Their Applications, vol. 1702. Springer, New York (1999)

    MATH  Google Scholar 

  30. Nourian, M., Caines, P.: \(\varepsilon \)-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM J. Control Optim. 51(4), 3302–3331 (2013)

    Article  MathSciNet  Google Scholar 

  31. Nourian, M., Caines, P., Malhamé, R., Huang, M.: Mean field LQG control in leader-follower stochastic multi-agent systems: likelihood ratio based adaptation. IEEE Trans. Autom. Control 57(11), 2801–2816 (2012)

    Article  MathSciNet  Google Scholar 

  32. Sîrbu, A., Loreto, V., Servedio, V.D., Tria, F.: Opinion dynamics: models, extensions and external effects. In: Participatory Sensing, Opinions and Collective Awareness, pp. 363–401. Springer, Cham (2017)

  33. Stella, L., Bagagiolo, F., Bauso, D., Como, G.: Opinion dynamics and stubbornness through mean-field games. In: 52nd IEEE Conference on Decision and Control. IEEE, pp. 2519–2524 (2013)

  34. Sun, J., Li, X., Yong, J.: Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. SIAM J. Control Optim. 54(5), 2274–2308 (2016)

    Article  MathSciNet  Google Scholar 

  35. Von Stackelberg, H.: Marktform Und Gleichgewicht. Springer, (1934). (An English translation appeared in The Theory of the Market Economy. Oxford University Press, Oxford, England (1952)

  36. Wei, J., Tang, J., Liu, Y., Jin, Y.: Game analysis of ordering model for batch between buyer and supplier in stochastic demand. J. Northeastern Univ. 30(10), 1390–1393 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Yong, J.: Linear forward-backward stochastic differential equations. Appl. Math. Optim. 39(1), 93–119 (1999)

    Article  MathSciNet  Google Scholar 

  38. Yong, J.: A leader-follower stochastic linear quadratic differential game. SIAM J. Control Optim. 41(4), 1015–1041 (2002)

    Article  MathSciNet  Google Scholar 

  39. Yong, J.: Forward-backward stochastic differential equations with mixed initial-terminal conditions. Trans. Am. Math. Soc. 362(2), 1047–1096 (2010)

    Article  MathSciNet  Google Scholar 

  40. Yong, J., Zhou, X.: Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43. Springer, New York (1999)

    Book  Google Scholar 

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Correspondence to Kehan Si.

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Jianhui Huang acknowledges the financial support by RGC Grant PolyU 153005/14P, 153275/16P. Kehan Si acknowledges the financial support from the Shandong University, and the present work constitutes a part of his work for his postgraduate dissertation. Zhen Wu acknowledges the financial support by NSFC (11831010,61573217).

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Huang, J., Si, K. & Wu, Z. Linear-Quadratic Mixed Stackelberg–Nash Stochastic Differential Game with Major–Minor Agents. Appl Math Optim 84, 2445–2494 (2021). https://doi.org/10.1007/s00245-020-09713-z

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