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Stochastic Stackelberg equilibria with applications to time dependent newsvendor models

Author

Listed:
  • Øksendal, Bernt

    (Dept. of Mathematics, University of Oslo)

  • Sandal, Leif K.

    (Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration)

  • Ubøe, Jan

    (Dept. of Finance and Management Science, Norwegian School of Economics and Business Administration)

Abstract

In this paper we prove a sufficient maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is an Ito-Levy process, and to increase realism information is delayed, e.g., due to production time. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time variable coefficients. Moreover, these results are operational because we are able to offer explicit solution formulas. An interesting finding is that more precise information may be a considerable disadvantage for the retailer.

Suggested Citation

  • Øksendal, Bernt & Sandal, Leif K. & Ubøe, Jan, 2011. "Stochastic Stackelberg equilibria with applications to time dependent newsvendor models," Discussion Papers 2011/9, Norwegian School of Economics, Department of Business and Management Science.
    • Handle: RePEc:hhs:nhhfms:2011_009
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    File URL: http://hdl.handle.net/11250/164022
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    Cited by:

    1. Zheng, Yueyang & Shi, Jingtao, 2022. "A linear-quadratic partially observed Stackelberg stochastic differential game with application," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    2. Yueyang Zheng & Jingtao Shi, 2020. "A Stackelberg Game of Backward Stochastic Differential Equations with Applications," Dynamic Games and Applications, Springer, vol. 10(4), pages 968-992, December.
    3. Chen, Lv & Shen, Yang & Su, Jianxi, 2020. "A continuous-time theory of reinsurance chains," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 129-146.
    4. Gholami, Reza Azad & Sandal, Leif Kristoffer & Ubøe, Jan, 2021. "A solution algorithm for multi-period bi-level channel optimization with dynamic price-dependent stochastic demand," Omega, Elsevier, vol. 102(C).
    5. Albrecht, E & Baum, Günter & Birsa, R & Bradamante, F & Bressan, A & Chapiro, A & Cicuttin, A & Ciliberti, P & Colavita, A & Costa, S & Crespo, M & Cristaudo, P & Dalla Torre, S & Diaz, V & Duic, V &, 2010. "Results from COMPASS RICH-1," Center for Mathematical Economics Working Papers 535, Center for Mathematical Economics, Bielefeld University.
    6. Reza Azad Gholami & Leif Kristoffer Sandal & Jan Ubøe, 2022. "Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts," Games, MDPI, vol. 13(6), pages 1-26, October.
    7. Azad Gholami, Reza & Sandal, Leif K. & Ubøe, Jan, 2019. "Markets With Memory: Dynamic Channel Optimization Models With Price-Dependent Stochastic Demand," Discussion Papers 2019/8, Norwegian School of Economics, Department of Business and Management Science.
    8. Ata Allah Taleizadeh & Mahsa Noori-daryan & Kannan Govindan, 2016. "Pricing and ordering decisions of two competing supply chains with different composite policies: a Stackelberg game-theoretic approach," International Journal of Production Research, Taylor & Francis Journals, vol. 54(9), pages 2807-2836, May.
    9. Azad Gholami, Reza & Sandal, Leif Kristoffer & Ubøe, Jan, 2024. "Who benefits from postponement in multi-period supply channel optimization?," Omega, Elsevier, vol. 123(C).
    10. Azad Gholami, Reza & Sandal, Leif K. & Ubøe, Jan, 2016. "Channel Coordination in a Multi-period Newsvendor Model with Dynamic, Price-dependent Stochastic Demand," Discussion Papers 2016/6, Norwegian School of Economics, Department of Business and Management Science.
    11. Alavi Fard, Farzad & He, Jian & Ivanov, Dmitry & Jie, Ferry, 2019. "A utility adjusted newsvendor model with stochastic demand," International Journal of Production Economics, Elsevier, vol. 211(C), pages 154-165.
    12. Li Chen & Peipei Zhou & Hua Xiao, 2023. "Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations," Mathematics, MDPI, vol. 11(13), pages 1-18, June.
    13. Weiwei Zhang & Zhongfei Li & Ke Fu & Fan Wang, 2017. "Effect of the Return Policy in a Continuous-Time Newsvendor Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-28, December.
    14. Sandal, Leif K. & Ubøe, Jan, 2012. "Stackelberg equilibria in a multiperiod vertical contracting model with uncertain and price-dependent demand," Discussion Papers 2012/2, Norwegian School of Economics, Department of Business and Management Science.
    15. Kogan, Konstantin & Chernonog, Tatyana, 2019. "Competition under industry-stock-driven prevailing market price: Environmental consequences and the effect of uncertainty," European Journal of Operational Research, Elsevier, vol. 276(3), pages 929-946.
    16. Jingtang Ma & Shan Yang, 2024. "High-dimensional stochastic control models for newsvendor problems and deep learning resolution," Annals of Operations Research, Springer, vol. 339(1), pages 789-811, August.
    17. Youkyung Won, 2016. "Dominance Relationship Among the Retailer’s Strategies Under the Semi-Stackelberg Newsvendor Situation with Quantity Discounts," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(02), pages 1-20, April.
    18. Wang, Yu & Yan, Zhiguo, 2023. "Pareto-based Stackelberg differential game for stochastic systems with multi-followers," Applied Mathematics and Computation, Elsevier, vol. 436(C).
    19. Yue Sun & Hongdan Li & Juanjuan Xu & Huanshui Zhang, 2023. "Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 199(3), pages 1158-1187, December.

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    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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