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Myopic Solutions of Markov Decision Processes and Stochastic Games

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  • Matthew J. Sobel

    (Georgia Institute of Technology, Atlanta, Georgia)

Abstract

Sufficient conditions are presented for a Markov decision process to have a myopic optimum and for a stochastic game to possess a myopic equilibrium point. An optimum (or an equilibrium point) is said to be “myopic” if it can be deduced from an optimum (or an equilibrium point) of a static optimization problem (or a static [Nash] game). The principal conditions are (a) each single period reward is the sum of terms due to the current state and action, (b) each transition probability depends on the action taken but not on the state from which the transition occurs, and (c) an appropriate static optimum (or equilibrium point) is ad infinitum repeatable. These conditions are satisfied by several dynamic oligopoly models and numerous Markov decision processes.

Suggested Citation

  • Matthew J. Sobel, 1981. "Myopic Solutions of Markov Decision Processes and Stochastic Games," Operations Research, INFORMS, vol. 29(5), pages 995-1009, October.
    • Handle: RePEc:inm:oropre:v:29:y:1981:i:5:p:995-1009
    DOI: 10.1287/opre.29.5.995
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    Cited by:

    1. K. Avrachenkov & V. Ejov & J. A. Filar & A. Moghaddam, 2019. "Zero-Sum Stochastic Games over the Field of Real Algebraic Numbers," Dynamic Games and Applications, Springer, vol. 9(4), pages 1026-1041, December.
    2. Li, Zhaolin (Erick) & Liang, Guitian & Fu, Qi (Grace) & Teo, Chung-Piaw, 2023. "Base-Stock Policies with Constant Lead Time: Closed-Form Solutions and Applications," Working Papers BAWP-2023-01, University of Sydney Business School, Discipline of Business Analytics.
    3. Boxiao Chen & Xiuli Chao & Cong Shi, 2021. "Nonparametric Learning Algorithms for Joint Pricing and Inventory Control with Lost Sales and Censored Demand," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 726-756, May.
    4. Jie Ning & Matthew J. Sobel, 2019. "Easy Affine Markov Decision Processes," Operations Research, INFORMS, vol. 67(6), pages 1719-1737, November.
    5. Yan, Xiaoming & Chao, Xiuli & Lu, Ye, 2024. "Optimal control policies for dynamic inventory systems with service level dependent demand," European Journal of Operational Research, Elsevier, vol. 314(3), pages 935-949.
    6. Gavirneni, Srinagesh & Bollapragada, Srinivas & E. Morton, Thomas, 1998. "Periodic review stochastic inventory problem with forecast updates: Worst-case bounds for the myopic solution," European Journal of Operational Research, Elsevier, vol. 111(2), pages 381-392, December.
    7. Xu, Ningxiong, 2008. "Myopic policy for a two-product and multi-period supply contract with different delivery lead times and storage limitation," International Journal of Production Economics, Elsevier, vol. 115(1), pages 179-188, September.
    8. Lode Li & Martin Shubik & Matthew J. Sobel, 2013. "Control of Dividends, Capital Subscriptions, and Physical Inventories," Management Science, INFORMS, vol. 59(5), pages 1107-1124, May.
    9. Zeynep Müge Avsar & Melike Baykal‐Gürsoy, 2002. "Inventory control under substitutable demand: A stochastic game application," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(4), pages 359-375, June.
    10. Matthew J. Sobel & Wei Wei, 2010. "Myopic Solutions of Homogeneous Sequential Decision Processes," Operations Research, INFORMS, vol. 58(4-part-2), pages 1235-1246, August.
    11. Cetinkaya, S. & Parlar, M., 1998. "Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs," European Journal of Operational Research, Elsevier, vol. 110(1), pages 20-41, October.
    12. Chiang, Chi, 2003. "Optimal replenishment for a periodic review inventory system with two supply modes," European Journal of Operational Research, Elsevier, vol. 149(1), pages 229-244, August.
    13. Torpong Cheevaprawatdomrong & Robert L. Smith, 2004. "Infinite Horizon Production Scheduling in Time-Varying Systems Under Stochastic Demand," Operations Research, INFORMS, vol. 52(1), pages 105-115, February.
    14. Boxiao Chen & Xiuli Chao, 2020. "Dynamic Inventory Control with Stockout Substitution and Demand Learning," Management Science, INFORMS, vol. 66(11), pages 5108-5127, November.
    15. Eriksson, Katarina, 2019. "An option mechanism to coordinate a dyadic supply chain bilaterally in a multi-period setting," Omega, Elsevier, vol. 88(C), pages 196-209.
    16. Victoria L. Zhang, 1996. "Ordering policies for an inventory system with three supply modes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(5), pages 691-708, August.
    17. Brian Downs & Richard Metters & John Semple, 2001. "Managing Inventory with Multiple Products, Lags in Delivery, Resource Constraints, and Lost Sales: A Mathematical Programming Approach," Management Science, INFORMS, vol. 47(3), pages 464-479, March.
    18. N. Krishnamurthy & S. K. Neogy, 2020. "On Lemke processibility of LCP formulations for solving discounted switching control stochastic games," Annals of Operations Research, Springer, vol. 295(2), pages 633-644, December.
    19. Jian Yang & Jim (Junmin) Shi, 2023. "Discrete‐item inventory control involving unknown censored demand and convex inventory costs," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 45-64, January.
    20. Ying Liang & Yingying Yi & Qiufen Sun, 2014. "The Impact of Migration on Fertility under China’s Underlying Restrictions: A Comparative Study Between Permanent and Temporary Migrants," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 116(1), pages 307-326, March.
    21. Amin H. Amershi & Joel S. Demski & John Fellingham, 1985. "Sequential Bayesian Analysis in accounting settings," Contemporary Accounting Research, John Wiley & Sons, vol. 1(2), pages 176-192, March.
    22. Greys Sošić, 2010. "Stability of information‐sharing alliances in a three‐level supply chain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(3), pages 279-295, April.
    23. David G. Lawson & Evan L. Porteus, 2000. "Multistage Inventory Management with Expediting," Operations Research, INFORMS, vol. 48(6), pages 878-893, December.
    24. Jie Ning, 2021. "Reducible Markov Decision Processes and Stochastic Games," Production and Operations Management, Production and Operations Management Society, vol. 30(8), pages 2726-2751, August.
    25. Felipe Caro & Victor Martínez-de-Albéniz, 2010. "The Impact of Quick Response in Inventory-Based Competition," Manufacturing & Service Operations Management, INFORMS, vol. 12(3), pages 409-429, January.

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