需求學習和定價策略一直是收益管理領域學術研究中非常重要的部分，在過往的文獻中已提出許多演算法。而我們以Yang, Lee, & Chen (2021) 所發表的競爭性需求學習為背景，在研究中作者考慮的是需求曲線未知的條件下公司必須透過觀察歷史資料估計需求曲線參數，每一間公司販賣相同產品並相互競爭，公司的需求取決於其他公司的價格，亦即需求由市場中所有參與者共同決定，每間公司的目標都是最大化他們的收入。正常情況下公司不斷根據最佳決策反應來調整自己的定價策略，最終所有公司的定價決策都達到納許均衡。其中提出一種數據驅動的學習演算法來解決公司處於競爭該如何定價，而為了達成此目標，演算法背後需要提出一些常用的假設性質，本研究主要針對假設性質以及經典案例進行一系列的討論分析，與最佳化條件高度相關的增量差異性也納入考慮，並在分析過程中提出超模和凸性兩個關鍵要素出現若且唯若的充要條件，本研究表明了在特殊範圍條件下這幾種關鍵性質之間如何相互影響並利用超模提出嚴謹的驗證。

Demand learning and pricing strategies have always been a very important part of academic research in the field of revenue management, and many algorithms have been proposed in the past literature. We take the competitive demand learning proposed by Yang, Lee, & Chen (2021) as the background. When the demand curve is unknown, the firm must estimate the parameters of the demand curve by observing historical data. Under normal circumstances, companies continuously adjust their pricing strategies according to the best decision response, and eventually all companies' pricing decisions reach a Nash equilibrium. To achieve this goal, some common assumptions and properties need to be developed behind the algorithm. This study focuses on these important properties. A series of discussion and analysis are carried out for the hypothetical properties and classic examples, and the increasing difference that is highly correlated with the optimization conditions is also taken into account. Our study demonstrates the interplay of several key properties under a special range of conditions.

References

Arnoud V. den Boer (2015). Dynamic pricing and learning: Historical origins, current research, and new directions. University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands.
Bertsimas, D., & Perakis, G. (2006). Dynamic pricing: A learning approach. In Mathematical and computational models for congestion charging (pp. 45-79). Springer, Boston, MA.
Bernstein F, Federgruen A (2004) Dynamic inventory and pricing models for competing retailers. Naval Research Logistics (NRL) 51(2):258–274.
Bernstein F, Federgruen A (2004) A general equilibrium model for industries with price and service competition. Operations research 52(6):868–886.
Besbes, O., and Zeevi, A. (2009). Dynamic pricing without knowing the demand function: Risk bounds and near-optimal algorithms. Operations Research, 57(6), 1407-1420.
Besbes, O. and Zeevi, A. (2015). On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. Management Science, 61(4):723–739.
Cheung, Simchi-Levi, and Wang (2017), Technical Note—Dynamic Pricing and Demand Learning with Limited Price Experimentation. Operations ResearchVol. 65, No. 6.
Federgruen, A. and Heching, A. (1999), “Combined Pricing and Inventory Control Under Uncertainty,” Operations Research, 47, 3, 454-475.
Ferreira, Kris, and Emily Mower (2021), Demand Learning and Pricing for Varying Assortments. manufacturing and service operations management.
Fisher, M., J. Hammond, W. Obermeyer and A. Raman (1994), “Making Supply Meet Demand in an Uncertain World,” Harvard Business Review, May-June 1994.
Harrison, J., Keskin, N., and Zeevi, A. (2012). Bayesian dynamic pricing policies: Learning and earning under a binary prior distribution. Management Science, 58(3):570–586.
Keskin and Zeevi (2014). “Dynamic pricing with an unknown demand model: Asymptotically optimal semi-myopic policies.” Operations Research 62 (5): 1142–1167.
Milgrom, P. and J. Roberts (1990), “Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities,” Econometrica, 58, 1255-1277.
Nash, J. F. (1950), Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36, 48–49.
Rothschild, M. (1974). A two-armed bandit theory of market pricing. Journal of Economic Theory, 9(2):185– 202.
Springer. (2005), Convexity and Supermodularity. In: The Logic of Logistics. Springer Series in Operations Research. Springer, New York.
Topkis, D. (1998), Supermodularity and Complementarity, Princeton University Press, Princeton, NJ.
Ted Loch-Temzelides (2020), “Walrasian equilibrium behavior in nature“ Proceedings of the National Academy of Sciences.
Yang, Lee, and Chen (2020) Yang, Yongge, Yu-Ching Lee, and Po-An Chen. 2020. “Competitive Demand Learning: a Data-Driven Pricing Algorithm.” arXiv preprint arXiv:2008.05195.
Zipkin, P. (2000), Foundations of Inventory Management, McGraw-Hill, New York.