動態定價能藉由調整定價決策提高公司收益。我們考慮在穩定需求環境中，存在多間公司販賣單一種商品的均衡定價系統，並且將每一期的問題視為非合作賽局的納許均衡問題。在需求曲線的參數是未知的條件下，每間公司必須透過歷史資料估計環境需求曲線之參數。我們提出一個利用乘法模型與正交擾動價格設置的需求學習演算法，其能夠求出每間公司之價格策略並同時學習需求曲線之參數以及最大化收益。在演算法中，每間公司的目標即透過依序調整價格決策來達成最佳收益。本研究進行一系列的數值實驗來分析在不同設定下之演算法表現，並利用後悔函數來衡量表現。我們發現到雖然演算法利用線性最小平方法估計非線性之需求曲線參數，但演算法的表現仍然為佳，並且隨著時間到達一萬期週期後，收斂比率均高於百分之九十以上。最後，本演算法可應用於競爭市場中販賣單一種產品之動態定價問題。

Dynamic pricing enables firms get more revenue by adjusting the price decision. We consider a periodic review equilibrium pricing system in which multiple firms competing in the market of single product in a stationary demand environment and regard the single-period problem as the Nash equilibrium problem under a non-cooperative game. The parameters of the demand model are not known a priori and each firm needs to estimate parameters of underlying demand model by observed data. We propose a demand learning algorithm assuming a multiplicative demand model with an equilibrium and orthogonally-perturbed price setup, which can learn and earn concurrently to solve for the pricing policy of each firm in the competition. In the algorithm, each firms' objective is to sequentially adjust prices to maximize revenues under demand uncertainty. This research conducts a series of numerical experiments to analyze the algorithm in different settings and measure the performance by regret. We observe that the performance of the algorithm is good even though the algorithm estimates the parameters of nonlinear demand curves, logit and exponential models, by the linear least square method and show that the fraction of optimal revenues are above 90% as time goes by 10000 periods. Our algorithm can be applied to dynamic pricing problem with unknown demand for selling single product in a competitive market.

Besbes, O., & Zeevi, A. (2015). On the (surprising) sufficiency of linear models for dynamic pricing with demand learning. Management Science, 61(4), 723-739.
Broder, J., & Rusmevichientong, P. (2012). Dynamic pricing under a general parametric choice model. Operations Research, 60(4), 965-980.
Chen, B., Chao, X., & Ahn, H. S. (2019). Coordinating pricing and inventory replenishment with nonparametric demand learning. Operations Research.
Chen, B., Chao, X., & Shi, C. (2015). Nonparametric algorithms for joint pricing and inventory control with lost-sales and censored demand. Mathematics of Operations Research, Major Revision.
Cooper, W. L., Homem-de-Mello, T., & Kleywegt, A. J. (2015). Learning and pricing with models that do not explicitly incorporate competition. Operations research, 63(1), 86-103.
Carvalho, A. X., & Puterman, M. L. (2005). Learning and pricing in an internet environment with binomial demands. Journal of Revenue and Pricing Management, 3(4), 320-336.
Chen, X., & Deng, X. (2006, October). Settling the complexity of two-player Nash equilibrium. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06) (pp. 261-272). IEEE.
Contreras, J., Klusch, M., & Krawczyk, J. B. (2004). Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Transactions on Power Systems, 19(1), 195-206.
Dai, Y., Chao, X., Fang, S. C., & Nuttle, H. L. (2005). Pricing in revenue management for multiple firms competing for customers. International Journal of Production Economics, 98(1), 1-16.
den Boer, A., & Zwart, B. (2014). Simultaneously learning and optimizing using controlled variance pricing. forthcoming in Management Science.
Keskin, N. B., & Zeevi, A. (2014). Dynamic pricing with an unknown demand model: Asymptotically optimal semi-myopic policies. Operations Research, 62(5), 1142-1167.
Lantseva, A., Mukhina, K., Nikishova, A., Ivanov, S., & Knyazkov, K. (2015). Data-driven modeling of airlines pricing. Procedia Computer Science, 66, 267-276.
Nash, J. (1951). Non-cooperative games. Annals of mathematics, 286-295.
Rothschild, M. (1974). A two-armed bandit theory of market pricing. Journal of Economic Theory, 9(2), 185-202.
Song, H., Liu, C. C., & Lawarrée, J. (2002). Nash equilibrium bidding strategies in a bilateral electricity market. IEEE transactions on Power Systems, 17(1), 73-79.
Saez-Gallego, J., Morales, J. M., Zugno, M., & Madsen, H. (2016). A data-driven bidding model for a cluster of price-responsive consumers of electricity. IEEE Transactions on Power Systems, 31(6), 5001-5011.
Wang, Z., Deng, S., & Ye, Y. (2014). Close the gaps: A learning-while-doing algorithm for single-product revenue management problems. Operations Research, 62(2), 318-331.