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Simultaneously Learning and Optimizing Using Controlled Variance Pricing


  • Arnoud V. den Boer

    () (Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands; and University of Amsterdam, 1098 XH Amsterdam, The Netherlands)

  • Bert Zwart

    () (Centrum Wiskunde and Informatica, 1098 XG Amsterdam, The Netherlands; and Department of Mathematics, VU University Amsterdam, 1081 HV Amsterdam, The Netherlands)


Price experimentation is an important tool for firms to find the optimal selling price of their products. It should be conducted properly, since experimenting with selling prices can be costly. A firm, therefore, needs to find a pricing policy that optimally balances between learning the optimal price and gaining revenue. In this paper, we propose such a pricing policy, called controlled variance pricing (CVP). The key idea of the policy is to enhance the certainty equivalent pricing policy with a taboo interval around the average of previously chosen prices. The width of the taboo interval shrinks at an appropriate rate as the amount of data gathered gets large; this guarantees sufficient price dispersion. For a large class of demand models, we show that this procedure is strongly consistent, which means that eventually the value of the optimal price will be learned, and derive upper bounds on the regret, which is the expected amount of money lost due to not using the optimal price. Numerical tests indicate that CVP performs well on different demand models and time scales. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Suggested Citation

  • Arnoud V. den Boer & Bert Zwart, 2014. "Simultaneously Learning and Optimizing Using Controlled Variance Pricing," Management Science, INFORMS, vol. 60(3), pages 770-783, March.
  • Handle: RePEc:inm:ormnsc:v:60:y:2014:i:3:p:770-783

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    References listed on IDEAS

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    9. McLennan, Andrew, 1984. "Price dispersion and incomplete learning in the long run," Journal of Economic Dynamics and Control, Elsevier, vol. 7(3), pages 331-347, September.
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    Cited by:

    1. Stefanus Jasin, 2014. "Reoptimization and Self-Adjusting Price Control for Network Revenue Management," Operations Research, INFORMS, vol. 62(5), pages 1168-1178, October.
    2. William L. Cooper & Tito Homem-de-Mello & Anton J. Kleywegt, 2015. "Learning and Pricing with Models That Do Not Explicitly Incorporate Competition," Operations Research, INFORMS, vol. 63(1), pages 86-103, February.
    3. Arnoud V. den Boer & Bert Zwart, 2015. "Dynamic Pricing and Learning with Finite Inventories," Operations Research, INFORMS, vol. 63(4), pages 965-978, August.
    4. repec:eee:touman:v:51:y:2015:i:c:p:96-102 is not listed on IDEAS
    5. Ningyuan Chen & Guillermo Gallego, 2018. "A Primal-dual Learning Algorithm for Personalized Dynamic Pricing with an Inventory Constraint," Papers 1812.09234,
    6. Omar Besbes & Denis Sauré, 2014. "Dynamic Pricing Strategies in the Presence of Demand Shifts," Manufacturing & Service Operations Management, INFORMS, vol. 16(4), pages 513-528, October.
    7. Omar Besbes & Yonatan Gur & Assaf Zeevi, 2015. "Non-Stationary Stochastic Optimization," Operations Research, INFORMS, vol. 63(5), pages 1227-1244, October.
    8. Omar Besbes & Assaf Zeevi, 2015. "On the (Surprising) Sufficiency of Linear Models for Dynamic Pricing with Demand Learning," Management Science, INFORMS, vol. 61(4), pages 723-739, April.
    9. Ruben van de Geer & Arnoud V. den Boer & Christopher Bayliss & Christine Currie & Andria Ellina & Malte Esders & Alwin Haensel & Xiao Lei & Kyle D. S. Maclean & Antonio Martinez-Sykora & Asbj{o}rn Nil, 2018. "Dynamic Pricing and Learning with Competition: Insights from the Dynamic Pricing Challenge at the 2017 INFORMS RM & Pricing Conference," Papers 1804.03219,
    10. Zizhuo Wang & Shiming Deng & Yinyu Ye, 2014. "Close the Gaps: A Learning-While-Doing Algorithm for Single-Product Revenue Management Problems," Operations Research, INFORMS, vol. 62(2), pages 318-331, April.


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