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Stochastic Optimal Control for Online Seller under Reputational Mechanisms


  • Milan Bradonjić

    () (Mathematics of Networks and Systems, Bell Labs, 600 Mountain Avenue, Murray Hill, NJ 07974, USA)

  • Matthew Causley

    () (Department of Mathematics, Kettering University, Flint, MI 48504, USA)

  • Albert Cohen

    () (Department of Statistics and Probability and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA)


In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e. , the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from [1] and used effectively in the Stochastic Optimal Control of Advertising [2]. This model of reputation is combined with the effect of online reputation on sales price empirically verified in [3]. We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller’s reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.

Suggested Citation

  • Milan Bradonjić & Matthew Causley & Albert Cohen, 2015. "Stochastic Optimal Control for Online Seller under Reputational Mechanisms," Risks, MDPI, Open Access Journal, vol. 3(4), pages 1-20, December.
  • Handle: RePEc:gam:jrisks:v:3:y:2015:i:4:p:553-572:d:60014

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    References listed on IDEAS

    1. Patrick Bajari & Ali Hortaçsu, 2004. "Economic Insights from Internet Auctions," Journal of Economic Literature, American Economic Association, vol. 42(2), pages 457-486, June.
    2. Paul Resnick & Richard Zeckhauser & John Swanson & Kate Lockwood, 2006. "The value of reputation on eBay: A controlled experiment," Experimental Economics, Springer;Economic Science Association, vol. 9(2), pages 79-101, June.
    3. Daniel Houser & John Wooders, 2006. "Reputation in Auctions: Theory, and Evidence from eBay," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 15(2), pages 353-369, June.
    4. Ram C. Rao, 1986. "Estimating Continuous Time Advertising-Sales Models," Marketing Science, INFORMS, vol. 5(2), pages 125-142.
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    More about this item


    Stochastic optimal control models; online stores; Hamilton-Jacobi-Bellman equation;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law


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