Robust pricing when the demand curve is unknown

© 2024 EPFL

© 2024 EPFL

When trying to price a new product a firm may have only a vague idea about the demand it is facing. In fact, many demand curves may be plausible. Prof. Weber’s forthcoming research paper on “Robust Monopoly Pricing with Unknown Demand” addresses this problem when the firm’s unknown demand may be nonlinear. The approach can be made entirely data-driven and it provides performance guarantees relative to the “ex post optimal profit” which could have been obtained if the demand curve was in fact perfectly known. The paper is forthcoming in the Scandinavian Journal of Economics, established over 125 years ago.

The relative robustness criterion minimizes the maximum relative regret a decision maker may experience relative to how well s/he could have done without model ambiguity. This criterion is equivalent to maximizing a performance index, which amounts to the worst-case ratio of achieved profit performance to ex post optimal profit performance. A distinguishing feature of this criterion over the more commonly used maximin payoff criterion (or worst-case optimization) and minimax absolute regret is that it tends not to be as costly, delivering reasonable expected profits and at the same time quite nontrivial performance guarantees, which balances performance deviations at both ends of the spectrum, “high” and “low.” Indeed, taking only the worst-case performance into account leads to expensive decisions, while absolute regret generically achieves zero profits with positive probability, thus providing no reasonable performance guarantee from a financial viewpoint. Using the data-driven version of the proposed approach mimics the variation of the ex post optimal price across different demand curve realizations.

Figure: Performance ratios against realized profits for the proposed relatively robust, maximin-payoff (WC), and certainty-equivalence (CE) solutions (maximizing expected profits). Source: [1, Fig. 11a]

The present paper [1] belongs to an ongoing research stream on “relatively robust decisions” at the Chair of Operations, Economics and Strategy (OES) at EPFL, described in [2]. The project relates to earlier work by Prof. Weber on robust price discrimination (with Jun Han) [3], and on fair welfare maximization [4].


The optimal pricing of goods, especially when they are new and the innovating firm is a monopolist, must proceed without precise knowledge of the demand curve. This paper provides a pricing method with a relative robustness guarantee by maximizing a performance index which amounts to a worst-case ratio of the obtained payoff to the best possible payoff. Assuming monotonicity and complementarity of demand in price and the unknown demand parameter, the performance index is fully determined by its behavior at the boundary of the parameter space. This allows for an efficient computation of an optimal robust price. In the linear case, which can also be used for nonlinear demand with bounded slope, the method provides a simple closed-form solution. A comparison with the standard worst-case payoff criterion reveals substantial improvements in both absolute and relative performance, at only a small cost relative to the maximized expected profit.



[1] Weber, T.A. (2024) “Monopoly Pricing with Unknown Demand,” Scandinavian Journal of Economics, Forthcoming.
[DOI: 10.1111/sjoe.12564; open access]

[2] Weber, T.A. (2023) “Relatively Robust Decisions,” Theory and Decision, Vol. 94, No. 1, pp. 35—62.  [DOI: 10.1007/s11238-022-09866-z; open access]

[3] Han, J., Weber, T.A. (2023) “Price Discrimination with Robust Beliefs,” European Journal of Operational Research, Vol. 306, No. 2, pp. 795—809.   
[DOI: 10.1016/j.ejor.2022.08.022; open access]

[4] Goel, A., Meyerson, A., Weber, T.A. (2009) “Fair Welfare Maximization,” Economic Theory, Vol. 41, No. 3, pp. 465—494. [DOI: 10.1007/s00199-008-0406-0]