Time-consistent, risk-averse dynamic pricing

• Published in: European journal of operational research Vol. 277; no. 2; pp. 587 - 603
• Main Authors: Schur, Rouven, Gönsch, Jochen, Hassler, Michael
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-09-2019
• Subjects: Conditional value-at-risk; Dynamic pricing; Revenue management; Risk aversion; Risk aversion; Dynamic pricing; Conditional value-at-risk; Revenue management
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2019.02.038

•Classical approaches are risk-neutral, but risk aversion is widespread in practice.•In dynamic problems, time consistency is important.•Nested conditional value at risk ensures decisions are actually implemented.•Transformation into a classical, risk-neutral dynamic pricing problem.•Numerical study: risk/revenue trade-off, comparison to approaches from literature.
Many industries use dynamic pricing on an operational level to maximize revenue from selling a fixed capacity over a finite horizon. Classical risk-neutral approaches do not accommodate the risk aversion often encountered in practice. When risk aversion is considered, time-consistency becomes an important issue. In this paper, we use a dynamic coherent risk-measure to ensure that decisions are actually implemented and only depend on states that may realize in the future. In particular, we use the risk measure Conditional Value-at-Risk (CVaR), which recently became popular in areas like finance, energy or supply chain management.
A result is that the risk-averse dynamic pricing problem can be transformed to a classical, risk-neutral problem. To do so, a surprisingly simple modification of the selling probabilities suffices. Thus, all structural properties carry over. Moreover, we show that the risk-averse and the risk-neutral solution of the original problem are proportional under certain conditions, that is, their optimal decision variable and objective values are proportional, respectively. In a small numerical study, we evaluate the risk vs. revenue trade-off and compare the new approach with existing approaches from literature.
This has straightforward implications for practice. On the one hand, it shows that existing dynamic pricing algorithms and systems can be kept in place and easily incorporate risk aversion. On the other hand, our results help to understand many risk-averse decision makers who often use “conservative” estimates of selling probabilities or discount optimal prices.