In many revenue management applications risk-averse decision-making is crucial. In dynamic settings, however, it is challenging to find the right balance between maximizing expected rewards and avoiding poor performances. In this paper, we consider time-consistent mean-semivariance (MSV) optimization for dynamic pricing problems within a discrete MDP framework, which are shown to be NP hard. We present a novel fixpoint-based dynamic programming approach to compute risk-sensitive feedback policies with Pareto-optimal combinations of mean and semivariance. We illustrate the effectiveness and the applicability of our concepts compared to state-of-the-art heuristics. For various numerical examples the results show that our approach clearly outperforms all other heuristics and obtains a performance guarantee with less then 0.2% optimality gap. Our approach is general and can be applied to MDPs beyond dynamic pricing.

Today’s technology facilitates selling strategies that were unthinkable only a few years ago. One increasingly popular strategy uses incompletely specified products (ICSPs). The seller retains the right to specify some details of the product or service after the sale. The selling strategies’ main advantages are an additional dimension for market segmentation and operational flexibility due to supply-side substitution possibilities. Since the strategy became popular with Priceline and Hotwire in the travel industry about two decades ago, it has increasingly been adopted by other industries with stochastic demand and limited capacity as well. At the same time, it is actively researched from the perspectives of strategic operations management, empirics, and revenue management.

This paper first describes the application of ICSPs in practice. Then, we introduce the different research communities that are active in this field and relate the terminology they use. The main part is an exhaustive review of the literature on selling ICSPs from the different perspectives. Here, we complement a tabular overview with an introduction into the community and a detailed description of each paper. Finally, possible directions for future research are outlined.

We see that strategic operations management has described advantages of ICSPs over other strategies in a variety of settings, but also identified countervailing effects. Today, empirical research is confined to hotels and airlines and largely disconnected from the other perspectives. Operational papers are ample, but mostly concerned with the availability of ICSPs. Research on operational (dynamic) pricing is surprisingly scarce.

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.

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. We add to the scarce literature on risk aversion by considering the risk measure conditional value-at-risk (CVaR), which recently became popular in areas like finance, energy, or supply chain management. A key aspect of this paper is selling a single unit of capacity, which is highly relevant in, for example, the real estate market. We analytically derive the optimal policy and obtain structural results. The most important managerial implication is that the risk-averse optimal price is constant over large parts of the selling horizon, whereas the price continuously declines in the standard setting of risk-neutral dynamic pricing. This offers a completely new explanation for the price-setting behavior often observed in practice. For arbitrary capacity, we develop two algorithms to efficiently compute the value function and evaluate them in a numerical study. Our results show that applying a risk-averse policy, even a static one, often yields a higher CVaR than applying a dynamic, but risk-neutral, policy.

Keywords

Revenue management Dynamic pricing Dynamic programming Risk management Service operations

This paper provides an overview of the literature on dynamic pricing with strategic customers. In the past, research on dynamic pricing was mostly concerned with optimally pricing products over time in a market with myopic customers. In recent years, the consideration of strategic customers, who can delay a purchase to take advantage of a future discount, has dramatically increased. This paper’s main contribution is the development of a comprehensive classification scheme to structure the field of research and, based upon this, a systematic overview of all relevant papers. We then present in detail the various aspects considered in the literature together with their motivation from industry and state the major findings of the most relevant papers. Further attention is given to important problem extensions proposed in the literature that have been considered in only a few papers and are usually motivated by specific practical applications. Finally, promising directions for future research are indicated.