Selling Different Versions of the Same Product – Robustly!

The idea of offering different versions of the same product dates back more than one hundred years, when publishers realized that selling the same book with different paper qualities and bindings could be done at very different prices, so as to attract very different types of customers. Thus, the practice of “price discrimination” has firmly taken hold in our economy, allowing companies to respond better to customer needs and at the same time to increase their ability to extract surplus, when maximizing their expected profits. In economics, this problem—sometimes referred to as “screening”—led to Joseph Stiglitz winning the Nobel Prize in Economics in 2001. It consists of extracting a piece of private information from an agent (such as his willingness-to-pay) by offering a menu of options, so that by selecting his most preferred option, the agent would end up revealing his private piece of information. The corresponding optimal (i.e., profit-maximizing) design of a “product menu” relies on the firm’s beliefs about how likely it is to encounter any given “type” of customer, where a “type” refers to a specific piece of private information (such as a low or high willingness-to-pay). The research paper tackles this problem when the designer of the economic mechanism (i.e., the company) does not really know what beliefs about the different types would be reasonable to have, that is, there is “belief ambiguity.” When comparing the results in the paper to a standard linear-quadratic two-type screening model widely used in the literature, it is remarkable that it is possible to guarantee that the optimal robust product menu (in the absence of valid beliefs) always achieves at least 75% of the optimal expected profits when the relative frequencies of consumer types are perfectly known.

The present research paper [1] is part of an ongoing project on “relatively robust decisions” at the Chair of Operations, Economics and Strategy (OES) at EPFL, as formalized in [2]. The project also relates to earlier work by Prof. Weber on fair welfare maximization [3], which featured a relative performance index quite similar to the one used in the latest research.

Abstract

This paper considers the problem of second-degree price discrimination when the type distribution is unknown or imperfectly specified by means of an ambiguity set. As robustness measure we use a performance index, equivalent to relative regret, which quantifies the worst-case attainment ratio between actual payoff and ex-post optimal payoff. We provide a simple representation of this performance index, as the lower envelope of two boundary performance ratios, relative to beliefs that lie at the boundary of the ambiguity set. A characterization of the solution to the underlying robust identification problem is given, which leads to a robust product portfolio, for which we also determine the worst-case performance over all possible consumer types. For a standard linear-quadratic specification of the robust screening model, a worst-case performance index of 75% guarantees that the robust product portfolio exhibits a profitability that lies within a 25%-band of an ex-post optimal product portfolio, over all possible model parameters and beliefs. Finally, a numerical comparison benchmarks the robust solution against a number of alternative belief heuristics.