Envy-Free Pricing in Versioning Problems
86 Pages Posted: 22 Dec 2022
Date Written: December 11, 2022
Abstract
In this paper, we treat versioning problems where the reservation prices satisfy the Spence-Mirlees condition, although we do not require that consumers derive a higher value from a higher product type. The prior versioning literature makes additional assumptions in order to decompose the problems by consumer types. We relax such assumptions and find different optimal structures. Moreover, we consider development costs, span of consumers costs, versioning costs, and limits on the number of offered products. We show that, with these costs, the problem can be solved as a shortest path problem resolving a previous complexity conjecture. Using this insight, we prove various structural results about how the optimal policy changes with cost changes. For example, we show that as development cost decreases, all consumers and seller are better off, the seller's profit is a convex decreasing function of versioning cost, and the size of the market served weakly reduces with an increase in span of consumers cost. Surprisingly, we find that seller profit can reduce when reservation prices increase. When the consumer and product types are continuous, we develop a discretization approach that identifies the optimal policy to any desired accuracy. We show that, for the functional forms of reservation prices studied before, the problem simplifies in that it decomposes by consumers. For this case, we derive new analytical solutions. Our results show that when span of consumers costs are present, the optimal product allocation profile may be discontinuous, a solution structure that has not been identified in previous studies.
Keywords: Versioning, envy-free pricing, product line design, shortest-path problem
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