Optimal Envy-Free Pricing with Metric Substitutability

Abstract

We study the unit-demand envy-free pricing problem faced by a profit-maximizing seller with unlimited supply when there is metric substitutability among the items—consumer $i$'s value for item $j$ is $v_i-c_{i,j}$, and the substitution costs, $\{c_{i,j}\}$, form a metric. Our model is motivated by the observation that sellers often sell the same product at different prices in different locations, and rational consumers optimize the tradeoff between prices and substitution costs. While the general envy-free pricing problem is hard to approximate, we show that the problem of maximizing revenue with metric substitutability among items can be solved exactly in polynomial time. We do this by first showing that in any optimal price vector, the set of nodes that pay exactly their value uniquely determines which nodes buy an item and what price they pay, and therefore the revenue. We transform the problem of finding an optimal set of such nodes to an instance of weighted independent set on a perfect graph which can be solved in polynomial time by the strong perfect graph theorem, proving the result. We then analyze the computational tractability of various extensions to our model. We begin with relaxing the metric substitutability requirement and show that when the substitution costs do not form a metric, even if a $(1+\epsilon)$-approximate triangle inequality holds, the problem becomes NP-hard. Thus the triangle inequality characterizes the threshold at which the problem goes from “tractable” to “hard.” We then relax assumptions on the supply and demand. We consider restricting supplies to a subset of locations, or the amount of supplies, or allowing buyers to demand more than one unit. In all cases, the problem becomes NP-hard. In addition, the multiunit demand case illustrates an interesting paradoxical nonmonotonicity: The optimal revenue the seller can extract can actually decrease when consumers' demands increase. We show the revenue maximization problem with multiunit demand is APX-hard even for the simplest valuations with equal marginal values for all items up to the demand constraint, and demands of at most 3.