Journal of the ACM (JACM)
On the complexity of price equilibria
Journal of Computer and System Sciences - STOC 2002
On profit-maximizing envy-free pricing
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Combination can be hard: approximability of the unique coverage problem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Single-minded unlimited supply pricing on sparse instances
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Approximation algorithms and online mechanisms for item pricing
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
A Nonparametric Approach to Multiproduct Pricing
Operations Research
Algorithmic pricing via virtual valuations
Proceedings of the 8th ACM conference on Electronic commerce
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Buying cheap is expensive: hardness of non-parametric multi-product pricing
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Item pricing for revenue maximization
Proceedings of the 9th ACM conference on Electronic commerce
Market equilibrium via a primal--dual algorithm for a convex program
Journal of the ACM (JACM)
Uniform Budgets and the Envy-Free Pricing Problem
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Envy, Multi Envy, and Revenue Maximization
WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Envy-free pricing in multi-item markets
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
The power of uncertainty: bundle-pricing for unit-demand customers
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
On envy-free pareto efficient pricing
FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
Envy-free pricing in multi-item markets
ACM Transactions on Algorithms (TALG)
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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.