Envy-free dynamic pricing schemes

TL;DR

This paper investigates the existence and computation of envy-free dynamic pricing schemes in unit-demand markets, proposing algorithms for social welfare maximization under various fairness notions and analyzing the complexity for revenue maximization.

Contribution

It introduces four notions of envy-freeness over different time periods and provides polynomial algorithms for three, while analyzing the hardness of revenue maximization with fixed agent order.

Findings

Polynomial algorithms for envy-free dynamic prices in three fairness notions.

Walrasian equilibria correspond to the strongest envy-freeness notion.

Revenue maximization problems are APX-hard with fixed agent order.

Abstract

A combinatorial market consists of a set of indivisible items and a set of agents, where each agent has a valuation function that specifies for each subset of items its value for the given agent. From an optimization point of view, the goal is usually to determine a pair of pricing and allocation of the items that provides an efficient distribution of the resources, i.e., maximizes the social welfare, or is as profitable as possible for the seller, i.e., maximizes the revenue. To overcome the weaknesses of mechanisms operating with static prices, a recent line of research has concentrated on dynamic pricing schemes. In this model, agents arrive in an unspecified sequential order, and the prices can be updated between two agent-arrivals. Though the dynamic setting is capable of maximizing social welfare in various scenarios, the assumption that the agents arrive one after the other eliminates the standard concept of fairness. In this paper, we study the existence of optimal dynamic prices under fairness constraints in unit-demand markets. We propose four possible notions of envy-freeness of different strength depending on the time period over which agents compare themselves to others: the entire time horizon, only the past, only the future, or only the present. For social welfare maximization, while the first definition leads to Walrasian equilibria, we give polynomial-time algorithms that always find envy-free optimal dynamic prices in the remaining three cases. In contrast, for revenue maximization, we show that the corresponding problems are APX-hard if the ordering of the agents is fixed. On the positive side, we give polynomial-time algorithms for the setting when the seller can choose the order in which agents arrive.