Welfare and Profit Maximization with Production Costs

TL;DR

This paper studies combinatorial auction pricing with increasing marginal costs, proposing algorithms that achieve constant or logarithmic approximation ratios for maximizing social welfare or profit under various cost functions.

Contribution

It introduces the first algorithmic framework for combinatorial pricing with increasing marginal costs, providing approximation guarantees for different cost function classes.

Findings

Constant factor approximations for linear, polynomial, and logarithmic costs.

Logarithmic approximations for general increasing marginal costs.

Bounds are essentially tight for the considered settings.

Abstract

Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been well-studied in the case of limited supply (one copy of each item), and in the case of digital goods (the seller can produce additional copies at no cost). Yet in the case of resources---oil, labor, computing cycles, etc.---neither of these abstractions is just right: additional supplies of these resources can be found, but at increasing difficulty (marginal cost) as resources are depleted. In this work, we initiate the study of the algorithmic mechanism design problem of combinatorial pricing under increasing marginal cost. The goal is to sell these goods to buyers with unknown and arbitrary combinatorial valuation functions to maximize either the social welfare, or the seller's profit; specifically we focus on the setting of \emph{posted item prices} with buyers arriving online. We give algorithms that achieve {\em constant factor} approximations for a class of natural cost functions---linear, low-degree polynomial, logarithmic---and that give logarithmic approximations for more general increasing marginal cost functions (along with a necessary additive loss). We show that these bounds are essentially best possible for these settings.