Welfare Maximization with Production Costs: A Primal Dual Approach

TL;DR

This paper analyzes online combinatorial auctions with production costs, demonstrating that incentive-compatible posted pricing mechanisms can nearly achieve optimal competitive ratios, and establishing lower bounds for all algorithms.

Contribution

It characterizes the optimal competitive ratio for online auctions with convex production costs and shows that incentive-compatible posted pricing mechanisms are nearly optimal.

Findings

Posted pricing mechanisms achieve near-optimal competitive ratios.

Lower bounds show no algorithms can do better than a certain threshold.

Results improve upon and match previous research in the field.

Abstract

We study online combinatorial auctions with production costs proposed by Blum et al. using the online primal dual framework. In this model, buyers arrive online, and the seller can produce multiple copies of each item subject to a non-decreasing marginal cost per copy. The goal is to allocate items to maximize social welfare less total production cost. For arbitrary (strictly convex and differentiable) production cost functions, we characterize the optimal competitive ratio achievable by online mechanisms/algorithms. We show that online posted pricing mechanisms, which are incentive compatible, can achieve competitive ratios arbitrarily close to the optimal, and construct lower bound instances on which no online algorithms, not necessarily incentive compatible, can do better. Our positive results improve or match the results in several previous work, e.g., Bartal et al., Blum et al., and Buchbinder and Gonen. Our lower bounds apply to randomized algorithms and resolve an open problem by Buchbinder and Gonen.