Combinatorial Auctions with Conflict-Based Externalities

TL;DR

This paper introduces models and algorithms for combinatorial auctions with conflict-based externalities, providing approximation guarantees that depend on the conflict graph's maximum out-degree, and applies these to adwords auctions.

Contribution

It develops new algorithms and mechanisms for combinatorial auctions with externalities, incorporating conflict graphs and achieving near-optimal approximation ratios.

Findings

Achieves an $ ext{O}( ext{alpha} imes ext{Delta})$-approximation for conflict-free mechanisms.

Designs a cone program-based rounding algorithm with an $ ext{O}(( ext{Delta} imes ext{log log Delta})/ ext{log Delta})$ ratio.

Provides truthful mechanisms with approximation ratios $o( ext{Delta})$ for adwords when items are few.

Abstract

Combinatorial auctions (CA) are a well-studied area in algorithmic mechanism design. However, contrary to the standard model, empirical studies suggest that a bidder's valuation often does not depend solely on the goods assigned to him. For instance, in adwords auctions an advertiser might not want his ads to be displayed next to his competitors' ads. In this paper, we propose and analyze several natural graph-theoretic models that incorporate such negative externalities, in which bidders form a directed conflict graph with maximum out-degree $\Delta$. We design algorithms and truthful mechanisms for social welfare maximization that attain approximation ratios depending on $\Delta$. For CA, our results are twofold: (1) A lottery that eliminates conflicts by discarding bidders/items independent of the bids. It allows to apply any truthful $\alpha$-approximation mechanism for conflict-free valuations and yields an $\mathcal{O}(\alpha\Delta)$-approximation mechanism. (2) For fractionally sub-additive valuations, we design a rounding algorithm via a novel combination of a semi-definite program and a linear program, resulting in a cone program; the approximation ratio is $\mathcal{O}((\Delta \log \log \Delta)/\log \Delta)$. The ratios are almost optimal given existing hardness results. For the prominent application of adwords auctions, we present several algorithms for the most relevant scenario when the number of items is small. In particular, we design a truthful mechanism with approximation ratio $o(\Delta)$ when the number of items is only logarithmic in the number of bidders.