Complexity of Auctions with Interdependence

TL;DR

This paper investigates the complexity of designing truthful auction mechanisms in interdependent valuation models, removing domain restrictions, and providing both algorithms and hardness results for value and cost optimization.

Contribution

It introduces a comprehensive analysis of the computational complexity of truthful mechanisms in general interdependent settings, including new algorithms and hardness proofs.

Findings

Identifies tractable cases reducing to classical combinatorial problems.

Provides efficient algorithms for certain restricted cases.

Proves NP-hardness and query complexity lower bounds for the general case.

Abstract

We study auction design in the celebrated interdependence model introduced by Milgrom and Weber [1982], where a mechanism designer allocates a good, maximizing the value of the agent who receives it, while inducing truthfulness using payments. In the lesser-studied procurement auctions, one allocates a chore, minimizing the cost incurred by the agent selected to perform it. Most of the past literature in theoretical computer science considers designing truthful mechanisms with constant approximation for the value setting, with restricted domains and monotone valuation functions. In this work, we study the general computational problems of optimizing the approximation ratio of truthful mechanism, for both value and cost, in the deterministic and randomized settings. Unlike most previous works, we remove the domain restriction and the monotonicity assumption imposed on value functions. We provide theoretical explanations for why some previously considered special cases are tractable, reducing them to classical combinatorial problems, and providing efficient algorithms and characterizations. We complement our positive results with hardness results for the general case, providing query complexity lower bounds, and proving the NP-Hardness of the general case.