On the Hardness of Dominant Strategy Mechanism Design

TL;DR

This paper investigates the communication complexity of dominant strategy mechanisms in combinatorial auctions, revealing fundamental limits on efficiency and approximation ratios, and contrasting deterministic and randomized approaches.

Contribution

It establishes polynomial lower bounds for optimal welfare in dominant strategies and introduces a deterministic FPTAS for multi-unit auctions, highlighting a gap with randomized mechanisms.

Findings

Optimal welfare in dominant strategies requires polynomial communication.

A deterministic FPTAS exists for multi-unit auctions with decreasing marginal values.

No polynomial communication mechanism can surpass a certain approximation ratio for general valuations.

Abstract

We study the communication complexity of dominant strategy implementations of combinatorial auctions. We start with two domains that are generally considered "easy": multi-unit auctions with decreasing marginal values and combinatorial auctions with gross substitutes valuations. For both domains we have fast algorithms that find the welfare-maximizing allocation with communication complexity that is poly-logarithmic in the input size. This immediately implies that welfare maximization can be achieved in ex-post equilibrium with no significant communication cost, by using VCG payments. In contrast, we show that in both domains the communication complexity of any dominant strategy implementation that achieves the optimal welfare is polynomial in the input size. We then move on to studying the approximation ratios achievable by dominant strategy mechanisms. For multi-unit auctions with decreasing marginal values, we provide a dominant-strategy communication FPTAS. For combinatorial auctions with general valuations, we show that there is no dominant strategy mechanism that achieves an approximation ratio better than $m^{1-\epsilon}$ that uses $poly(m,n)$ bits of communication, where $m$ is the number of items and $n$ is the number of bidders. In contrast, a \emph{randomized} dominant strategy mechanism that achieves an $O(\sqrt m)$ approximation with $poly(m,n)$ communication is known. This proves the first gap between computationally efficient deterministic dominant strategy mechanisms and randomized ones. En route, we answer an open question on the communication cost of implementing dominant strategy mechanisms for more than two players, and also solve some open problems in the area of simultaneous combinatorial auctions.