Complexity of Mechanism Design

TL;DR

This paper explores the computational complexity of designing mechanisms for preference aggregation in multiagent systems, showing NP-completeness for deterministic mechanisms and tractability for randomized ones, with implications for practical mechanism design.

Contribution

It demonstrates that the mechanism design problem is NP-complete for deterministic mechanisms but becomes tractable when randomized mechanisms are allowed, offering a new approach to preference aggregation.

Findings

Deterministic mechanism design is NP-complete.

Randomized mechanisms make the design problem tractable.

Randomization can improve social outcomes without loss.

Abstract

The aggregation of conflicting preferences is a central problem in multiagent systems. The key difficulty is that the agents may report their preferences insincerely. Mechanism design is the art of designing the rules of the game so that the agents are motivated to report their preferences truthfully and a (socially) desirable outcome is chosen. We propose an approach where a mechanism is automatically created for the preference aggregation setting at hand. This has several advantages, but the downside is that the mechanism design optimization problem needs to be solved anew each time. Focusing on settings where side payments are not possible, we show that the mechanism design problem is NP-complete for deterministic mechanisms. This holds both for dominant-strategy implementation and for Bayes-Nash implementation. We then show that if we allow randomized mechanisms, the mechanism design problem becomes tractable. In other words, the coordinator can tackle the computational complexity introduced by its uncertainty about the agents' preferences by making the agents face additional uncertainty. This comes at no loss, and in some cases at a gain, in the (social) objective.