A Graph-Theoretical Perspective on Law Design for Multiagent Systems

TL;DR

This paper analyzes the complexity of designing laws in multiagent systems to prevent undesirable outcomes, showing NP-hardness and proposing approximation methods based on hypergraph vertex cover algorithms.

Contribution

It introduces a formal framework for law design in multiagent systems and proves NP-hardness of minimal law synthesis, offering approximation solutions.

Findings

NP-hardness of minimal law design for both useful and gap-free laws

Approximation algorithms from hypergraph vertex cover can be applied

Formal analysis of law effectiveness in multiagent systems

Abstract

A law in a multiagent system is a set of constraints imposed on agents' behaviours to avoid undesirable outcomes. The paper considers two types of laws: useful laws that, if followed, completely eliminate the undesirable outcomes and gap-free laws that guarantee that at least one agent can be held responsible each time an undesirable outcome occurs. In both cases, we study the problem of finding a law that achieves the desired result by imposing the minimum restrictions. We prove that, for both types of laws, the minimisation problem is NP-hard even in the simple case of one-shot concurrent interactions. We also show that the approximation algorithm for the vertex cover problem in hypergraphs could be used to efficiently approximate the minimum laws in both cases.