A Graph-Theoretical Perspective on Law Design for Multiagent Systems
Qi Shi, Pavel Naumov
TL;DR
The paper tackles the problem of designing minimally restrictive laws to prevent undesirable outcomes in one-shot multiagent systems. It models laws as prohibitions within one-shot concurrent games and shows that both useful-law design and gap-free law design are NP-hard, even in simple settings. The authors build a unified, graph-theoretic framework by reducing law-design problems to vertex cover in hypergraphs and related graphs, enabling approximation via known VC algorithms and providing two-way reductions between VC and both useful-law and gap-free-law problems. A key by-product is addressing responsibility gaps through the gap-free design framework. The results offer a practical, scalable approach to normative design in multiagent settings by leveraging established VC techniques, with future directions including weighted actions and broader normative logics.
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.