The regulation of systems composed of many interacting decentralised agents is a long-standing problem. This problem has typically been resolved by the imposition of rules. Rules resolve the limitations of sub-optimal agents whilst simultaneously aligning the behaviour of the agents with the demands of society. Traffic laws, for example, let a driver cross a junction without seeing around the corner, and transport protocols such as TCP let applications share a network without knowing of one another. The problem is more pressing today, as autonomous systems are deployed in society. In this work, we establish the basis of a system-agnostic theory of rule design for multi-agent systems. We first state the rule design problem (RDP) in its generality. It relates a rule to the set of system performances, defined by the regulator, that the agents deployed under it can attain. However, it is not yet possible to solve the RDP in full generality. As such, we introduce the homogeneous rule design problem (HRDP), in which the agents in each environment are identical, and among other interesting properties, show that under worst-case aggregation the two problems have the same solution, bypassing the combinatorial complexity of the RDP. To solve the RDP under worst-case aggregation we propose Rule Branch and Bound (RuleBnB), an anytime branch-and-bound algorithm. Finally, we illustrate the value of solving the RDP on an autonomous driving case study.

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