In the context of rapid discoveries by leaders in AI, governments must consider how to design regulation that matches the increasing pace of new AI capabilities. Regulatory Markets for AI is a proposal designed with adaptability in mind. It involves governments setting outcomebased targets for AI companies to achieve, which they can show by purchasing services from a market of private regulators. We use an evolutionary game theory model to explore the role governments can play in building a Regulatory Market for AI systems that deters reckless behaviour. We warn that it is alarmingly easy to stumble on incentives which would prevent Regulatory Markets from achieving this goal. These "Bounty Incentives" only reward private regulators for catching unsafe behaviour. We argue that AI companies will likely learn to tailor their behaviour to how much effort regulators invest, discouraging regulators from innovating. Instead, we recommend that governments always reward regulators, except when they find that those regulators failed to detect unsafe behaviour that they should have. These "Vigilant Incentives" could encourage private regulators to find innovative ways to evaluate cutting-edge AI systems.

Society could soon see transformative artificial intelligence (TAI). Models of competition for TAI show firms face strong competitive pressure to deploy TAI systems before they are safe. This paper explores a proposed solution to this problem, a Windfall Clause, where developers commit to donating a significant portion of any eventual extremely large profits to good causes. However, a key challenge for a Windfall Clause is that firms must have reason to join one. Firms must also believe these commitments are credible. We extend a model of TAI competition with a Windfall Clause to show how firms and policymakers can design a Windfall Clause which overcomes these challenges. Encouragingly, firms benefit from joining a Windfall Clause under a wide range of scenarios. We also find that firms join the Windfall Clause more often when the competition is more dangerous. Even when firms learn each other's capabilities, firms rarely wish to withdraw their support for the Windfall Clause. These three findings strengthen the case for using a Windfall Clause to promote the safe development of TAI.