A race for technological supremacy in AI could lead to serious negative consequences, especially whenever ethical and safety procedures are underestimated or even ignored, leading potentially to the rejection of AI in general. For all to enjoy the benefits provided by safe, ethical and trustworthy AI systems, it is crucial to incentivise participants with appropriate strategies that ensure mutually beneficial normative behaviour and safety-compliance from all parties involved. Little attention has been given to understanding the dynamics and emergent behaviours arising from this AI bidding war, and moreover, how to influence it to achieve certain desirable outcomes (e.g. AI for public good and participant compliance). To bridge this gap, this paper proposes a research agenda to develop theoretical models that capture key factors of the AI race, revealing which strategic behaviours may emerge and hypothetical scenarios therein. Strategies from incentive and agreement modelling are directly applicable to systematically analyse how different types of incentives (namely, positive vs. negative, peer vs. institutional, and their combinations) influence safety-compliant behaviours over time, and how such behaviours should be configured to ensure desired global outcomes, studying at the same time how these mechanisms influence AI development. This agenda will provide actionable policies, showing how they need to be employed and deployed in order to achieve compliance and thereby avoid disasters as well as loosing confidence and trust in AI in general.