Abstract: Which variables does an agent have an incentive to control with its decision, and which variables does it have an incentive to respond to? We formalise these incentives, and demonstrate unique graphical criteria for detecting them in any single decision causal influence diagram. To this end, we introduce structural causal influence models, a hybrid of the influence diagram and structural causal model frameworks. Finally, we illustrate how these incentives predict agent incentives in both fairness and AI safety applications.