Faced with data-driven policies, individuals will manipulate their features to obtain favorable decisions. While earlier works cast these manipulations as undesirable gaming, recent works have adopted a more nuanced causal framing in which manipulations can improve outcomes of interest, and setting coherent mechanisms requires accounting for both predictive accuracy and improvement of the outcome. Typically, these works focus on known causal graphs, consisting only of an outcome and its parents. In this paper, we introduce a general framework in which an outcome and n observed features are related by an arbitrary unknown graph and manipulations are restricted by a fixed budget and cost structure. We develop algorithms that leverage strategic responses to discover the causal graph in a finite number of steps. Given this graph structure, we can then derive mechanisms that trade off between accuracy and improvement. Altogether, our work deepens links between causal discovery and incentive design and provides a more nuanced view of learning under causal strategic prediction.
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