  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.
