  The intersection of causal inference and machine learning for decision-making
is rapidly expanding, but the default decision criterion remains an
\textit{average} of individual causal outcomes across a population. In
practice, various operational restrictions ensure that a decision-maker's
utility is not realized as an \textit{average} but rather as an \textit{output}
of a downstream decision-making problem (such as matching, assignment, network
flow, minimizing predictive risk). In this work, we develop a new framework for
off-policy evaluation with \textit{policy-dependent} linear optimization
responses: causal outcomes introduce stochasticity in objective function
coefficients. Under this framework, a decision-maker's utility depends on the
policy-dependent optimization, which introduces a fundamental challenge of
\textit{optimization} bias even for the case of policy evaluation. We construct
unbiased estimators for the policy-dependent estimand by a perturbation method,
and discuss asymptotic variance properties for a set of adjusted plug-in
estimators. Lastly, attaining unbiased policy evaluation allows for policy
optimization: we provide a general algorithm for optimizing causal
interventions. We corroborate our theoretical results with numerical
simulations.
