  In causal inference, the joint law of a set of counterfactual random
variables is generally not identified. We show that a conservative version of
the joint law - corresponding to the smallest treatment effect - is identified.
Finding this law uses recent results from optimal transport theory. Under this
conservative law we can bound causal effects and we may construct inferences
for each individual's counterfactual dose-response curve. Intuitively, this is
the flattest counterfactual curve for each subject that is consistent with the
distribution of the observables. If the outcome is univariate then, under mild
conditions, this curve is simply the quantile function of the counterfactual
distribution that passes through the observed point. This curve corresponds to
a nonparametric rank preserving structural model.
