  The fundamental problem of causal inference -- that we never observe
counterfactuals -- prevents us from identifying how many might be negatively
affected by a proposed intervention. If, in an A/B test, half of users click
(or buy, or watch, or renew, etc.), whether exposed to the standard experience
A or a new one B, hypothetically it could be because the change affects no one,
because the change positively affects half the user population to go from
no-click to click while negatively affecting the other half, or something in
between. While unknowable, this impact is clearly of material importance to the
decision to implement a change or not, whether due to fairness, long-term,
systemic, or operational considerations. We therefore derive the
tightest-possible (i.e., sharp) bounds on the fraction negatively affected (and
other related estimands) given data with only factual observations, whether
experimental or observational. Naturally, the more we can stratify individuals
by observable covariates, the tighter the sharp bounds. Since these bounds
involve unknown functions that must be learned from data, we develop a robust
inference algorithm that is efficient almost regardless of how and how fast
these functions are learned, remains consistent when some are mislearned, and
still gives valid conservative bounds when most are mislearned. Our methodology
altogether therefore strongly supports credible conclusions: it avoids
spuriously point-identifying this unknowable impact, focusing on the best
bounds instead, and it permits exceedingly robust inference on these. We
demonstrate our method in simulation studies and in a case study of career
counseling for the unemployed.
