  Conventionally, the construction of a pair-matched sample selects treated and
control units and pairs them in a single step with a view to balancing observed
covariates $\mathbf{x}$ and reducing the heterogeneity or dispersion of
treated-minus-control response differences, $Y$. In contrast, the method of
cardinality matching developed here first selects the maximum number of units
subject to covariate balance constraints and, with a balanced sample for
$\mathbf{x}$ in hand, then separately pairs the units to minimize heterogeneity
in $Y$. Reduced heterogeneity of pair differences in responses $Y$ is known to
reduce sensitivity to unmeasured biases, so one might hope that cardinality
matching would succeed at both tasks, balancing $\mathbf{x}$, stabilizing $Y$.
We use cardinality matching in an observational study of the effectiveness of
for-profit and not-for-profit private high schools in Chile - a controversial
subject in Chile - focusing on students who were in government run primary
schools in 2004 but then switched to private high schools. By pairing to
minimize heterogeneity in a cardinality match that has balanced covariates, a
meaningful reduction in sensitivity to unmeasured biases is obtained.
