  Distributionally robust chance constrained programs minimize a deterministic
cost function subject to the satisfaction of one or more safety conditions with
high probability, given that the probability distribution of the uncertain
problem parameters affecting the safety condition(s) is only known to belong to
some ambiguity set. We study three popular approximation schemes for
distributionally robust chance constrained programs over Wasserstein balls,
where the ambiguity set contains all probability distributions within a certain
Wasserstein distance to a reference distribution. The first approximation
replaces the chance constraint with a bound on the conditional value-at-risk,
the second approximation decouples different safety conditions via Bonferroni's
inequality, and the third approximation restricts the expected violation of the
safety condition(s) so that the chance constraint is satisfied. We show that
the conditional value-at-risk approximation can be characterized as a tight
convex approximation, which complements earlier findings on classical
(non-robust) chance constraints, and we offer a novel interpretation in terms
of transportation savings. We also show that the three approximations can
perform arbitrarily poorly in data-driven settings, and that they are generally
incomparable with each other.
