  I provide a novel approach to characterizing the set of interim realizable
allocations, in the spirit of Matthews (1984) and Border (1991). The approach
allows me to identify precisely why exact characterizations are difficult to
obtain in some settings. The main results of the paper then show how to adapt
the approach in order to obtain approximate characterizations of the interim
realizable set in such settings.
  As an application, I study multi-item allocation problems when agents have
capacity constraints. I identify necessary conditions for interim
realizability, and show that these conditions are sufficient for realizability
when the interim allocation in question is scaled by 1/2. I then characterize a
subset of the realizable polytope which contains all such scaled allocations.
This polytope is generated by a majorization relationship between the scaled
interim allocations and allocations induced by a certain ``greedy algorithm''.
I use these results to study mechanism design with equity concerns and model
ambiguity. I also relate optimal mechanisms to the commonly used deferred
acceptance and serial dictatorship matching algorithms. For example, I provide
conditions on the principal's objective such that by carefully choosing school
priorities and running deferred acceptance, the principal can guarantee at
least half of the optimal (full information) payoff.
