  We consider the problem of probabilistic allocation of objects under ordinal
preferences. We devise an allocation mechanism, called the vigilant eating rule
(VER), that applies to nearly arbitrary feasibility constraints. It is
constrained ordinally efficient, can be computed efficiently for a large class
of constraints, and treats agents equally if they have the same preferences and
are subject to the same constraints. When the set of feasible allocations is
convex, we also present a characterization of our rule based on ordinal
egalitarianism. Our results about VER do not just apply to allocation problems
but to all collective choice problems in which agents have ordinal preferences
over discrete outcomes. As a case study, we assume objects have priorities for
agents and apply VER to sets of probabilistic allocations that are constrained
by stability. VER coincides with the (extended) probabilistic serial rule when
priorities are flat and the agent proposing deterministic deferred acceptance
algorithm when preferences and priorities are strict. While VER always returns
a stable and constrained efficient allocation, it fails to be strategyproof,
unconstrained efficient, and envy-free. We show, however, that each of these
three properties is incompatible with stability and constrained efficiency.
