  In many contexts involving ranked preferences, agents submit partial orders
over available alternatives. Statistical models often treat these as marginal
in the space of total orders, but this approach overlooks information contained
in the list length itself. In this work, we introduce and taxonomize approaches
for jointly modeling distributions over top-$k$ partial orders and list lengths
$k$, considering two classes of approaches: composite models that view a
partial order as a truncation of a total order, and augmented ranking models
that model the construction of the list as a sequence of choice decisions,
including the decision to stop. For composite models, we consider three
dependency structures for joint modeling of order and truncation length. For
augmented ranking models, we consider different assumptions on how the
stop-token choice is modeled. Using data consisting of partial rankings from
San Francisco school choice and San Francisco ranked choice elections, we
evaluate how well the models predict observed data and generate realistic
synthetic datasets. We find that composite models, explicitly modeling length
as a categorical variable, produce synthetic datasets with accurate length
distributions, and an augmented model with position-dependent item utilities
jointly models length and preferences in the training data best, as measured by
negative log loss. Methods from this work have significant implications on the
simulation and evaluation of real-world social systems that solicit ranked
preferences.
