  Elections and opinion polls often have many candidates, with the aim to
either rank the candidates or identify a small set of winners according to
voters' preferences. In practice, voters do not provide a full ranking;
instead, each voter provides their favorite K candidates, potentially in ranked
order. The election organizer must choose K and an aggregation rule.
  We provide a theoretical framework to make these choices. Each K-Approval or
K-partial ranking mechanism (with a corresponding positional scoring rule)
induces a learning rate for the speed at which the election correctly recovers
the asymptotic outcome. Given the voter choice distribution, the election
planner can thus identify the rate optimal mechanism. Earlier work in this area
provides coarse order-of-magnitude guaranties which are not sufficient to make
such choices. Our framework further resolves questions of when randomizing
between multiple mechanisms may improve learning, for arbitrary voter noise
models.
  Finally, we use data from 5 large participatory budgeting elections that we
organized across several US cities, along with other ranking data, to
demonstrate the utility of our methods. In particular, we find that
historically such elections have set K too low and that picking the right
mechanism can be the difference between identifying the ultimate winner with
only a 80% probability or a 99.9% probability after 400 voters.
