  Motivated by the difficulty of specifying complete ordinal preferences over a
large set of $m$ candidates, we study voting rules that are computable by
querying voters about $t < m$ candidates. Generalizing prior works that focused
on specific instances of this problem, our paper fully characterizes the set of
positional scoring rules that can be computed for any $1 \leq t < m$, which
notably does not include plurality. We then extend this to show a similar
impossibility result for single transferable vote (elimination voting). These
negative results are information-theoretic and agnostic to the number of
queries. Finally, for scoring rules that are computable with limited-sized
queries, we give parameterized upper and lower bounds on the number of such
queries a deterministic or randomized algorithm must make to determine the
score-maximizing candidate. While there is no gap between our bounds for
deterministic algorithms, identifying the exact query complexity for randomized
algorithms is a challenging open problem, of which we solve one special case.
