  Election rules are formal processes that aggregate voters preferences,
typically to select a single candidate, called the winner. Most of the election
rules studied in the literature require the voters to rank the candidates from
the most to the least preferred one. This method of eliciting preferences is
impractical when the number of candidates to be ranked is large. We ask how
well certain election rules (focusing on positional scoring rules and the
Minimax rule) can be approximated from partial preferences collected through
one of the following procedures: (i) randomized-we ask each voter to rank a
random subset of $\ell$ candidates, and (ii) deterministic-we ask each voter to
provide a ranking of her $\ell$ most preferred candidates (the $\ell$-truncated
ballot). We establish theoretical bounds on the approximation ratios and we
complement our theoretical analysis with computer simulations. We find that
mostly (apart from the cases when the preferences have no or very little
structure) it is better to use the randomized approach. While we obtain fairly
good approximation guarantees for the Borda rule already for $\ell = 2$, for
approximating the Minimax rule one needs to ask each voter to compare a larger
set of candidates in order to obtain good guarantees.
