  Impartial selection has recently received much attention within the
multi-agent systems community. The task is, given a directed graph representing
nominations to the members of a community by other members, to select the
member with the highest number of nominations. This seemingly trivial goal
becomes challenging when there is an additional impartiality constraint,
requiring that no single member can influence her chance of being selected.
Recent progress has identified impartial selection rules with optimal
approximation ratios. Moreover, it was noted that worst-case instances are
graphs with few vertices. Motivated by this fact, we propose the study of
additive approximation, the difference between the highest number of
nominations and the number of nominations of the selected member, as an
alternative measure of the quality of impartial selection. Our positive results
include two randomized impartial selection mechanisms which have additive
approximation guarantees of $\Theta(\sqrt{n})$ and $\Theta(n^{2/3}\ln^{1/3}n)$
for the two most studied models in the literature, where $n$ denotes the
community size. We complement our positive results by providing negative
results for various cases. First, we provide a characterization for the
interesting class of strong sample mechanisms, which allows us to obtain lower
bounds of $n-2$, and of $\Omega(\sqrt{n})$ for their deterministic and
randomized variants respectively. Finally, we present a general lower bound of
$2$ for all deterministic impartial mechanisms.
