  Impartial selection is the selection of an individual from a group based on
nominations by other members of the group, in such a way that individuals
cannot influence their own chance of selection. We give a deterministic
mechanism with an additive performance guarantee of $O(n^{(1+\kappa)/2})$ in a
setting with $n$ individuals where each individual casts $O(n^{\kappa})$
nominations, where $\kappa\in[0,1]$. For $\kappa=0$, i.e. when each individual
casts at most a constant number of nominations, this bound is $O(\sqrt{n})$.
This matches the best-known guarantee for randomized mechanisms and a single
nomination. For $\kappa=1$ the bound is $O(n)$. This is trivial, as even a
mechanism that never selects provides an additive guarantee of $n-1$. We show,
however, that it is also best possible: for every deterministic impartial
mechanism there exists a situation in which some individual is nominated by
every other individual and the mechanism either does not select or selects an
individual not nominated by anyone.
