A group of peers must choose one of them to receive a prize; everyone cares only about winning, not about who gets the prize if someone else. An award rule is impartial if one's message never influences whether or not one wins the prize. We explore the consequences of impartiality when each agent nominates a single (other) agent for the prize.
On the positive side, we construct impartial nomination rules where both the influence of individual messages and the requirements to win the prize are not very different across agents. Partition the agents in two or more districts, each of size at least 3, and call an agent a local winner if he is nominated by a majority of members of his own district; the rule selects a local winner with the largest support from nonlocal winners, or a fixed default agent in case there is no local winner.
On the negative side, impartiality implies that ballots cannot be processed anonymously as in plurality voting. Moreover, we cannot simultaneously guarantee that the winner always gets at least one nomination, and that an agent nominated by everyone else always wins.