Three-agent Peer Evaluation

I show that every rule for dividing a dollar among three agents impartially (so that each agent's share depends only on her evaluation by her associates) underpays some agent by at least one-third of a dollar for some consistent profile of evaluations. I then produce an impartial division rule that never underpays or overpays any agent by more than one-third of a dollar, and for most consistent evaluation profiles does much better.