Among a group of selfish agents, we consider nomination correspondences that determine who should get a prize on the basis of each agent’s nomination. Holzman and Moulin (Econometrica 81:173–196, <CitationRef CitationID="CR4">2013</CitationRef>) show that (i) there is no nomination function that satisfies the axioms of impartiality, positive unanimity, and negative unanimity, and (ii) any impartial nomination function that satisfies the axiom of anonymous ballots is constant (and thus violates positive unanimity). In this article, we show that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(\mathrm {i})^\prime $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">i</mi> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> </math> </EquationSource> </InlineEquation> there exists a nomination correspondence, named plurality with runners-up, that satisfies impartiality, positive unanimity, and negative unanimity, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(\mathrm {ii})^\prime $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">ii</mi> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> </math> </EquationSource> </InlineEquation> any impartial nomination correspondence that satisfies anonymous ballots is not necessarily constant, but violates positive unanimity. Copyright Springer-Verlag Berlin Heidelberg 2014
