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It is common that a jury must grade a set of candidates in a cardinal scalesuch as {1,2,3,4,5} or an ordinal scale such as {Great, Good, Average, Bad }.When the number of candidates is very large such as hotels (BOOKING),restaurants (GOOGLE), apartments (AIRBNB), drivers (UBER), or papers (EC), itis unreasonable to assume that each jury member will provide a separate gradefor each candidate. Each jury member is more likely to abstain for somecandidates, cast a blank vote, or be associated at random, or as a function ofits expertise, with only a small subset of the candidates and is asked to gradeeach of those. Extending the classical theory, we study aggregation methods inwhich a voter will not be eligible to grade all the candidates, and thecandidates are not eligible for the same sets of voters. Moreover, eachcandidate on which they are eligible, the voter will have the choice between: ablank vote, grade the candidate, or abstain. Assuming single-peaked preferencesover the grades, we axiomatically characterise a broad class of strategy-proofgrading mechanisms satisfying axioms such as unanimity, anonymity, neutrality,participation or consistency. Finally, when a strict ranking is necessary (todistinguish let say between two borderline papers in a conference), sometie-breaking rules, extending the leximin and majority judgment, are definedand are shown to be equivalent to some strategy-proof grading functions on aricher space of outcome. Our paper will propose new rules, called phantom-proxymechanisms, to aggregate the votes in the examples above or others, whichdiffer from the usual average mark, that are easily manipulable. Moreover, thephantom-proxy are able to reduce the injustices caused by some candidatesjuries too generous or severe.

Grading and Ranking Large number of candidates