Grading and Ranking Large number of candidates

It is common that a jury must grade a set of candidates in a cardinal scale such 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), it is unreasonable to assume that each jury member will provide a separate grade for each candidate. Each jury member is more likely to abstain for some candidates, cast a blank vote, or be associated at random, or as a function of its expertise, with only a small subset of the candidates and is asked to grade each of those. Extending the classical theory, we study aggregation methods in which a voter will not be eligible to grade all the candidates, and the candidates are not eligible for the same sets of voters. Moreover, each candidate on which they are eligible, the voter will have the choice between: a blank vote, grade the candidate, or abstain. Assuming single-peaked preferences over the grades, we axiomatically characterise a broad class of strategy-proof grading mechanisms satisfying axioms such as unanimity, anonymity, neutrality, participation or consistency. Finally, when a strict ranking is necessary (to distinguish let say between two borderline papers in a conference), some tie-breaking rules, extending the leximin and majority judgment, are defined and are shown to be equivalent to some strategy-proof grading functions on a richer space of outcome. Our paper will propose new rules, called phantom-proxy mechanisms, to aggregate the votes in the examples above or others, which differ from the usual average mark, that are easily manipulable. Moreover, the phantom-proxy are able to reduce the injustices caused by some candidates juries too generous or severe.