Notions of anonymity for object assignment: impossibility theorems

We search for impartiality in the allocation of objects when monetary transfers are not possible. Our main focus is anonymity. The standard definition requires that if agents’ names are permuted, their assignments be permuted in the same way. Since no rule satisfies this definition, we introduce weaker variants, “anonymity for distinct preferences,” “pairwise-anonymity for distinct preferences,” “pairwise-anonymity for fully differentiated profiles,” and “independence of others’ permutations.” We show that for more than two agents and two objects, no rule is pairwise-anonymous for distinct preferences and Pareto-efficient (Theorem 1), no rule is pairwise-anonymous for distinct preferences and independent of others’ permutations (Theorem 2), and no rule is pairwise-anonymous for fully differentiated profiles and strategy-proof (Theorem 3).