Abstract:We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6.
Agents partition deterministic outcomes into good or bad. A direct revelation mechanism selects a lottery over outcomes -also interpreted as time-shares. Under such dichotomous preferences, the probability that the lottery outcome be a good one is a canonical utility representation.The utilitarian mechanism averages over all deterministic outcomes "approved" by the largest number of agents. It is efficient, strategyproof and treats equally agents and outcomes.We reach the impossibility frontier if we also place the lower bound 1 n on each agent's utility, where n is the number of agents; or if this lower bound is the fraction of good outcomes to feasible outcomes.We conjecture that no ex-ante efficient and strategyproof mechanism guarantees a strictly positive utility to all agents at all profiles, and prove a weaker version of this conjecture.
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