From multi-allocations to allocations, with subadditive valuations

We consider the problem of fair allocation of m indivisible items to n agents with monotone subadditive valuations. For integer d ≥ 2, a d-multi-allocation is an allocation in which each item is allocated to at most d different agents. We show that d-multi-allocations can be transformed into allocations, while not losing much more than a factor of d in the value that each agent receives. One consequence of this result is that for allocation instances with equal entitlements and subadditive valuations, if ρ-MMS d-multi-allocations exist, then so do fracρ4d-MMS allocations. Combined with recent results of Seddighin and Seddighin [EC 2025], this implies the existence of Omega(frac1loglog n)-MMS allocations.