What is the best α for which α-MMS (maximin share) allocations always exist?
The maximin share (MMS) is the value an agent can guarantee by dividing items into n bundles and receiving the worst bundle. An α-MMS allocation guarantees each agent at least α times their MMS value.
Known bounds: - α ≥ 10/13 ≈ 0.769 HKSS25 - α ≤ 39/40 = 0.975 FST21 - For 3 agents: α ≥ 11/12 ≈ 0.917 FN22
The gap between the upper and lower bounds remains significant, and closing it is a major open problem in fair division.