Conclusions
In general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?
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Brams, S.J., Taylor, A.D. & Zwicker, W.S. Old and new moving-knife schemes. The Mathematical Intelligencer 17, 30–35 (1995). https://doi.org/10.1007/BF03024785
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DOI: https://doi.org/10.1007/BF03024785