Super-proportional division
In the context of fair cake-cutting, a super-proportional division is a division in which each partner receives strictly more than 1/n of the resource by their own subjective valuation ().
A super-proportional division is better than a proportional division, in which each partner is guaranteed to receive at least 1/n (). However, in contrast to proportional division, a super-proportional division does not always exist. When all partners have exactly the same value functions, the best we can do is give each partner exactly 1/n.
A necessary condition for the existence of a super-proportional division is, therefore, that not all partners have the same value measure.
A surprising fact is that, when the valuations are additive and non-atomic, this condition is also sufficient. I.e., when there are at least two partners whose value function is even slightly different, then there is a super-proportional division in which all partners receive more than 1/n.
The existence of a super-proportional division was first conjectured as early as 1948:
The first published proof to the existence of super-proportional division was as a corollary to the Dubins–Spanier convexity theorem. This was a purely existential proof based on convexity arguments.
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