Nevertheless, such a division exists with high probability for indivisible items under certain assumptions on the valuations of the agents.
A more lenient fairness criterion is partial proportionality, in which each partner receives a certain fraction f(n) of the total value, where f(n) ≤ 1/n.
When all valuations are additive set functions and the entire cake is divided, the following implications hold: When the valuations are only subadditive, EF still implies PR, but PR no longer implies EF even with two partners: it is possible that Alice's share is worth 1/2 in her eyes, but Bob's share is worth even more.
The implications are summarized in the following table: One advantage of the proportionality criterion over envy-freeness and similar criteria is that it is stable with regards to voluntary exchanges.
As an example, assume that a certain land is divided among 3 partners: Alice, Bob and George, in a division that is both proportional and envy-free.
From Bob's point of view, the division is still proportional, since he still holds a subjective value of at least 1/3 of the total, regardless of what Alice and George do with their plots.
Additionally, there should be at least a possibility (if not a guarantee) that the partner receives more than 1/n; this explains the importance of the existence theorems of super-proportional division.