Strongly proportional division

A strongly proportional division^[1] (sometimes called super-proportional division) is a kind of a fair division. It is a division of resources among n partners, in which the value received by each partner is strictly more than his/her due share of 1/n of the total value. Formally, in a strongly proportional division of a resource C among n partners, each partner i, with value measure V_i, receives a share X_i such that

${\displaystyle V_{i}(X_{i})>V_{i}(C)/n}$.

Obviously, a strongly proportional division does not exist when all partners have the same value measure. The best condition that can always be guaranteed is ${\displaystyle V_{i}(X_{i})\geq V_{i}(C)/n}$, which is the condition for a plain proportional division. However, one may hope that, when different agents have different valuations, it may be possible to use this fact for the benefit of all players, and give each of them strictly more than their due share.