Fair economic division : how to modify Shapley value to take into account the fact that different people have different productivity

When several participants, working together, gained some amount of money, what is the fair way to distribute this amount between them? This is the problem that the future Nobelist Lloyd Shapley was working on when he proposed what is now called the Shapley value - a division uniquely determined by natural fairness assumptions. However, this solutions is not universal: it assumes that all participants are equal - in particular, that they have equal productivity. In practice, people have different productivity levels, and these productivity levels can differ a lot: for example, some software engineers are several times more productive than others. It is desirable to take this difference in productivity into account.Shapley value is based on an axiomatic approach: it is uniquely determined by the appropriate fairness assumptions. To generalize Shapley value to the case of different productivity, we modified these assumptions appropriately, and analyzed what can be derived from these modified assumptions.We prove that there is a unique division scheme that satisfies all the resulting assumptions. This scheme is thus a generalization of Shapley value to this more general and more realistic situation, when different participants have different productivities.Both the formulation of the problem and the result are new. The resulting division scheme can be used to more adequately distribute the common gains - by explicitly taking into account the fact that different participants have, in general, different productivities.