Conflict Situations Are Inevitable When There Are Many Participants: A Proof Based on the Analysis of Aumann-Shapley Value
When collaboration of several people results in a business success, an important issue is how to fairly divide the gain between the participants. In principle, the solution to this problem is known since the 1950s: natural fairness requirements lead to the so-called Shapley value. However, the computation of Shapley value requires that we can estimate, for each subset of the set of all participants, how much gain they would have gained if they worked together without others. It is possible to perform such estimates when we have a small group of participants, but for a big company with thousands of employers this is not realistic. To deal with such situations, Nobelists Aumann and Shapley came up with a natural continuous approximation to Shapley value -- just like a continuous model of a solid body helps, since we cannot take into account all individual atoms. Specifically, they defined the Aumann-Shapley value as a limit of the Shapley value of discrete approximations: in some cases this limit exists, in some it does not. In this paper, we show that, in some reasonable sense, for almost all continuous situations the limit does not exist: we get different values depending on how we refine the discrete approximations. Our conclusion is that in such situations, since computing of fair division is not feasible, conflicts are inevitable.
Technical Report: UTEP-CS-23-11