Fair split of profit generated by n parties

Abstract

The authors studied the process of merging insured groups, and the splitting of the profit that arises in the process due to the fact that the risk for the merged group is essentially reduced. There emerges a profit and there are various ways of splitting this profit between the combined groups. Techniques from game theory, in particular cooperative game theory turn out to be useful in splitting of the profit. The authors proceed in this paper to apply techniques of utility theory to study the possibility of a fair split of that profit. In this research, the authors consider a group of n parties 1,...,n such that each of them has a corresponding utility function u1(x),...,un(x) . Given a positive amount of money C, a fair split of C is a vector (c1,...,cn) in Rn, such that c1 +...cn = C and u1(c1) = u2(c2) = ... = un(cn). The authors presume the utility functions to be normalized, that is ui(c) = 1 for each party i, i = 1, ... ,n. The authors show that a fair split exists and is unique for any given set of utility functions u1(x), ..., un(x), and for any given amount of money C. The existence theorem follows from observing simplexes. The uniqueness follows from the utility functions being strictly increasing. An example is given of normalizing some utility functions, and evaluating the fair split in special cases. In this article, the authors study the case of merging two groups (or more) of insured members, they provide an evaluation of the emerging benefit in the process, and the splitting of the benefit between the groups.