Abraham Zaks

In the process of evaluating the premium of an insurance plan, one considers the risk arising from various uncertainties. The authors suppose for a plan whose net premium is p and the standard deviation is σ the premium including the risk factor will be p + 3σ for a given member, and 3σ reflects the risk. For a group of n members with the same premium p and with standard deviation σ, the premium including the risk factor will be p + 3σ/√n where 3σ/√n reflects the risk for each member of the group. The authors study the emerging profit in case of n insured groups each with its own premium and its own risk when all the n insured groups merge into a single group uniting all insured members. They prove that there emerge a profit due to joining the n groups into a single one due to a reduced total risk of the n separate insured groups when merging into a single group. The emerging profit between the various groups may be divided using the Shapley values method or using utility functions for each group. The auhors discuss various reasonable ways to split the emerging profit between the n groups and show that the split of the profit depends on the chosen method. The main tools are techniques of game theory, in particular those of cooperative games.

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),...,u_n(x) . Given a positive amount of money C, a fair split of C is a vector (c₁,...,c_n) in Rⁿ, such that c₁ +...c_n = C and u₁(c₁) = u₂(c₂) = ... = u_n(c_n). The authors presume the utility functions to be normalized, that is u_i(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), ..., u_n(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.