Fair split of profit generated by n parties ”

We consider n parties with n corresponding utility functions, denoted by u1,…,un. Given a positive amount of money C, a fair split of C is a vector (c1,…,cn)∈Rn such that c1 ⋯ cn=C and u1(c1) = u2(c2) = ⋯ = un(cn). In this paper we show the existence and uniqueness of a fair split to any given amount of money C.

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.

INTRODUCTION
In the process of merging insured groups, there emerges a profit that arises in the process due to the fact that the risk for the merged groups is reduced. There arise various ways to split the emerging profit between the merging groups. Techniques from game theory, in particular cooperative games, will be applied to consider various ways to split the emerging profit. The authors apply techniques of utility theory to investigate various possible ways to split this profit.
The proof of theorem 2.2 uses the famous theorem: ( )

MAIN RESULTS
In the rest the authors presume n utility functions and a positive amount of money . After proving the existence of a fair split the authors will show its uniqueness.
The authors suggest a more general result then the sole existence one. for all pairs , ij so that 1 i jn ≤< ≤ and so that the following equality holds 1 . n c cC ++=  Notice that we omitted the assumption that the functions are strictly increasing, hence the result that a fair split exist will follow from the following proof of theorem 2.  Next the authors turn to the uniqueness that follows from the utility functions being strictly increasing, which is a monotonicity property. This property turns out to be crucial in deriving the proof of the uniquness of a fair split.
Proof of the uniquness of a fair split: Using the fact that a utility function is strictly increasing, we will show that this implies that a fair split is unique.

EXAMPLE
Assume that three persons join in a common interest to split between them in a fair way the sum of ,

CONCLUSION
The ideas and the method as described in this article provide a model for an insurance company to bid for an insurance portfolio of a group of inssured members, in which case one expects the energence of benefit that results from adding the group's and the company's portfolio, and one may derive options to improve the winning prospect of the company's offer for the bid.
This article also suggests how to study the case of merging of insurance portfolios, as in the case of one insurance company bidding to buy another insurance company, or two insurance ompanies that consider merging to one, or two insurance ompanies that consider to manage jointly theit portfolios. The article discusses options of evaluating the emerging benefit in the process of joining the two insurance portfolios to one. The results may provide a range of offers to bid that may improve the winning prospects, or of how to split the emerging benefit between two merging companies.
In a similar way this article provides ideas for a model for two groups of inssured members, each with an inssurance portfolio, that consider merging their portfolios. The mergence is reasonable if there re-sults a benefit in the process. Here the authors consider evaluations of emerging benefit and how to split it between the two groups.
Notice that the ideas and the results of this article may turn usefull to split its overall expenses (e.g. administrative load, water and electricity expenses, various taxes etc.) in a company fairly between its several departments by charging overhead for each of its various departments to fairly determine the part of each of the various departments of the company in the overall expenses of the company. One starts by considerring the overall expenses for each department separately, then one derives the emerging saving of the overall expenses for all departments when operating united under the company within a single administration. Finally, one uses the results of this article and the ideas as presented in it to split the emerging saving to the various departments.