Functional Equations and Inequalities in ‘Rational Group Decision Making’

This paper consists of a reformulation and generalization of results in [2–4]. A problem of ‘rational group decision making’ is the following: A fixed amount s is to be allocated to a fixed number m of competing projects. Each member of a group of n decision makers makes recommendations, the jth allocating, say, xij to the ith project, in order to establish the ‘consensus’ allocation fi(x i) [we write x i = (xil,..., xin)]. We suppose only that fi(0) = 0 [‘consensus of rejection’; 0 = (0,...,0)] and that the allocations are non-negative. In the case m > 2, we prove that each fi is the same weighted arithmetic mean. There are other solutions for m = 2, even if f1 = f2 is supposed. We determine all of them. The solutions are also established for variable s.