Aggregation Theorems for Allocation Problems

Suppose that n individuals assign values to a sequence of m numerical decision variables subject to the constraints that the m values assigned by each individual must be nonnegative and sum to some fixed positive $\sigma $. Suppose that we wish to aggregate their individual assignments to produce consensual values of these variables satisfying the aforementioned constraints. Aczél and Wagner have shown that if $m\geqq 3$, then a method of aggregation is based on weighted arithmetic averaging iff (a) the consensual value assigned to each variable depends only on the values assigned by individuals to that variable and (b) the consensual value is zero if all individuals assign that variable the value zero. In the present paper we extend this result in various ways, dropping the unanimity condition (b) and allowing individual and consensual values to be restricted to some subinterval of $[0,\sigma ]$.