Unsorting the Proportional Fairness Relation

Typical problems related to the application of proportional fairness are sparsity of the relation with increasing dimension, and the operator confusion problem. Here, we propose a new fairness relation derived from proportional fairness to handle these problems. The design principle behind this relation is relational unsorting: if there is a relation x(R)y between elements x and y from n-dimensional Euclidian space, the unsorted relation x(uR)y holds whenever there is a permutation x* of the elements of x for which x*(R)y holds. We apply this concept to proportional fairness, study the properties of the new relation, contrast with another relation based on over-sorting proportional fairness, and provide simulations to demonstrate the ease of ordered proportional fairness for meta-heuristic search.