Unsorting the Proportional Fairness Relation

Abstract

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.