In general, the set of users utilities is boundedbecause of the limitation of resources. There may existmany Pareto optimal points in the set of users utilities.For selecting a Pareto optimum point, a family of fairnesscriteria, that contains the max-min fairness and aparameterized family of fairness (by Mo andWalrand), hasbeen proposed and examined in some concrete networkingcontexts that result in specific convex utility sets. We newlyexamine general compact (closed and bounded) utility setswhich include the specific utility sets as special cases. Wefirst prove that each of the family of fairness criteria givesa unique fair (Pareto optimum) point if the utility set isconvex. We find, however, counter-examples where each ofthe family of fairness criteria gives multiple fair points ifthe utility set is not convex. We propose an extention of thefamily of fairness criteria such that each of them gives onlya unique fair point regardless of whether the utility set isconvex or not, to which we give proofs. By using a specificload balancing model, we illustrate the counter-examplesand how each criterion of our extended fair family givesa unique fair point
