Optimal tradeoff between efficiency and Jain´s fairness index in resource allocation

In this paper, we study tradeoff policies between efficiency and the Jain´s fairness index of the benefits received by M users in general resource allocation scenarios. Analyzing the commonly-used α-fair tradeoff policy, it is shown that, except for the case of M =2 users, this policy does not necessarily achieve the optimal Efficiency-Jain tradeoff. In particular, it is shown that, when the number of users M >;2, the gap between the efficiency achieved by the α-fair and the optimal Efficiency-Jain tradeoff policy can be unbounded, for the same Jain´s index. Finding the optimal Efficiency-Jain tradeoff for arbitrary set of admissible benefits is generally difficult. To alleviate this difficulty, we derive sufficient conditions, which, when satisfied by the set of admissible benefits, lead to efficiently computable optimal tradeoff and benefit vectors. Numerical results for a typical communication network scenario are provided to confirm analytical findings.