## Jain's fairness index

Jainâ€™s fairness index [1] is a fairness metric for static resource allocation between \(n\) participants â€“ for instance, dividing a cake into \(n\) slices or allocating bandwidth among network flows. If each participant \(i\) receives \(x_i \in \mathbb{R}^+\), Jainâ€™s index is a value between \(0\) and \(1\) defined by: $$f(x_1, \dots, x_n) = \frac{\left(\sum_{i=1}^n x_i\right)^2}{n \sum_{i=1}^n x_i^2}$$ Some examples: If all the agents receive the same share (completely fair): \(x_i = 1/n\) and \(f(\vec{x}) = 1\) If one agent receives all the resource (most unfair): \(f(\vec{x}) = 1/n\) If \(k\) users receive \(1/k\) and \(n-k\) users receive \(0\): \(f(\vec{x}) = k/n\) [1] R....