Jain’s fairness index [1] is a fairness metric for static resource allocation between \(n\) participants – for instance, dividing a cake into \(n\) slices or allocating bandwidth among network flows. If each participant \(i\) receives \(x_i \in \mathbb{R}^+\), Jain’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. Jain, D.-M. Chiu, and W. R. Hawe. A Quantitative Measure of Fairness and Discrimination for Resource Allocation in Shared Computer Systems. DEC Research Report TR-301, 1984.