A measure of fairness
Nathan, Hippu Salk Kristle (2018). 'A Measure of Fairness' Paper presented at the annual conference of the HDCA, Buenos Aires, Argentina 2018.
Fairness in allocation of resources is a challenge at all levels; from a household distributing food among its members to a federal government distributing finances to its provinces. This study proposes a measure to assess fairness of any allocation by constructing a geometric allocation space and using Minkowski distance function and min-max normalization technique. Every recipients (henceforth referred to as individuals) are posited to have two limits of allocation; sustenance and saturation, representing minimum and maximum allocation allowed in a normative sense.
An n-dimensional space is conceptualised constituting the loci of all the possible allocations among individuals. Every vertex of this space indicates the resource is allocated to one individual; every edge (excluding the vertices) indicates the allocation is shared between two individuals; every triangular face (excluding the edges) indicates the allocation shared among three individuals; and so on. In this space, which would contain the most fair and most unfair allocations, one can assume that there can be no allocation which can be farther from most fair allocation than the most unfair one. In this sense, fairness of any allocation is computed by taking the additive inverse of the ratio of the distance of the actual allocation from most fair to the maximum distance.
The study outlines the desired properties of the proposed fairness index. A well defined utility function is assumed to obtain solutions for different conditions of resource allocation, such as, Majoritarian (greatest utility to greatest number of people), Utilitarian (maximum utility), Bergson-Samuelson (equal marginal utility) and Rawlsian (maximin principle). An empirical illustration has been provided to demonstrate fairness index under different conditions. The study concludes with applications of the proposed measure and the policy relevance.