Feasible Shared Destiny Risk Distributions

Social risk equity is concerned with the comparative evaluation of social risk distributions, which are probability distributions over the potential sets of fatalities. In the approach to the evaluation of social risk equity introduced by Gajdos, Weymark, and Zoli (Shared destinies and the measurement of social risk equity, Annals of Operations Research 176:409-424, 2010), the only information about such a distribution that is used in the evaluation is that contained in a shared destiny risk matrix whose entry in the kth row and ith column is the probability that person i dies in a group containing k individuals. Such a matrix is admissible if it satisfies a set of restrictions implied by its definition. It is feasible if it can be generated by a social risk distribution. It is shown that admissibility is equivalent to feasibility. Admissibility is much easier to directly verify than feasibility, so this result provides a simply way to identify which matrices to consider when the objective is to socially rank the feasible shared destiny risk matrices.