leximin

Leximin is a principle used in fair division and welfare economics to select allocations of resources by comparing individuals’ utilities in a lexicographic, or stepwise, order. For any feasible allocation, each agent receives a utility value. The leximin procedure sorts these utilities in nondecreasing order and treats the resulting vector as a whole; an allocation is leximin-optimal if its sorted utility vector is lexicographically greater than that of every other feasible allocation. In other words, it first maximizes the welfare of the worst-off individual, and among allocations that achieve the same minimum, it maximizes the welfare of the second-worst-off, and so on.

Leximin is often described as a refinement of the Rawlsian maximin principle. By continuing the lexicographic comparison beyond the minimum, leximin seeks not only to improve the worst-off person but also to improve the overall distribution in a hierarchical manner, emphasizing equality in the lower end of the distribution while still allowing differences among better-off individuals.

Properties and considerations: leximin allocations promote a strong form of fairness by prioritizing the least advantaged, but they rely on cardinal utilities to rank allocations and may be sensitive to the scale and transformation of those utilities. In some models, leximin-optimal allocations align with Pareto efficiency, but this is not guaranteed in every setting. The approach can be computationally demanding and may yield allocations that sacrifice total welfare in favor of fairness.

Applications include fair division of indivisible goods, rent division, cost-sharing, and other resource allocation problems where reducing inequality at the bottom end is a priority.