Evaluating social welfare when people disagree about inequality

Many approaches to social welfare assume that every pair of alternatives can be ranked. One must be better, worse or equally good. Economists call this requirement completeness. But disagreement about the importance of inequality can make a complete ranking difficult to justify.

In the arXiv preprint “Separable social welfare evaluation without completeness,” Kaname Miyagishima, Eisuke Uchida and Kenta Yoshida ask which principles support welfare comparisons that deliberately leave some alternatives unranked. The supplied record identifies the work as a preprint and provides no journal publication reference. Its results are mathematical, not findings from an experiment or population survey.

The model represents an allocation as a utility vector: a list assigning each person a numerical level of well-being. It considers finite societies with at least two people. These utility numbers are inputs to the theory, not measurements collected by the authors.

The proposed multi-prioritarian criterion uses a family of evaluation functions rather than a single welfare formula. Each function transforms everyone’s utility, and the transformed values are added. One allocation ranks at least as highly as another only if every function in the family agrees.

Each function is continuous, strictly increasing and concave. Strict increase means that improving someone’s utility raises their evaluated contribution. Concavity means that an additional unit of utility has a weakly smaller evaluated benefit at a higher starting level. This allows priority to the worse-off, although the permitted functions also include linear evaluation, which simply values utility gains equally.

If different functions support opposing rankings, neither allocation need rank above the other. This incomparability is different from indifference: it records unresolved disagreement rather than a judgment that the alternatives are equally good. Agreement here means unanimity among mathematical evaluators, not a vote taken among citizens.

The authors establish a characterization theorem connecting this representation to four welfare principles. Strong Pareto requires an allocation to be strictly better if nobody loses utility and someone gains. Impartiality makes identities irrelevant: exchanging people’s utility levels does not change the evaluation. The Pigou–Dalton principle accepts transfers that reduce inequality while preserving total utility and not reversing the recipients’ relative positions.

Separability, the fourth principle, says that people whose utilities remain unchanged between two alternatives should not affect the comparison through their common utility levels. This is what allows an assessment to focus on the people whose situations differ.

Crucially, those four principles are not enough. The theorem also assumes Continuous Divisible Completion. Roughly, this condition considers accepted differences between utility distributions, combines and rationally rescales them, and takes suitable limits. Whenever the resulting difference corresponds to two allocations in the original society, that comparison must already be accepted. Rescaling is a mathematical operation, not an assumption that people can be divided.

The proof constructs a closed convex cone—a set of distributional differences stable under addition and nonnegative scaling—and uses a mathematical separation theorem to obtain the evaluation functions. The completion condition is stronger than ordinary continuity of allocation comparisons. The paper supplies a counterexample demonstrating that the four welfare principles alone do not guarantee the proposed representation.

The framework also connects to generalized Lorenz comparisons. These rank people from lowest to highest utility and compare cumulative totals for the worst-off person, the worst-off two people, and so on. Requiring every such total to be at least as high gives the same ranking as agreement across all increasing concave evaluators.

The paper illustrates incomparability with two-person allocations of (0, 10) and (2, 2). The first has more total utility, but the second gives the worst-off person more. Generalized Lorenz comparisons leave them incomparable, whereas adding utilities alone favors the first.

Further results examine when different representing families encode the same rankings and how to compare their inequality aversion. Important uniqueness results require the same evaluation family across societies, together with finiteness or finite-generation assumptions. Under additional assumptions about permissible changes to utility scales, the authors obtain more restricted forms, sometimes reducing the criterion to total utility.

These qualifications define the contribution’s limits. The work does not establish which inequality attitudes society should adopt, measure actual disagreement, or identify a preferred redistribution policy. It shows how a specified set of welfare principles can support partial judgments—and precisely which extra assumptions are needed to derive that structure.