Existence of a largest social welfare relation on real-valued infinite populations under Strong Pareto, Permutation Invariance, and Quasi-Independence

Determine whether, when the set of worlds is W = R^X (real-valued utility assignments over a countably infinite population), there exists a largest social welfare relation—i.e., a reflexive, transitive preorder that weakly extends every other preorder in the class—that satisfies Strong Pareto, Permutation Invariance, and Quasi-Independence.

Background

The paper studies social welfare relations (SWRs) on infinite populations under two impartiality-efficiency axioms—Strong Pareto and Permutation Invariance—and introduces a further axiom, Quasi-Independence. It shows that completeness is incompatible with these axioms in some settings, motivating a focus on incomplete SWRs and on largest relations that maximize decisiveness subject to the axioms.

The main results establish that, for finite-valued worlds (W = W_F), the Sum Preorder (SP) is the largest relation satisfying Strong Pareto, Permutation Invariance, and Quasi-Independence. However, the authors also show that SP is not maximal on the broader domain W = RX, leaving unresolved whether any largest relation exists under the same axioms for real-valued worlds.

References

First, most obviously, we have not resolved the question of whether, if W=\mathbb RX there exists a largest preorder which satisfies our axioms. We have provided some partial results on this question, and shown that the Sum Preorder is not largest or even maximal on this set.

— Maximal Social Welfare Relations on Infinite Populations Satisfying Permutation Invariance  (2408.05851 - Goodman et al., 2024) in Conclusion