Working paper

Title: Nash Social Ordering, Its Dual and Generalizations over Normalized Well-being Profiles.

Author: Biung-Ghi Ju and Hocheol SHIN (Seoul National University)

Abstract:
This paper explores social choice theory by examining new methods of aggregating agents’ normalized well-being profiles and constructing social welfare functions. In particular, it introduces the Dual Nash ordering rule and formalizes it as the Dual Nash social welfare function. The paper then shows that this concept can be generalized into a broader class of mathematically well-defined social welfare functions. A key advantage of the Dual Nash approach over existing frameworks is its probabilistic interpretation. By shifting from a traditional utility-based viewpoint to a probability-based perspective, this work tackles the classic question of how to combine individual well-being into an overall measure of social welfare, thereby suggesting a novel way to handle uncertainty or interdependence across individuals.

The study axiomatizes the Dual Nash Ordering as a symmetric counterpart to the traditional Nash social welfare function. By incorporating copula and co-copula functions, the analysis extends classical Nash criteria to account for interdependencies among individuals’ well-being indices, thus going beyond the usual assumption of independence. Moreover, the paper shows that the concept of copulas in statistics is closely linked to the concept of Harsanyi dividends in cooperative game theory, even though these ideas have been studied in separate academic fields. 

Overall, this probabilistic approach to social welfare provides a more comprehensive framework for maximizing collective outcomes—especially in complex scenarios where individuals’ fates are highly interconnected—and opens new pathways for theoretical developments and practical policy design.