A new social welfare function with a number of desirable properties
By relaxing the dominating set in three ways (e.g., from "each member beats every non-member" to "each member beats or ties every non-member, with an additional requirement that at least one member beat every non-member"), we propose a new social welfare function, which satisfies a number of desirable properties including Condorcet winner principle, Condorcet loser principle, strong Gehrlein-stability (hence Smith set principle), anonymity, neutrality, weak Pareto, strong Pareto, non-dictatorship, and [independence of irrelevant alternatives (IIA) when the pairwise majority relation is an ordering on the alternative set]. If the pairwise majority relation is complete and transitive, the proposed method yields a collective preference relation that coincides with the input majority relation. It thus shares the same collective preference function on the dichotomous domain with the approval voting and the majority voting. It runs in polynomial time and thus possesses a competitive advantage over a number of computationally intractable voting rules such as the Dodgson's rule, the Kemeny's rule, the Slater's rule, the Banks rule, and the Schwartz's tournament equilibrium set (TEQ) rule. When it is used in tournaments, its winner belongs to the uncovered set, the top cycle set, the Smith set, and the Schwartz set. In addition, in a tournament where the number of alternatives is not more than 4, its winner set is a subset, sometimes proper, of the Copeland winner set. Whether this attractive argument is still valid in four-more-alternative tournaments remains an open question.