A generalization to networks of Young's characterization of the Borda rule
Daniela Bubboloni, Michele Gori
TL;DR
The paper develops a general framework to extend Young’s characterization of the Borda rule from voters’ linear preferences to networks of pairwise evaluations. By introducing networks with a capacity function and defining net-outdegree, net-indegree, and total network solutions, the authors prove that on any regular domain, these three network solutions are the unique neutral, consistent, and cancellation-respecting rules; this result also yields analogous characterizations for social choice correspondences. The approach hinges on linear algebra and network theory, establishing a robust decomposition of networks into balanced, cyclic, and pseudo-symmetric components and showing regularity conditions that guarantee uniqueness. Consequently, the work unifies and extends a broad class of classic social choice results (including Borda, Partial Borda, Averaged Borda, Approval Voting, Plurality, and anti-Plurality) and provides a versatile toolkit for deriving new characterization theorems. The findings have broad implications for both theoretical and applied voting systems, offering a universal lens to study scoring-type rules via networks.
Abstract
We prove that, for any given set of networks satisfying suitable conditions, the net-oudegree network solution, the net-indegree network solution, and the total network solution are the unique network solutions on that set satisfying neutrality, consistency and cancellation. The generality of the result obtained allows to get an analogous result for social choice correspondences: for any given set of preference profiles satisfying suitable conditions, the net-oudegree social choice correspondence, the net-indegree social choice correspondence and the total social choice correspondence are the unique social choice correspondences on that set satisfying neutrality, consistency and cancellation. Using the notable fact that several well-known voting rules coincide with the restriction of net-oudegree social choice correspondence to appropriate sets of preference profiles, we are able to deduce a variety of new and known characterization theorems for the Borda rule, the Partial Borda rule, the Averaged Borda rule, the Approval Voting, the Plurality rule and the anti-Plurality rule, among which Young's characterization of the Borda rule and Fishburn's characterization of the Approval Voting.