An Integrated Approach to Preferential Voting Models with Variable Weights for Rank Positions

In a ranked voting system, voters select a subset of candidates and rank them from most to least preferred. Data envelopment analysis (DEA)-based voting models, among others, are used to determine the rank-position weights most favorable for each candidate, with the goal of achieving the highest aggregate score. However, concerns have been raised about the weights assigned to each rank position, as well as the potential for rank reversal of some candidates resulting from changes in votes earned by other candidates. To address these issues, some authors have developed two improved models. These models aim to incorporate the constraints of candidates that are not being evaluated into a single restriction, preventing inefficient candidates from influencing the order of efficient candidates. Moreover, these models treat the parameters used to make the distance between successive ranks as variable weights, and calculate average efficiency scores of candidates while considering the entire range of parameters. In this study, we revisit the two improved models and explore an alternative approach based on results from linear algebra and convex analysis, which is more intuitive and easier to understand. Furthermore, we provide closed-form optimal solutions for DEA-based voting models that share the common goal of maximizing the distance between successive ranks while considering both efficiency-related and weight constraints. The analysis of these four models offers a better understanding of their similarities and differences.