A general model for dealing with ranking voting systems

Abstract

A key problem in decision-making is selecting a winning candidate or establishing a global ranking for a set of candidates when individuals' preferences are expressed through linear orders. Scoring rules are a specific case of positional voting systems (PVSs) that are widely used in sports competitions. Likewise, some scoring rules, such as the Borda rule and plurality, have also been extensively analyzed in the field of social choice. However, the choice of the scoring vector may significantly influence the results, leading to the development of models that avoid subjective vector selection. In this paper, we introduce a general model that encompasses some previous proposals present in the literature. Our model does not have an important deficiency that some other models do, such as the fact that the relative order between two candidates may change even if there is no variation in the positions obtained by those candidates. We give an explicit formula for calculating candidate scores, enabling direct determination of winners or rankings without solving the model for each candidate, and we also analyze the fulfillment of some well-known properties. Likewise, through theoretical analysis and examples, we identify and rule out specific PVSs that may yield questionable outcomes.