Ranking voting systems and surrogate weights: Explicit formulas for centroid weights

One of the most important issues in the field of ranking voting systems is the choice of the weighting vector. This issue has been addressed in the literature from different approaches, and one of them has been to obtain the weighting vector as a solution to a linear programming problem. In this paper we analyze some models proposed in the literature and show that one of their main shortcomings is that they cannot guarantee the uniqueness of the solution, so the winner or the final ranking of the candidates may depend on the chosen weighting vector. An alternative to these models is the use of surrogate weights, among which rank order centroid (ROC) weights stand out as the centroid of a specific simplex. Following this idea, in this paper we show the explicit expression for the weights that form the centroid of diverse simplices utilized in ranking voting systems, and we also see that certain surrogate weights frequently employed in literature can be derived as extreme cases where the simplices collapse into a single vector. Moreover, we argue that averaging two weighting vectors can be a valid approach in some cases and, in this way, we can get weighting vectors that closely resemble those used in some sports competitions.