Aggregated threshold functions or how to measure the performance of an electoral system

This article introduces a set of functions that measures the mechanical performance of an electoral system. Aggregated threshold functions offer the necessary and sufficient share of the vote nationwide to win a given number of seats. Traditionally, electoral systems have been measured taking into account the share of the vote required to win one seat given a district. In the approach used here, the values obtained are calculated taking into account all districts in which a country is divided and for any number of seats. This article offers the definition and formalization of these functions. Once the aggregated threshold functions are defined in all their terms, I show some data validation to test their capacity of prediction. The main goal of the article is to provide with a tool that can be used, for example, to develop a measure that summarizes in a single value the functioning of an electoral system. This value can be used by electoral reformers as well as by students of electoral systems to test the consequences of electoral systems as a whole.