Modular Quasi-Pseudo Metrics and the Aggregation Problem

Abstract

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed.