Modular Quasi-Pseudo Metrics and the Aggregation Problem

Maria del Mar Bibiloni-Femenias and Oscar Valero ()
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Maria del Mar Bibiloni-Femenias: Mathematics and Computer Science Department, Universitat de les Illes Balears, Carr. Valldemossa km. 7.5, 07122 Palma, Spain
Oscar Valero: Mathematics and Computer Science Department, Universitat de les Illes Balears, Carr. Valldemossa km. 7.5, 07122 Palma, Spain

Mathematics, 2024, vol. 12, issue 12, 1-20

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.

Keywords: modular quasi-pseudo metric; quasi-pseudo metric; aggregation; monotony; subadditivity; triangle triplet (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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