Penalty-Based Aggregation of Strings

International Summer School on Aggregation Operators (2019. Olomouc, Czech Republic

The acclamation consensus state and an associated ranking rule

The study of conditions, under which the existence of an “absolute” best winner can be assured, is a hot topic in the field of social choice. Unanimity is an evident example of a condition under which the winner is obvious. However, many more properties weaker than unanimity have been analysed in literature: the presence of a Condorcet winner, strong stochastic transitivity, the presence of a candidate that Borda dominates all other candidates, etc. Unfortunately, one could easily find a prominent ranking rule, for which the outcome does not agree with these relaxed conditions. In this study, we aim to identify a condition weaker than unanimity, but under which the social outcome is still obvious. This condition, defined as the conjunction of three properties already studied by the present authors and hereinafter referred to as acclamation, will be proven to be a meeting point for the most prominent ranking rules in social choice theory, and will be used for introducing an intuitively appealing ranking rule

Fuzzy compatibility relations and pseudo-monometrics: Some correspondences

De Baets and Mesiar in a seminal work showed that Fuzzy Compatibility Relations (FCR) - reflexive and symmetric fuzzy relations - that are also T-transitive w.r.t. an Archimedean t-norm are in a one-to-one correspondence with pseudo-metrics. However, FCRs that are not T-transitive have not merited as much scrutiny. Recently, ternary relations on a set known as betweenness relations and monometrics on the obtained betweenness set, or a B-set, have garnered a lot of attention, especially for their role in decision making and penalty based data aggregation. In this work we show that there is a one-to-one correspondence between FCRs and distance functions, not necessarily metrics, through these betweenness relations. Through interesting correspondences between FCRs and pseudo-monometrics on a given B-set, we also give some pointers to one of the major challenges in this field - that of obtaining pseudo-monometrics on a given B-set

The role of betweenness relations, monometrics and penalty functions in data aggregation

Penalty functions have been used for data aggregation since Yager introduced his "general theory of information aggregation" back in 1993. Obviously, the ideas of Yager have long time been surpassed. However, over the years, the use of penalty functions has shifted towards the aggregation of values in a closed interval. Here, we propose to return to the origin of penalty-based data aggregation and to expand the current definition of penalty functions beyond the confinement to closed intervals. We do this by bringing to the field of data aggregation an old acquaintance of the early scholars of geometry: betweenness relations