Davide Martinetti

A study on the transitivity of probabilistic and fuzzy relations

Given a set of alternatives we consider a fuzzy relation and a probabilistic relation defined on such a set. We investigate the relation between the T-transitivity of the fuzzy relation and the cycle-transitivity of the associated probabilistic relation. We provide a general result, valid for any t-norm and we later provide explicit expressions for important particular cases. We also apply the results obtained to explore the transitivity satisfied by the probabilistic relation defined on a set of random variables. We focus on uniform continuous random variables.

On the role of acyclicity in the study of rationality of fuzzy choice functions

The role of the acyclicity property of fuzzy preference relations is studied in the framework of rationality of fuzzy choice functions. Two standard ways of constructing a fuzzy choice function from a given fuzzy preference relation are considered and properties such as acyclicity and completeness are shown to be sufficient to ensure the rationality of the fuzzy choice function. Special attention is paid to the triangular norm used for modelling the conjunction. The results obtained are compared to the classical results on rationality of crisp choice functions. Finally, the well-known Richter theorem is investigated in the fuzzy setting.

On Arrow---Sen style equivalences between rationality conditions for fuzzy choice functions

The Arrow–Sen Theorem establishes the equivalence between different rationality conditions for a choice function. In this contribution we deal with fuzzy versions of these rationality conditions and we study the connection between them. We recall results found in the literature and we prove that they are valid under weaker conditions. We also present new implications and counterexamples that show that our results cannot be obtained under weaker conditions.

From Preference Relations to Fuzzy Choice Functions

This is a first approach to the study of the connection between fuzzy preference relations and fuzzy choice functions. In particular we depart from a fuzzy preference relation and we study the conditions it must satisfy in order to get a fuzzy choice function from it. We are particulary interested in one function: G-rationalization. We discuss the relevance of the completeness condition on the departing preference relation. We prove that not every non-complete fuzzy preference relation leads to a choice function.

ON THE PRESERVATION OF SEMIORDERS FROM THE FUZZY TO THE CRISP SETTING

Different definitions of the concept of a fuzzy semiorder are compared. It is proved that their α-cuts are crisp binary relations that may fail to be Ferrers and semitransitive, in general. Consequently, we analyze the preservation of semiorders when coming back from the fuzzy to the crisp setting using α-cuts. In the final sections, a discussion is developed about the extension to the fuzzy setting of the concept of a threshold of utility discrimination, and its corresponding numerical representability of fuzzy semiorders by means of the representability of their α-cuts as crisp binary relations.

Statistical Preference as a Tool in Consensus Processes

In a consensus process, the intensities of preference can be expressed by means of probability distributions instead of single values. In that case, it is necessary to compare, in a simple way, pairs of probability distributions. Since classical methods do not assure the possibility of comparing any pair of distributions, a modern method is considered in this paper. It is called statistical preference. One of its most remarkable advantages is that it allows to compare any pair of probability distributions.

Min-transitivity of graded comparisons for random variables

Classically, the comparison of random variables have been done by means of a crisp order, which is known as stochastic dominance. In the last years, the classical stochastic dominance have been extended to a graded version by means of a probabilistic relation. In this work we propose different ways of measuring the gradual order among random variables by using fuzzy relations instead of probabilistic relations. The connection between the cycle-transitivity of the probabilistic relation and the T-transitivity of the associated fuzzy weak preference relation is characterized in the particular case of the minimum t-norm.

Interpretation of Statistical Preference in Terms of Location Parameters

Abstract Stochastic orders are methods that allow the comparison of random quantities. One of the most used stochastic orders is stochastic dominance. This method is based on the direct comparison of the cumulative distribution functions of the random variables, and it is characterized by comparing the expectations of the adequate transformation of the variables. Statistical preference is another alternative based on a probabilistic relation that provides preference degrees between the variables. This paper proves that statistical preference is connected to another location parameter different from the expectation: the median. Then, both stochastic orders have different interpretations, in the same way as mean and median are two different location parameters for describing random samples. Nevertheless, we prove that stochastic dominance and statistical preference are connected when the random variables are independent.

Bridging Probabilistic and Fuzzy Approaches to Choice Under Uncertainty

Imprecise choices can be described using either a probabilistic or a fuzzy formalism. No relation between them has been studied so far. In this contribution we present a connection between the two formalisms that strongly makes use of fuzzy implication operators and t-norms. In this framework, Luce’s Choice Axiom turns out to be a special case when the product t-norm is considered and other similar choice axioms can be stated, according to the t-norm in use. Also a new family of operators for transforming bipolar relations into unipolar ones is presented.

Uncertain Choices: A Comparison of Fuzzy and Probabilistic Approaches

Choices among alternatives in a set can be expressed in three different ways: by means of choice functions, by means of preference relations or using choice probabilities. The connection between the two first formalizations has been widely studied in the literature, both in the crisp or classical context and in the setting of fuzzy relations. However, the connection between probabilistic choice functions and fuzzy choice functions seems to have been forgotten and as far as we know, no literature can be found about it.