Narrow Results

A fuzzy preference relation is a popular model to represent both individual and group preferences. However, what is often sought is a subset of alternatives that is an ultimate solution of a decision problem. In order to arrive at such a final solution individal and/or group choice rules may be employed. There is a wealth of such rules devised in the context of the classical, crisp preference relations. Originally, most of the popular group decision making rules were conceived for classical (crisp) preference relations (orderings), and then extended to the case of traditional fuzzy preference relations. Moreover, they often differ in their assumptions about the properties o the preference relations to be processed. In the paper we pursue the path towards a universal representation of such rules that provides an effective generalization of the classical rules for the fuzzy case. Moreover, it leads to a meaningful extension to the linguistic preferences, in the spirit of the computing with words paradigm.

In group decision making (GDM) processes, prior to the selection of the best alternative(s), it would be desirable that experts achieve a high degree of consensus or agreement between them. Due to the complexity of most decision making problems, individuals' preferences may not satisfy formal properties. ‘Consistency’ is one of such properties, and it is associated with the transitivity property. Obviously, when carrying out a rational decision making, consistent information, i.e. information which does not imply any kind of contradiction, is more appropriate than information containing some contradictions. Therefore, in a GDM process, consistency should also be sought after. In this paper we present a consensus model for GDM problems that proceeds from consistency to consensus. This model integrates a novel consistency reaching module based on consistency measures. In particular, the model generates advice on how experts should change their preferences in order to increase their consistency. Also, the consensus model is considered adaptive because the search for consensus is adapted to the level of agreement achieved at each consensus round.

In a previous paper, the authors proposed an alternative approach to classical dimension theory, based upon a general representation of strict preferences not being restricted to partial order sets. Without any relevant restriction, the proposed approach was conceived as a potential powerful tool for decision making problems where basic information has been modeled by means of valued binary preference relations. In fact, assuming that each decision maker is able to consistently manage intensity values for preferences is a strong assumption even when there are few alternatives being involved (if the number of alternatives is large, the same criticism applies to crisp preferences). Any representation tool, as the one proposed by the authors, will in principle play a key role in order to help decision makers to understand their preference structure. In this paper we introduce an alternative approach in order to avoid certain complexity issues of the initial proposal, allowing a close representation easier to be obtained in practice.

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.

In a binary choice voting scenario, voters may have fuzzy preferences but are required to make crisp choices. In order to compare a crisp voting procedure with more general mechanisms of fuzzy preference aggregation, we first focus on the latter. We present a formulation of strategy-proofness in this setting and study its consequences. On one hand, we achieve an axiomatic recommendation of the median as the aggregation rule for fuzzy preferences. On the other hand, we present conditions under which strategic concerns imply the optimality of a crisp voting procedure and argue that there is a potential gain in the integration of the preference and choice aggregation programs — namely that an underlying fuzzy preference structure may also help inform the selection of a choice aggregation rule.

Under certain aggregation rules, particular subsets of the voting population fully characterize the social preference relation, and the preferences of the remaining voters become irrelevant. In the traditional literature, these types of rules, i.e. voting and simple rules, have received considerable attention because they produce non-empty social maximal sets under single-peaked preference profiles but are particularly poorly behaved in multi-dimensional space. However, the effects of fuzzy preference relations on these types of rules is largely unexplored. This paper extends the analysis of voting and simple rules in the fuzzy framework. In doing so, we contribute to this literature by relaxing previous assumptions about strict preference and by illustrating that Black's Median Voter Theorem does not hold under all conceptualizations of the fuzzy maximal set.