A review of soft consensus models in a fuzzy environment

Abstract

In the consensus reaching processes developed in group decision making problems we need to measure the closeness among experts’ opinions in order to obtain a consensus degree. As it is known, to achieve a full and unanimous consensus is often not reachable in practice. An alternative approach is to use softer consensus measures, which reflect better all possible partial agreements, guiding the consensus process until high agreement is achieved among individuals. Consensus models based on soft consensus measures have been widely used because these measures represent better the human perception of the essence of consensus. This paper presents an overview of consensus models based on soft consensus measures, showing the pioneering and prominent papers, the main existing approaches and the new trends and challenges.

Introduction

The term “consensus” has been used for years, even centuries, in a variety of context and areas. It generally concerns various situations in which there is a set of experts who present their testimonies which basically concern their opinions on some alternatives, topics, courses of action, etc., in question. The experts can be both individuals as such, smaller groups, larger groups, even organizations if they can be considered uniform with respect to the issues considered and testimonies [1], [2].

Consensus can be meant in various ways and group decision making (GDM) contexts. First, consensus is related with the state of agreement in a group in the sense that the individuals exhibit a state of common feeling as to the values in question. Strictly speaking, consensus has been meant from this perspective as a full and unanimous agreement [3], [4], though it has been deemed questionable if such a state is possible in virtually all real world situations [5]. Second, which is related to the first sense given above, consensus is meant as a way to reach consensus. This involves an evolution of the testimonies of the group towards consensus with respect to their testimonies; this evolution can be free or facilitated (moderated) by a special individual [1], [6]. Third, consensus can be meant as a way in which decisions should be meant in multi-person settings [7]. Basically, consensus decision making aims at attaining the consent, not necessarily the agreement, of the participants by accommodating views of all parties involved to attain a decision that will yield what will be beneficial to the entire group, not necessarily to the particular individuals who may give consent to what will not necessarily be their first choice but because, for instance, they wish to cooperate with the group. The full consent, however, does not mean that each individual is in full agreement [1]. Therefore, consensus boils down to cooperation in contrast to most GDM setting, notably voting, gaming, etc., which boil down to a competition.

It is clear that, ideally, consensus should refer to unanimity of individuals because the option or course of action attained will be best representative for the entire group. Obviously, unanimity may be difficult to attain, in particular in large and diversified groups of individuals as is the case in real world settings, and that is why milder benchmarks (definitions) of consensus have been employed exemplified by [1], [8]: unanimity minus one (U − 1), i.e., that all individuals but one support the decision, unanimity minus two (U − 2), i.e., all but two support the decision, unanimity minus three, (U − 3), etc. Moreover, some measures like 80%, 2/3, etc., can be employed, and even the so called rough consensus may be assumed which does not assume any specific rule which is to be determined later. Notice that all these milder definitions of consensus are still crisp and do not involve any imprecise (fuzzy) specification. We can well imagine here a fuzzy majority exemplified by “most”, “almost all”, “much more than a half” and so on [9], [10]. In such a way, we could introduce the concept of soft consensus as more adequate to model the GDM problems. We will discuss such fuzzy specifications, notably the concepts of a fuzzy majority and soft consensus [11], later in detail because they constitute a main topic here.

Virtually, all consensus reaching processes proceed in a multistage setting, i.e., the individuals change their opinions step by step until, possibly, some consensus is reached [1], [2]. Of course, this presupposes that the individuals are committed to those changes. At this junction, one can clearly see two situations. First, the process of changes of the testimonies proceeds in one way or another and can be modeled. Second, that process is moderated (facilitated) by a special person (“super-individual”) called a facilitator, moderator, etc., who is responsible for running the consensus reaching session in question by persuading the individuals to change their testimonies by rational argument, persuasion, etc., and keeping the process within a period of time considered [1]. The second option of a moderator running a consensus reaching process is usually more effective and efficient. However, this paradigm has been predominant in recent times only [5], [11], [12], [13].

Given the importance of obtaining an accepted solution by the whole group, the consensus has attained a great attention and it is virtually a major goal of GDM problems. The objective of this paper is to present a comprehensive presentation of the state of the art of all known consensus approaches, with an in-depth analysis of the respective problems and solutions as well as more relevant challenges. In particular, we focus on the consensus approaches in which there is a moderator, as they are more promising in practice and the most used in the literature, and based on the concept of fuzzy majority, which is more human-consistent and suitable for reflecting human perceptions of the meaning of consensus, i.e., the soft consensus.

This contribution is set out as follows. In Section 2, we describe the consensus processes based on moderator and the usual fuzzy GDM framework. In Section 3, we highlight the pioneer and most relevant contributions existing on consensus. In Section 4, the main fuzzy consensus approaches are described. The current trends and prospects in the development of consensus models are shown in Section 5. Finally, in Section 6, we present some concluding remarks.