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a group opinion (collective preference) is a very old problem. The first systematic
approaches to the problem were pioneered by Borda [3] and Condorcet [6], who
initiated the formal discipline of Social Choice in terms of voting. For an extended
review see Nurmi [23].
The subject of Group Decision Making (GDM), traditionally equated with Social
Choice, was revived in the twentieth century by Kenneth Arrow [1], who in his book
titled “Social Choice and Individual Values” was concerned with the difficulties of
group decisions and the inconsistencies they can generate leading to the well-known
Impossibility Theorem.
More specifically, Arrow has proved that in the context of ordinal and symmetric
(all decision makers with equal weight) preferences it is not possible to construct
a collective preference structure without this being imposed by a single individual,
the so-called “Arrow's dictator”. In the following 50ies and 60ies many axiomatic
variants to Arrow's hypothesis have been proposed, see for instance Fishburn [13,
14] and Kelly [21].
It was in the context of group decision theory that the traditional models of con-
sensual dynamics have been formulated, from De Groot's classical consensus model
[7] to various extended or alternative proposals: French [15], Lehrer and Wagner
[22], and Sen [27], mostly in the probabilistic framework. In the spirit of “hard”
consensus, De Groot's classical model of consensual dynamics [7] acts on the in-
dividual preference structures by combining them iteratively on the basis of a spe-
cific transition matrix of reciprocal weights which the decision makers assign to
each other, thereby quantifying the reciprocal influence in the process of consensus
reaching.
The basic framework within which most of the consensus processes are modelled
can be depicted in the following way. There is a set of decision makers or experts
who present their opinions concerning a set of alternatives and these alternatives
may initially differ to a large extent. If the individuals are rationally committed
to consensus, via some exchange of information, bargaining, etc. the individuals'
opinions can be modified and the group may get closer to consensus.
Almost all of these approaches treat consensus as a strict and unanimous agree-
ment, however, since various decision makers have different more or less conflict-
ing opinions the traditional strict meaning of consensus is unrealistic. The human
perception of consensus is much “softer”, and people are willing to accept that a
consensus has been reached when most or the more predominant decision makers
agree on the preferences associated to the most relevant alternatives.
This degree of consensus takes on it values in the unit interval, and it's more
realistic and human consistent than conventional degrees, mostly developed in the
probabilistic framework.
The “soft” consensus paradigm developed in Kacprzyk and Fedrizzi [18], Fedriz-
zi et al. [9], Carlsson et al. [5], Kacprzyk et al. [19] in the standard framework of
numerical fuzzy preferences was extended to a more dynamical context in Fedrizzi,
Fedrizzi, Marques Pereira [10, 11]. The new model combines a soft measure of
collective disagreement with an inertial mechanism of opinion changing aversion.
It acts on the network of single preference structures by a combination of a collective
