Group Conflict as Social Contradiction - Conflict and Multimodal Communication: Social Research and Machine Intelligence - page 35

for any pair of alternatives, agents know how to rank them, x is preferred to y or
y is preferred to x . 1 Profiles are lists of the divergent points of view of the three
individuals, as in the following example:

a>b>c

1:

b>a>c

2:

a>c>b

3:

In the scenario above, the agents have conflicting preferences and there is no
agreement on which is the best policy to be implemented. Since the policies are
alternative, 1 and 3 would pursue a , whereas 2 would pursue b . The example is
supposed to model a parliament, thus the possible conflicts have to be solved, as we
assume that the parliament as a whole should pursue one of the alternative policies.
Thus, we have to ask what is the preference of the group, namely the preference
that we can ascribe to the parliament composed by 1, 2, and 3. However, at this
point, we cannot ascribe a single preference to the group without assuming a rule to
settle disagreement. Suppose now that the individuals agree on a procedure to settle
their differences; for example, they agree on voting by majority on pairs of options.
Thus, agents elect the collective option by pairwise comparisons of alternatives. In
our example, a over b gets two votes (by 1 and 3), b over c gets two votes (by
1 and 2), and a over c gets three votes. The majority rule defines then a social
preference a>b>c that can be ascribed to the group as the group preference.
The famous Condorcet's paradox shows that it is not always the case that
individual preferences can be aggregated into a collective preference. Take the
following example.

1: a>b>c

2: b>c>a

3: c>a>b

Suppose agents again vote by majority on pairwise comparisons. In this case, a
is preferred to b because of 1 and 3, b is preferred to c because of 1 and 2, thus,
by transitivity, a has to be preferred to c . However, by majority also c is preferred
to a . Thus, the social preference is not “rational”, according to our definition of
rationality, as it violates transitivity.
Kenneth Arrow's famous impossibility theorem states that Condorcet's para-
doxes are not an unfortunate case of majority aggregation, rather they may occur for
any aggregation procedure that respects some intuitive fairness constraint (Arrow
1963 ). In the next section, we shall discuss in more detail the formal treatment of
the intuitions concerning fairness and we shall define a number of properties that
provide normative desiderata for the aggregation procedure.

1 These conditions are to be taken in a normative way. They are not of course descriptively adequate,
as several results in behavioral game theory show. However, the point of this approach is to show
that even when individuals are fully rational, i.e. they conform to the rationality criteria that we
have just introduced, the aggregation of their preferences is problematic.