conditions (1), (2), and (3) of Theorem 1. Since F is non-dictatorial the
minimal Cylindric decomposition of Rf cannot consist of just one section
with strictly more than two active components (by (3)). Now, notice that
when preferences are separable but not additively represent able, the active
components of a section can be ordered differently among themselves, de-
pending on which objects are present in another section. This can now be
used to show that we cannot have a section with more than two active com-
ponents together with another section with more than one active component.
To prove it, it is enough to construct profiles where the presence of an object
affects the ordering of the active components in another section. Therefore,
all sections have either only one active component (the objects that are al-
ways selected) or they have just two active components which are of the form
{{0}, {^}}. Hence, all sections in the minimal Cylindric decomposition of Rf
are singletons.
4 Final Remark
Until now, we have taken the dimension of our problems (i.e., the number
of objects), as well as the feasibility constraints, as given data. Our analysis
admits another reading without any formal change, except for its interpreta-
tion.
Consider a situation where society faces four alternatives, ɑ, b, c, and d.
One possibility is that each of these alternatives might be described by two
characteristics, and that identifying a = (0,0), b = (1,0), c = (0,1), and
d = (1,1) provides a good description of the actual choices (this particular
choice would indicate that a and c are similar in the first characteristic but
differ on the second, etc.). It may also be, in another extreme, that these four
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