# [EM] cyclic preferences

Jobst Heitzig heitzig-j at web.de
Tue Aug 3 08:12:48 PDT 2004

```Here's two other examples in which the expression of cyclic preferences
can be "rational" (or, if you prefer that term, "sensible") for a voter:

1. Consider a voter who evaluates the candidates according to a number
of aspects (or dimensions, criteria, issues, perspectives, whatever).
Assume that these aspects are not "measurable" in a numerical way but
that s/he can only tell whether a candidate is better than another
according to that aspect or not. Also, assume that s/he cannot assign
priorities to those aspects but considers them equally important. Assume
further that for almost every pair X,Y of candidates there is an aspect
in which X is better and another aspect in which Y is better. Most of
you will agree that this situation is quite realistic, insn't it?
Now, what preferences shall the voter express in this situation? There
are two natural ways: S/he will express the preference X>Y if and only
if X is better than Y according to either ALL aspects, or according to
MOST aspects. The preference relations which can result from the first
rule  include all quasi-orders (= reflexive and transitive but not
neccessarily total (="complete") relations), and those which can result
from the second rule include all reflexive relations whatsoever, in
particular, cyclic relations.
To give a concrete example: 3 candidates X,Y,Z, 3 aspects A1,A2,A3, and
orderings X>Y>Z according to A1, Y>Z>X according to A2, Z>X>Y according
to A3 (you all know this of course :-) The voter can either express no
preference at all, or the cyclic preference X>Y>Z>X. Which gives us more
information about his/her preferences? The latter, of course.

2. Consider a voter who has children to care for who have no right to
vote however. Don't you think it would be responsible of the voter to
vote not only in his/her own interest but also in the interest of
his/her children? If so, s/he faces essentially the same problem as
society does: s/he has to accumulate preferences in some way. So when
X>Y>Z for herself, Y>Z>X for her son and Z>X>Y for her daughter, the
most responsible thing to vote would of course be X>Y>Z>X.

```