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316 Decisions involving groups of individuals
get two votes and C only one, which implies that B > C. Finally, if we
were to compare A with C, C would get two votes and A only one, which
implies C > A. So not only do we have A > B > C but we also have C
> A, which means that the preferences of the group are not transitive.
This result is known as Condorcet’s paradox.
In many practical problems alternatives are not compared simultaneously,
as above, but sequentially. For example, the committee might
compare A with B first, eliminate the inferior option and then compare
the preferred option with C. Unfortunately, the order of comparison has
a direct effect on the option which is chosen as shown below.
If the group compared A with B first then A would survive the first
round. If A was then compared with C, C would be the location chosen.
Alternatively, if the group compared B with C first, B would survive
the first round. If B was then compared with A then location A would
be chosen.
Moreover, a clever group member could cheat by being dishonest
about his preferences if the preferences of the other members are already
known. Suppose that locations A and B are to be compared first. Edwards
realizes that this will make C, his least-preferred location, the final choice.
He would prefer B to be selected, so he dishonestly states his preferences
as B > A > C. This ensures that B, not A, will survive the first round
andgoonto‘defeat’Cinthesecond.
These sorts of problems led Arrow 10 to ask whether there is a
satisfactory method for determining group preferences when the preferences
of individual members are expressed as orderings. He identified
four conditions which he considered that a satisfactory procedure
should meet:
(1) The method must produce a transitive group preference order for
the options being considered.
(2) If every member of the group prefers one option to another then
so must the group. (You will recall that this condition was not
fulfilled in the production manager/accountant’s problem which we
considered earlier.)
(3) The group choice between two options, A and B, depends only
upon the preferences of members between these options and not on
preferences for any other option. (If this is not the case then, as we
saw above, an individual can influence the group ordering by lying
about his preferences.)
(4) There is no dictator. No individual is able to impose his or her
preferences on the group.