Now suppose that Pivotal Voter moves
B
above
A
, but keeps
C
in the same position and imagine
that any number (or all!) of the other voters change their ballots to move
C
above
B
, without
changing the position of
A
. Then aside from a repositioning of
C
this is the same as
Profile k
from

Part One and hence the societal outcome ranks
B
above
A
. Furthermore, by
IIA
the societal
outcome must rank
A
above
C
, as in the previous case. In particular, the societal outcome
ranks
B
above
C
, even though Pivotal Voter may have been the
only
voter to rank
B
above
C
.
By
IIA
this conclusion holds independently of how
A
is positioned on the ballots, so Pivotal Voter is
a dictator for
B
over
C
.
Part Three: There can be at most one dictator
[
edit
]
Part Three: Since voter
k
is the dictator for
B
over
C
, the pivotal voter for
B
over
C
must appear among the
first
k
voters. That is,
outside
of Segment Two. Likewise, the pivotal voter for
C
over
B
must appear among
voters
k
through
N
. That is, outside of Segment One.
In this part of the argument we refer back to the original ordering of voters, and compare the
positions of the different pivotal voters (identified by applying Parts One and Two to the other pairs of
candidates). First, the pivotal voter for
B
over
C
must appear earlier (or at the same position) in the
line than the dictator for
B
over
C
: As we consider the argument of Part One applied to
B
and
C
,
successively moving
B
to the top of voters' ballots, the pivot point where society
ranks
B
above
C
must come at or before we reach the dictator for
B
over
C
. Likewise, reversing the
roles of
B
and
C
, the pivotal voter for
C
over
B
must at or later in line than the dictator for
B
over
C
.
In short, if
k
X/Y
denotes the position of the pivotal voter for
X
over
Y
(for any two candidates
X
and
Y
),
then we have shown
k
B/C
≤ k
B/A
≤
k
C/B
.
Now repeating the entire argument above with
B
and
C
switched, we also have
k
C/B
≤
k
B/C
.
Therefore we have
k
B/C
= k
B/A
=
k
C/B
and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators)
occur at the same position in the list of voters. This voter is the dictator for the whole election.
Interpretations of the theorem
[
edit
]
Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way
with a statement such as
"No voting method is fair,"
"Every ranked voting method is flawed,"
or
"The
only voting method that isn't flawed is a dictatorship"
. These statements are simplifications of Arrow's
result which are not universally considered to be true. What Arrow's theorem does state is that a

deterministic preferential voting mechanism - that is, one where a preference order is the only
information in a vote, and any possible set of votes gives a unique result - cannot comply with all of
the conditions given above simultaneously.

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- Spring '17
- Reshmi
- Economics, Social Choice and Individual Values, Voting system, Social choice theory, Arrow's impossibility theorem, aggregate production, pivotal voter