The Plurality Method Fails
The Independence of Irrelevant
Alternatives Criterion
The Independence of Irrelevant Alternatives criterion
states:
If a re-election is held and the only change is that
some non-winning candidate drops out, the previous
winner should still win the election.
This is a condition of fairness in voting in the sense that
it would seem unfair that a candidate that otherwise
would have won, ends up losing in an election because
some other losing candidate drops out of the election.
• Suppose there are 9 faculty members in the math
department at a community college and they have
decided to vote on the choice of the next textbook for
College Algebra.
• Suppose there are three choices for the textbook –
books authored by Woodbury, Blitzer, and McKeague.
•
Suppose all 9 faculty members have reviewed or used
these books in the past and can rank their preferences
as to which they favor 1st, 2nd, and 3rd, in their own
opinion.
• The table below shows the preferences of the faculty
Number
of faculty
First choice
Second choice
Third choice
4
3
2
Woodbury
Blitzer
McKeauge
Blitzer
Woodbury
Blitzer
McKeague
McKeague
Woodbury
Number
of faculty
First choice
Second
choice
Third choice
4
3
2
Woodbury
Blitzer
McKeauge
Blitzer
Woodbury
Blitzer
McKeague
McKeague
Woodbury
Based on these preferences, and using the Plurality method,
the winner of the vote would be Woodbury.
If the vote were held with these preferences, Woodbury wins
because Woodbury has more votes than either of the other two
candidates.
Number
of faculty
First choice
Second
choice
Third choice
4
3
2
Woodbury
Blitzer
McKeauge
Blitzer
Woodbury
Blitzer
McKeague
McKeague
Woodbury
Suppose that the 2 faculty members that prefer McKeague as
their first choice realize that the majority of the faculty dislike the
McKeauge book.
Suppose the other 7 faculty agree that they would never vote for
McKeague and based on a concession from the 2 McKeauge
supporters, a decision is made to remove McKeague from the
ballot.
Assuming everyone’s preferences remain the same, even if
McKeague is removed, the table appears as follows…
Number
of faculty
First choice
Second
choice
Third choice
4
3
2
Woodbury
Blitzer
McKeauge
Blitzer
Woodbury
Blitzer
McKeague
McKeague
Woodbury
Assuming everyone’s preferences remain the same, even if
McKeague is removed, the table appears as follows…
Number
of faculty
First choice
Second
choice
Third choice
4
3
Woodbury
Blitzer
Blitzer
Woodbury
2
Blitzer
Woodbury
Now we can align those voters whose preferences are the
same.
Number
of faculty
First choice
Second
choice
4
5
Woodbury
Blitzer
Blitzer
Woodbury
Third choice
And now the new winner of the election, still using the same method,
is the Blitzer book.
Number
of faculty
First choice
Second
choice
Third choice
4
3
2
Woodbury
Blitzer
McKeauge
Blitzer
Woodbury
Blitzer
McKeague
McKeague
Woodbury
Perhaps those 2 faculty that were in the minority had an
agenda! Perhaps they knew that the McKeague book could
never win. Perhaps by suggesting the elimination of a choice
that may have seemed irrelevant to the majority of voters, those
2 faculty got their second choice and avoided what would have
been their last choice.
By eliminating what might have seemed an irrelevant choice,
because Mckeauge was not going to win, the results of the
election have changed. The winner is now Blitzer.
Number
of faculty
First choice
Second
choice
4
5
Woodbury
Blitzer
Blitzer
Woodbury
Third choice
This illustrates how the Plurality method fails the independence of
irrelevant alternatives criterion: Woodbury was going to win in the
original vote. By removing a losing candidate, the results of the
election have changed and Woodbury is no longer the winner.
• A similar situation may have occurred in the 2000 Presidential
election in Florida. George Bush won the Presidency in spite of
losing the nationwide popular vote because he had more electoral
votes than Al Gore.
• After the votes were counted, George Bush had 271 electoral votes
and Al Gore had 266 electoral votes. By winning Florida, Bush
received 25 electoral votes which made the difference.
• In 2000, Ralph Nader was the Green Party candidate. What would
have happened if Ralph Nader, who obviously did not win Florida,
had been removed from the ballot a few months or even weeks
before the election? Would that have made a difference in the very
close Florida count?
• In fact, the Florida vote was decided by 537 votes in favor of Bush.
Ralph Nader received 97,421 votes in Florida.
• Would some of Ralph Nader’s 97,421 votes in Florida have gone to
Al Gore if Nader had not been on the ballot?
• In his own book, Crashing the Party, Nader himself estimates that
had he not been on the ballot, 38% of his supporters would have
supported Gore. If that were true, that’s over 37,000 more votes for
Al Gore.
• In defense of Ralph Nader though, the following should be noted:
– There were 7 other third party candidates (besides Nader) in the 2000
election that received more votes in Florida than the 537 vote difference
between Bush and Gore. Ever heard of Harry Browne? He got over
16,000 votes for President in Florida that year.
– If Al Gore had won his home state of Tennessee, he would have had
enough electoral votes to have won the Presidency.
• There is a joke based on the failure of IIA in plurality voting:
• A waiter says to a customer “We have Apple, Blueberry and
Rhubarb pie for desert.”
• To which the customer replies, “I’d like Apple pie, and please bring
the check.”
• When the waiter returns he mentions that the Rhubarb pie was
actually sold out anyway. “In that case”, says the customer, “I’ll take
the Blueberry pie.”