**What do we mean by ***fair*?
Over the years, those who study voting theory have proposed numerous criteria which most
people would expect a 'fair' preferential election method to satisfy. In this course, we will consider four *Fairness
Criteria*. (There are others but we will only consider these four.)
**4 Fairness Criteria**
The Majority Criterion
The Condorcet Criterion
The Monotonicity Criterion
The
Independence of Irrelevant Alternatives Criterion.
For many years, mathematicians and others interested in
voting theory searched for a preferential voting procedure that would satisfy a reasonable set of
fairness criteria (such as the set of 4 above). Arrow's work in 1952 (mentioned
previously) abruptly ended these efforts when he showed that such a search was in vain. In
terms of the 4 fairness criteria above, Arrow's result means that there is NO consisent
preferential
voting method that can satisfy all four of them.
Thus, for example, a preferential procedure designed to satisfy the
Majority Criterion will always violate at least one of the other criteria. This, in turn,
means that there is no 'perfect' preferential voting procedure and the decision about the
procedure to be used is, by necessity, subjective. The best one can hope for is to be able
to objectively analyze the strengths and weaknesses of various preferential procedures and to apply
that knowledge to the task of selecting a 'good' one for a particular situation.
**The Majority
Criterion**
Any candidate receiving a majority of first place votes should be the winner.
*In other words, it would seem unfair to most people if
Candidate A got 51 first place votes, Candidate B got 40 first place votes, and Candidate
C got 9 first place votes (100 votes in all) but Candidate B was declared the winner of
the election. Such an outcome would violate the Majority Criterion.*
Back to the 4
Fairness Criteria
**The Condorcet
Criterion **
(named after the Marquis de Condorcet)
A candidate who wins head-to-head matchups with all other candidates should be
the winner.
*Suppose 4 candidates A, B, C, and D run for mayor of a
small town (a very small town!). There are 20 registered voters. The local newspaper
performed a post-election survey of each of the 20 registered voters. Among other things,
the survey asked the voters who they preferred in a two-way race between candidate C (the
one endorsed by the paper's editorial staff) and each of the other candidates. Here are
the results: *
11 voters preferred candidate C over candidate A
11 voters preferred candidate C over candidate B
17 voters preferred candidate C over candidate D
So, in head-to-head competition, candidate C won against each of the other candidates.
Wouldn't it seem unfair if candidate C was not declared the winner?
When the actual votes were tabulated, candidate A got 9 first place votes, candidate B got
no first place votes, candidate C got 8 first place votes, and candidate D got 3 first
place votes. If candidate C is not declared the winner, this would be a violation of the
Condorcet Criterion.
Back to the 4
Fairness Criteria
**The
Monotonicity Criterion**
If an election is held and a winner is declared, this winning candidate should
remain the winner in any revote in which all preference changes are in favor of the winner
of the original election.
*Three students, Al, Bob, and Carrie are running for
class president. The class will vote in rounds. The student with the fewest votes in the
first round will drop out and a new vote will be taken between the two remaining
candidates. The student with the most votes in this final round will be declared the
winner of the election. In the first round, Al gets 11 first place votes, Bob gets 8 first
place votes, and Carrie gets 10 first place votes. Bob drops out (since he had the fewest
votes in the first round). In the final round, Al gets 11 first place votes and Carrie
gets 18 first place votes. Carrie wins the election!*
But wait! The chairman of the election oversight committee destroyed the ballots before
the results had been officially certified by the administration. You guessed it! The
election had to be repeated. In the first round of the repeated election everyone voted
exactly as in the first round of the original election except for 4 voters who decided to
jump on the bandwagon and vote for Carrie instead of Al. As a result, Al gets 7 first
place votes, Bob gets 8 first place votes, and Carrie gets 14 first place votes. This
causes Al to drop out instead of Bob so that the final round of the repeated election is
between Bob and Carrie. BUT the 7 students who originally voted for Al prefer Bob over
Carrie. So, all 7 of them cast their votes for Bob in the final round. This gives Bob 15
votes and Carrie 14 votes. Bob wins the repeated election EVEN THOUGH THE ONLY CHANGES IN
VOTER PREFERENCE WERE THE 4 VOTES THAT CHANGED FROM AL TO CARRIE (THE ORIGINAL WINNER).
This illustrates a violation of the Monotonicity Criterion.
Back to the 4
Fairness Criteria
**The Independence of
Irrelevant Alternatives Criterion**
If an election is held and a winner is declared, this winning candidate should
remain the winner in any recalculation of votes as a result of one or more of the losing
candidates dropping out..
*As a publicity stunt for its soon-to-be-published
cookbook, the Culinary Club of Smallville decided to have a 'Best Pie' contest. The
entries were narrowed down to three for the final round of the contest. In this final
round, each club member ranked the 3 pies. Each first place vote is worth 3 points, each
second place vote is worth 2 points, and each third place vote is worth 1 point. Here is a
summary of the results:*
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie
third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second, and Al's Apple Pie
third;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third.
Based on the rules above,
Al's Apple Pie got 27 first place votes at 3 points each (81 points) and 26 third place
votes at 1 point each (26 points) for a grand total of 107 points.
Chris' Cream Pie got 2 first place votes at 3 points each (6 points) and 51 second place
votes at 2 points each (102 points) for a grand total of 108 points.
Pat's Peach Pie got 24 first place votes at 3 points each (72 points), 2 second place
votes at 2 points each (4 points), and 27 third place votes at 1 point each (27 points)
for a grand total of 103 points
**Thus, Chris' Cream Pie gets first place (108 points), Al's Apple Pie gets second
place (107 points), and Pat's Peach Pie gets third place (103 points).**
Before the results can be publicized, Pat (who is upset about a third place finish)
demands that her Peach Pie entry be retroactively withdrawn from the contest. Bowing to
her wishes, the club removes the Peach Pie and recalculates points (2 points for each
first place, 1 point for each second place).
Since Pat's Peach Pie is now out, the rankings are:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and ~~Pat's Peach
Pie third~~;
24 members ranked ~~Pat's Peach Pie first~~, Chris' Cream Pie ~~second~~
first, and Al's Apple Pie ~~third~~ second;
2 members ranked Chris' Cream Pie first, ~~Pat's Peach Pie second~~, and Al's
Apple Pie ~~third~~ second.
The new results are,
Al's Apple Pie gets 27 first place votes at 2 points each (54 points) and 26 second place
votes at 1 point each (26 points) for a grand total of 80 points;
Chris' Cream Pie gets 26 first place votes at 2 points each (52 points) and 27 second
place votes at 1 point each (27 points) for a grand total of 79 points.
**Thus, Al's Apple Pie gets first place (80 points), and Chris' Cream Pie gets
second place (79 points).**
Notice what happened. Because a loser (Pat) dropped out, the winner changed (from Chris to
Al). This is a violation of the the Independence of Irrelevant Alternatives Criterion.
Back to the 4
Fairness Criteria |