Transcript MGF 1107 Social Choice Math

Voting Methods
• Examples of Voting Methods (other than
majority rules)
– Plurality
– Borda Count
– Hare System
– Sequential Pairwise
– Approval Voting
Examples of Voting Methods
Consider the following preference schedule for 9 voters
considering 4 candidates A, B, C and D.
Number of Voters (9 total)
Rank
3
1
1
1
1
1
1
1st
A
A
B
B
C
C
D
2nd
D
B
C
C
B
D
C
3rd
B
C
D
A
D
B
B
4th
C
D
A
D
A
A
A
(a) Who wins by the plurality method?
With the plurality method, we consider only each voter’s 1st
place ranking.
A voter can only vote for his or her first choice.
Rank
1st
2nd
3
A
D
3rd
4th
B
C
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
D
C
D
D
A
A
D
D
A
B
A
1
D
C
B
A
(a) Who wins by the plurality method?
We consider only each voter’s 1st place ranking.
Because A has the most first place votes, A is the winner
by the plurality method.
Rank
1st
3
A
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
1
D
(b) Who wins by the Borda Count method of voting?
• Note that A has 13 points, B has 15 points, C has 13 points, and D has 13
total points.
• Because B has the most total points, B is the winner by the Borda
Count method.
Number of Voters (9 total)
Rank
3
1
1
1
1
1
1
1st
A
(9)
A
(3)
B
(3)
B
(3)
C
(3)
C
(3)
D
(3)
2nd
D
(6)
B
(2)
C
(2)
C
(2)
B
(2)
D
(2)
C
(2)
3rd
B
(3)
C
(1)
D
(1)
A
(1)
D
(1)
B
(1)
B
(1)
C
(0)
D
(0)
A
(0)
D
(0)
A
(0)
A
(0)
A
(0)
(3points)
(2points)
(1 point)
4th
(0 points)
(c) Who wins by the Hare System voting method?
• First, we must eliminate candidate D because the least number of
voters rank D as a first choice.
Rank
1st
2nd
3rd
4th
3
A
D
B
C
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
D
C
D
A
D
B
D
A
D
A
A
1
D
C
B
A
(c) Who wins by the Hare System voting method?
We cross out every instance of candidate D in the
preference lists of all voters.
Rank
1st
2nd
3
A
D
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
D
3rd
4th
B
C
C
D
D
A
A
D
D
A
B
A
1
D
C
B
A
(c) Who wins by the Hare System voting method?
Once candidate D is removed as an option, we assume
individual voter’s preferences remained unchanged with
respect to the other candidates, and so we move
candidates up into any open spaces.
Rank
1st
2nd
3
A
3rd
4th
B
C
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
C
A
A
A
B
A
1
C
B
A
(c) Who wins by the Hare System voting method?
Once candidate D is removed as an option, we assume
individual voter’s preferences remained unchanged with
respect to the other candidates, and so we move
candidates up into any open spaces.
Rank
1st
2nd
3
A
B
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
B
3rd
4th
C
C
A
A
A
A
1
C
B
A
(c) Who wins by the Hare System voting method?
There are now some repetitions, so we can re-organize
the table.
Rank
1st
2nd
3rd
4th (none)
4
A
B
C
Number of Voters (9 total)
2
3
B
C
C
B
A
A
(c) Who wins by the Hare System Method of Voting?
• Now, we begin again, by identifying any candidates which have the least number of
voters ranking them in first place.
• Remember there could be a tie for least, for example if candidate B and C were both
ranked first by 2 voters. If that had been the case then both B and C would be
eliminated and A would be declared the winner.
• In this example, we eliminate candidate B because B tops only 2 voters’ preference
lists.
Rank
1st
2nd
3rd
Number of Voters (9 total)
4
2
3
A
B
C
B
C
B
C
A
A
(c) Who wins by the Hare System Voting Method?
• Now, we begin again, by identifying any candidates which have the least number of
voters ranking them in first place.
• Remember there could be a tie for least, for example if candidate B and C were both
ranked first by 2 voters. If that had been the case then both B and C would be
eliminated and A would be declared the winner.
• In this example, we eliminate candidate B because B tops only 2 voters’ preference
lists.
Rank
1st
2nd
3rd
Number of Voters (9 total)
4
2
3
A
B
C
B
C
B
C
A
A
(c) Who wins by the Hare System Voting Method?
Now we have eliminated candidate B from contention.
Again, we assume voter preferences will remain
unchanged with respect to the other candidates and we
will move the remaining candidates up into any open
spaces.
Rank
1st
2nd
3rd
Number of Voters (9 total)
4
2
3
A
C
C
C
A
A
(c) Who wins by the Hare System Voting Method?
Now we have eliminated candidate B from contention.
Again, we assume voter preferences will remain
unchanged with respect to the other candidates and we
will move the remaining candidates up into any open
spaces.
Rank
1st
2nd
3rd
Number of Voters (9 total)
4
2
3
A
C
C
C
A
A
(c) Who wins by the Hare System Voting Method?
Now we re-organize the table because of the repetitions in the preference
lists. At this point, A is eliminated because A tops the least number of
preference lists (4) compared to C (with 5 lists). Because C is the only
candidate remaining after the sequence of eliminations, C is declared the
winner.
Note that if both A and C where ranked first by the same number of voters (for
example if both lists represented 5 voters) we would declare the two
remaining candidates tied.
Rank
1st
2nd
Number of Voters (9 total)
4
5
A
C
C
A
(d) Who wins by the sequential pairwise voting method?
•
•
•
•
We must determine the order in which we will compare candidates head-to-head.
This order is called the agenda. Suppose we will compare them in alphabetical
order: A, B, C, then D.
With the voter preferences listed below, we begin by comparing candidates A and
B.
We consider all voter lists and note that B ranks above A on 5 lists whereas A
ranks over B on only 4 lists. Therefore, B wins over A by a vote of 5 to 4.
Because B wins over A, based on the agenda, we now compare B with C.
Rank
1st
2nd
3rd
4th
3
A
D
B
C
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
D
C
D
A
D
B
D
A
D
A
A
1
D
C
B
A
(d) Who wins by the sequential pairwise voting method?
•
•
•
Comparing B and C head-to-head, we recognize that B ranks above C for 6 out
of 9 voters. That is, B beats C by a vote of 6 to 3.
Following the agenda A, B, C, and D, we now compare B with D.
We find that D is preferred to B among voters by a ratio of 5 to 4 and because we
have reached the last candidate, D is declared the winner by sequential
pairwise voting.
Rank
1st
2nd
3rd
4th
3
A
D
B
C
Number of Voters (9 total)
1
1
1
1
1
A
B
B
C
C
B
C
C
B
D
C
D
A
D
B
D
A
D
A
A
1
D
C
B
A
Is this at all disconcerting? Different voting methods
producing different results?
•
•
•
•
Candidate A prefers plurality
Candidate B prefers Borda Count
Candidate C prefers the Hare System
Candidate D prefers sequential pairwise voting (but with
agenda A,B,C,D)
Which voting method is the best? How do we decide?
How about approval voting?
An example of Approval Voting
• When voters use approval voting they cast votes only for candidates
they find acceptable.
• This means we can imagine voters will “draw a line” between
acceptable and unacceptable candidates as they vote.
• Voters need not decide on a preference list and rank all of the
candidates, however we may continue using preference lists without
loss of generality. In other words, it does no harm to suppose the
voters actually do rank the candidates and then choose to separate
them into two categories: acceptable and unacceptable.
• For our examples of approval voting we will continue using
preference lists and actually draw a line to represent the point of
division between acceptable and unacceptable candidates for each
voter.
An example of Approval Voting
•
•
Consider the preference list shown with lines added to indicate an
“approval line”
Voters will cast approval votes for candidates above the approval line and
will not cast approval votes for candidates below that line.
Number of Voters (100 total)
33
33
34
1st
A
B
C
2nd
B
A
A
3rd
C
C
B
Notice the lines indicate 33 voters approve of both candidates A and B, 33 voters
Approve of only candidate B and 34 candidates approve of only candidate C.
If the election were held with these preferences using approval voting, the winner
would be candidate B who would get 66 approval votes. Note that A gets 33
approval votes and C gets 34 approval votes.
Which Method is Best?
• Is approval voting the answer ?
– Is this the best method of voting ?
• We saw 4 different voting methods each give us a
different winner for the same election.
• There are two problems we must face:
– First, there is the problem that different voting methods can
produce different winners and we must decide which method to
use.
– Second, each of our voting methods have problems in and of
themselves. Even approval voting has its share of problems.