![Page 1: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/1.jpg)
Section 2.4: Rank MethodsMath for Liberal Studies
![Page 2: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/2.jpg)
Another Voting Method
We have studied the plurality and Condorcet methods so far
In this method, once again voters will be allowed to express their complete preference order
Unlike the Condorcet method, we will assign points to the candidates based on each ballot
![Page 3: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/3.jpg)
Rank Method
We assign points to the candidates based on where they are ranked on each ballot
The points we assign should be the same for all of the ballots in a given election, but can vary from one election to another
The points must be assigned nonincreasingly: the points cannot go up as we go down the ballot
![Page 4: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/4.jpg)
An Example
Suppose we assign points like this: 5 points for 1st place 3 points for 2nd place 1 point for 3rd place
Number of Voters Preference Order
6 Milk > Soda > Juice
5 Soda > Juice > Milk
4 Juice > Soda > Milk
![Page 5: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/5.jpg)
An Example
Determine the winner by multiplying the number of ballots of each type by the number of points each candidate receives
Number of Voters Preference Order
6 Milk > Soda > Juice
5 Soda > Juice > Milk
4 Juice > Soda > Milk
![Page 6: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/6.jpg)
An Example
5 points for 1st place 3 points for 2nd place 1 point for 3rd place
Number of Voters Preference Order Milk Soda Juice
6 Milk > Soda > Juice
5 Soda > Juice > Milk
4 Juice > Soda > Milk
![Page 7: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/7.jpg)
An Example
5 points for 1st place 3 points for 2nd place 1 point for 3rd place
Number of Voters Preference Order Milk Soda Juice
6 Milk > Soda > Juice 30
5 Soda > Juice > Milk 5
4 Juice > Soda > Milk 4
![Page 8: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/8.jpg)
An Example
5 points for 1st place 3 points for 2nd place 1 point for 3rd place
Number of Voters Preference Order Milk Soda Juice
6 Milk > Soda > Juice 30 18
5 Soda > Juice > Milk 5 25
4 Juice > Soda > Milk 4 12
![Page 9: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/9.jpg)
An Example
5 points for 1st place 3 points for 2nd place 1 point for 3rd place
Number of Voters Preference Order Milk Soda Juice
6 Milk > Soda > Juice 30 18 6
5 Soda > Juice > Milk 5 25 15
4 Juice > Soda > Milk 4 12 20
![Page 10: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/10.jpg)
An Example
Milk gets 39 points Soda gets 55 points Juice gets 41 points
Soda wins!
Number of Voters Preference Order Milk Soda Juice
6 Milk > Soda > Juice 30 18 6
5 Soda > Juice > Milk 5 25 15
4 Juice > Soda > Milk 4 12 20
![Page 11: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/11.jpg)
Rank Methods are Common
Sports Major League Baseball MVP NCAA rankings Heisman Trophy
Education Used by many universities (including Michigan and UCLA) to
elect student representatives Others
A form of rank voting was used by the Roman Senate beginning around the year 105
![Page 12: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/12.jpg)
A Special Kind of Rank Method
The Borda Count is a special kind of rank method
With 3 candidates, the scoring is 2, 1, 0 With 4 candidates, the scoring is 3, 2, 1, 0 With 5 candidates, the scoring is 4, 3, 2, 1, 0 etc.
Last place is always worth 0
![Page 13: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/13.jpg)
Are Rank Methods “Fair”?
Rank methods do not satisfy the Condorcet winner criterion
In this profile, the Condorcet winner is A
However, the Borda count winner is B
Voters Preference Order
4 A > B > C
3 B > C > A
![Page 14: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/14.jpg)
Are Rank Methods “Fair”?
Notice that C is a loser either way
If we get rid of C, noticewhat happens…
Voters Preference Order
4 A > B > C
3 B > C > A
![Page 15: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/15.jpg)
Are Rank Methods “Fair”?
Notice that C is a loser either way
If we get rid of C, noticewhat happens…
…now the Borda countwinner is A
Voters Preference Order
4 A > B
3 B > A
![Page 16: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/16.jpg)
Are Rank Methods “Fair”?
If we start with this profile, A is the clear winner
But adding C into the mixcauses A to lose using theBorda count
In this way, C is a “spoiler”
Voters Preference Order
4 A > B
3 B > A
![Page 17: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/17.jpg)
The Spoiler Effect
Voters prefer A over B
A third candidate C shows up
Now voters prefer B over A
![Page 18: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/18.jpg)
The Spoiler Effect With Pies
After finishing dinner, you and your friends decide to order dessert.
The waiter tells you he has two choices: apple pie and blueberry pie.
You order the apple pie. After a few minutes the waiter returns and says that
he forgot to tell you that they also have cherry pie. You and your friends talk it over and decide to have
blueberry pie.
![Page 19: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/19.jpg)
Another Example
In the 2000 Presidential election, if the election had been between only Al Gore and George W. Bush, the winner would have been Al Gore
However, when we add Ralph Nader into the election, the winner switches to George W. Bush
![Page 20: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/20.jpg)
Independence of Irrelevant Alternatives Condition (IIA)
The spoiler effect is sometimes called the independence of irrelevant of alternatives condition, or IIA for short
In a sense, the third candidate (the “spoiler”) is irrelevant in the sense that he or she cannot win the election
![Page 21: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/21.jpg)
How do we tell if a method satisfies the IIA condition?
Look at a particular profile and try to identify a candidate you think might be a spoiler
Determine the winner of the election with the spoiler, and also determine the winner if the spoiler is removed
If the winner switches between two non-spoiler candidates, then the method you are using suffers from the spoiler effect
![Page 22: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/22.jpg)
How do we tell if a method satisfies the IIA condition?
A beats B, but when C shows up, B winsC is a spoiler!
A beats B, but when C shows up, A still winsNo spoiler!
A beats B, but when C shows up, C winsNo spoiler!
![Page 23: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/23.jpg)
Still Searching
We now have two criteria for judging the fairness of an election method Condorcet winner criterion (CWC) Independence of irrelevant alternatives (IIA)
We still haven’t found an election method that satisfies both of these conditions
![Page 24: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/24.jpg)
Still Searching… No, Really!
Well, actually, the Condorcet method satisfies both conditions
But as we have seen, Condorcet’s method will often fail to decide a winner, so it’s not really usable
![Page 25: Math for Liberal Studies. We have studied the plurality and Condorcet methods so far In this method, once again voters will be allowed to express](https://reader035.vdocuments.mx/reader035/viewer/2022062712/56649c8f5503460f94948b0e/html5/thumbnails/25.jpg)
Still Searching… No, Really!
Ideally, we want an election method that always gives a winner, and satisfies our fairness conditions
In the next section we will consider several alternative voting methods, and test them using these and other conditions