Math for Liberal Studies Section 2 4 Rank

Math for Liberal Studies Section 2. 4: Rank Methods

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

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

An Example �Suppose we assign points like this: 5 points for 1 st place 3 points for 2 nd place 1 point for 3 rd place Number of Voters Preference Order 6 Milk > Soda > Juice 5 Soda > Juice > Milk 4 Juice > Soda > Milk

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

An Example � 5 points for 1 st place � 3 points for 2 nd place � 1 point for 3 rd place Number of Voters Preference Order 6 Milk > Soda > Juice 5 Soda > Juice > Milk 4 Juice > Soda > Milk Soda Juice

An Example � 5 points for 1 st place � 3 points for 2 nd place � 1 point for 3 rd place Number of Voters Preference Order Milk 6 Milk > Soda > Juice 30 5 Soda > Juice > Milk 5 4 Juice > Soda > Milk 4 Soda Juice

An Example � 5 points for 1 st place � 3 points for 2 nd place � 1 point for 3 rd place Number of Voters Preference Order Milk Soda 6 Milk > Soda > Juice 30 18 5 Soda > Juice > Milk 5 25 4 Juice > Soda > Milk 4 12 Juice

An Example � 5 points for 1 st place � 3 points for 2 nd place � 1 point for 3 rd 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

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

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

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

Are Rank Methods “Fair”? �Rank methods do not satisfy the Condorcet winner criterion �In this profile, the Condorcet winner is A Voters Preference Order 4 A>B>C 3 B>C>A �However, the Borda count winner is B

Are Rank Methods “Fair”? �Notice that C is a loser either way �If we get rid of C, notice what happens… Voters Preference Order 4 A>B>C 3 B>C>A

Are Rank Methods “Fair”? �Notice that C is a loser either way �If we get rid of C, notice what happens… �…now the Borda count winner is A Voters Preference Order 4 A>B 3 B>A

Are Rank Methods “Fair”? �If we start with this profile, A is the clear winner �But adding C into the mix causes A to lose using the Borda count �In this way, C is a “spoiler” Voters Preference Order 4 A>B 3 B>A

The Spoiler Effect �Voters prefer A over B �A third candidate C shows up �Now voters prefer B over A

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.

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

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

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

How do we tell if a method satisfies the IIA condition? �A beats B, but when C shows up, B wins C is a spoiler! �A beats B, but when C shows up, A still wins No spoiler! �A beats B, but when C shows up, C wins No spoiler!

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

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

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