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 their complete preference order  Unlike the Condorcet method, we will assign points to the candidates based on each ballot

 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

 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 6Milk > Soda > Juice 5Soda > Juice > Milk 4Juice > Soda > Milk

 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 6Milk > Soda > Juice 5Soda > Juice > Milk 4Juice > Soda > Milk

 5 points for 1 st place  3 points for 2 nd place  1 point for 3 rd place Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice 5Soda > Juice > Milk 4Juice > Soda > Milk

 5 points for 1 st place  3 points for 2 nd place  1 point for 3 rd place Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice30 5Soda > Juice > Milk5 4Juice > Soda > Milk4

 5 points for 1 st place  3 points for 2 nd place  1 point for 3 rd place Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice3018 5Soda > Juice > Milk525 4Juice > Soda > Milk412

 5 points for 1 st place  3 points for 2 nd place  1 point for 3 rd place Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice Soda > Juice > Milk Juice > Soda > Milk41220

 Milk gets 39 points  Soda gets 55 points  Juice gets 41 points  Soda wins! Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice Soda > Juice > Milk Juice > Soda > Milk41220

 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

 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

 Rank methods do not satisfy the Condorcet winner criterion  In this profile, the Condorcet winner is A  However, the Borda count winner is B VotersPreference Order 4A > B > C 3B > C > A

 Notice that C is a loser either way  If we get rid of C, notice what happens… VotersPreference Order 4A > B > C 3B > C > A

 Notice that C is a loser either way  If we get rid of C, notice what happens…  …now the Borda count winner is A VotersPreference Order 4A > B 3B > A

 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” VotersPreference Order 4A > B 3B > A

 Voters prefer A over B  A third candidate C shows up  Now voters prefer B over A

 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.

 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

 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

 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

 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!

 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

 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

 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