1. MAT 105 Spring 2008

2.  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

3.  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

4.  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 MilkSodaJuice 30186 52515 41220 395541 Soda wins with 55 points!

5.  Sports  Major League Baseball MVP  NCAA rankings  Heisman Trophy  Education  Used by many universities (including Michigan and UCLA) to elect student representatives  Used by some academic departments to elect members to committees  Others  A form of rank voting was used by the Roman Senate beginning around the year 105

6.  The Borda Count is a special kind of rank method  Each candidate is given a number of points equal to the number of candidates ranked below them  So with 3 candidates, in the Borda count 1 st place is worth 2 points, 2 nd place is worth 1 point, and 3 rd place is worth 0 points  With 4 candidates, the scoring is 3, 2, 1, 0

7.  Suppose we have an election where A is the winner, B is not, and there are possibly other candidates  Suppose now that we have a new election, and some of the voters change their ballots  However, no one who had A ranked above B changed their ballot to have B above A  What should the outcome of the new election be?

8.  Let’s look at an example  We’ll use the Borda count to find the winner of this election  A gets 11 points  B gets 6 points  C gets 4 points  A is the winner, and B is not  We will have a new election, and no one who had A above B will change to have B above A VotersPreference Order 4A > C > B 3B > A > C

9.  Notice that every voter changed his ballot  However, no one changed the order that they had A and B ranked, they only moved C  B wins the new election!  We say that C was “irrelevant” to the question of A versus B, but moving C around affected the outcome VotersPreference Order 4A > C > B 3B > A > C VotersPreference Order 4A > B > C 3B > C > A

10.  After finishing dinner, Sidney decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Sidney says "In that case I'll have the blueberry pie.“  In our example, A is apple pie, B is blueberry pie, and C is cherry pie

11.  This gives us a way to tell if a voting system is fair  Here’s the process:  We have an original election, where A wins and B does not  We hold a new election, and while the voters can change their ballots, no one changes from having A above B to having B above A  The outcome of the election should not change

12.  If it is not possible to change the outcome of the election by this process, we say the voting method satisfies IIA  If it is possible to change the outcome of the election by this process, we say the voting method does not satisfy IIA  Borda count does not satisfy IIA because of the example we had (so Borda count is “unfair” in this way)

13.  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 voters did not change their preference regarding Bush vs. Gore, but the winner changed  Plurality also does not satisfy IIA