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MAT 105 Spring 2008. Rank Voting. 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
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MAT 105 Spring 2008 Rank Voting
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 1st place • 3 points for 2nd place • 1 point for 3rd place Soda wins with 55 points!
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 • 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
A Special Kind of Rank Method • 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 1st place is worth 2 points, 2nd place is worth 1 point, and 3rd place is worth 0 points • With 4 candidates, the scoring is 3, 2, 1, 0
Problems with Rank Methods • 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?
Irrelevant Alternatives • Let’s look at an example • We’ll use the Borda countto find the winner of thiselection • 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
Irrelevant Alternatives • 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
Independence of Irrelevant Alternatives (IIA) • 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
Independence of Irrelevant Alternatives (IIA) • 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
Independence of Irrelevant Alternatives (IIA) • 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)
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 • The voters did not change their preference regarding Bush vs. Gore, but the winner changed • Plurality also does not satisfy IIA