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Math for Liberal Studies

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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

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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

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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

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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

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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

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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

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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

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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 > Juice30186 5Soda > Juice > Milk52515 4Juice > Soda > Milk41220

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Milk gets 39 points Soda gets 55 points Juice gets 41 points Soda wins! Number of Voters Preference OrderMilkSodaJuice 6Milk > Soda > Juice30186 5Soda > Juice > Milk52515 4Juice > Soda > Milk41220

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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

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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

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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

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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

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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

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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

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

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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.

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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

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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

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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

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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!

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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

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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

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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

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1.1, 1.2 Ballots and Plurality Method

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