Quick Review of Preference Schedules

Transcription

1 Quick Review of Preference Schedules An election is held between four candidates, A, B, C, and D. The preference ballots of 8 voters are shown. A A A C A C C A B C B D B D D C C D C B C B B D D B D A D A A B Recording each individual ballot can be inefficient. A preference schedule is a more compact way of representing the same information A A C B C D C D B D B A Paul Koester () MA 111, Plurality Method January 18, / 13

2 Quick Review of Preference Schedules A A C B C D C D B D B A The ballot [A, B, C, D] appeared in the original list three times. This same ballot is recorded once in the preference schedule, along with the multiplicity, 3. We can easily read information from the preference schedule. For example, A receives the most first place votes (3 + 2 = 5). Paul Koester () MA 111, Plurality Method January 18, / 13

3 Quick Review of Preference Schedules We can also record the information from the ballots in a different way: A B C D First Second Third Fourth There doesn t seem to be an official name attached to this type of table. I will refer to this as a Gary Table, in honor of a former student who suggested using this method instead of preference schedules. Last time, we saw that the sum across each row, and the sum along each column, will always be equal to the number of voters. Paul Koester () MA 111, Plurality Method January 18, / 13

4 How much information is contained in preference schedules? Suppose an election involves 3 candidates, X, Y, and Z. Any preference ballot must be one of six types. For an election with three candidate, say A, B, and C, voters can cast one of 6 types of linear preference ballots: X X Y Y Z Z Y, Z, X, Z, X, Y Z Y Z X Y X This means that, regardless of the number of voters (even if there are thousands or voters!), all the information in ballots can be recorded in just 6 numbers which appear along the top row of X X Y Y Z Z Y Z X Z X Y Z Y Z X Y X (We usually exclude columns if their multiplicity is 0.) Paul Koester () MA 111, Plurality Method January 18, / 13

5 How much information is contained in preference schedules? Last time, you discovered that in an election involving 4 candidates, there were 24 distinct preference ballots. Where did the 24 come from? How many preference ballots are there in a 5 candidate election? 6 candidate election? N candidate election? You can count the number of ballots in a 4 candidate election as follows: 4: Choose one of the four candidates to be listed at the top 3: Choose one of the remaining three to be listed second 2: Choose one of the remaining two to be list third 1: The remaining candidate must be listed last. Thus, the total number of choices is = 24 Paul Koester () MA 111, Plurality Method January 18, / 13

6 How much information is contained in preference schedules? You can count the number of ballots in a 5 candidate election as follows: 5: Choose one of the five candidates to be listed at the top 4: Choose one of the four candidates to be listed second 3: Choose one of the remaining three to be listed third 2: Choose one of the remaining two to be list fourth 1: The remaining candidate must be listed last. Thus, the total number of choices is = 120 Products of successive whole numbers are called factorials and are denoted with an!. 3! = = 6, 4! = = 24, 5! = In general, a preference schedule for an N candidate election has N! columns. (Again, we usually only list the columns that have multiplicity greater than 0.) Paul Koester () MA 111, Plurality Method January 18, / 13

7 An exercise for next time: It turns out that the Gary Schedules contain LESS information than the traditional preference schedule. As homework, you are asked to construct two DIFFERENT preference schedules which give rise to the SAME Gary schedule: A B C First Second Third WARNING: Be sure to allot enough time for this assignment. You may get lucky and get it right away, or you may need to spend a lot of time doing trial and error. Paul Koester () MA 111, Plurality Method January 18, / 13

8 Determining the Winner of an Election: It turns out that determining the winner of an election depends on BOTH the ballots and how that ballots are tallied. For instance, there was disagreement within our class on the first day concerning which game the ΛΩ fraternity should watch. If you determine the winner with a point system, then they should watch basketball. If you base the winner off of which candidate has the most first place votes, then they should watch football. Which method is the correct one? I m going to dodge that question for a little while. Before passing judgements on different voting methods, we should try to classify the different methods. First, by a voting method, I mean a procedure for determining the winner of an election. In this course, we will focus primarily on voting methods that use preference ballots. Paul Koester () MA 111, Plurality Method January 18, / 13

9 The Plurarlity Method: The simplest voting method is to determine the winner based on who has the most first place votes. This is the Plurality Method. Consider the preference schedule from the beginning of class: A A C B C D C D B D B A In this election, A has 5 first place votes, C has three first place votes, and neither B nor D have any first place votes. Thus, A is the winner, provided we determine the winner using the Plurality Method. Paul Koester () MA 111, Plurality Method January 18, / 13

10 The Plurarlity Method: More formally, we say that a candidate is a plurality candidate if that candidate receives the most number of first place votes. Notice that plurality is being used in two different definitions: a plurality candidate is a candidate that receives the most number of first place votes, whereas the Plurality Method is a method for determining the winner of an election. In the above election, not only did A receive more votes than any other candidate, A actually received more than half of the total first place votes. More formally, we say that a candidate is a majority candidate if that candidate receives more than half of the first place votes. The distinction between a majority and a plurality is very important. Paul Koester () MA 111, Plurality Method January 18, / 13

11 How are Majority and Plurality different? A B C B A B C C A C receives more first place votes than either A or B, so C is a Plurality Candidate. No candidate received more than half of the first place votes (exactly half is 7/2 = 3.5, and since a candidate must receive a whole number of votes, a Majority Candidate would need to receive at least 4 votes) Paul Koester () MA 111, Plurality Method January 18, / 13

12 How are Majority and Plurality different? An election can have multiple plurality candidates (Example?) An election can have AT MOST ONE majority candidate (Why?) A majority candidate is automatically a plurality candidate (Why?) A plurality candidate is not necessarily a majority candidate (like in the previous slide) Paul Koester () MA 111, Plurality Method January 18, / 13

13 Why an election can have at most one majority candidate: Suppose A is the majority candidate. Thus, A receives more than half of the first place votes. This means that LESS THAN HALF of the first place votes are left to be divided amongst the remaining candidates. Thus even if candidate B receives all of the remaining votes, B will still have fewer than half of the first place votes. Why a majority candidate is automatically a plurality candidate: Suppose A is the majority candidate. Thus, A receives more than half of the first place votes. This means that LESS THAN HALF of the first place votes are left to be divided amongst the remaining candidates. Each of the remaining candidates have less than half of the first place votes, and in particular, they have fewer votes than A Paul Koester () MA 111, Plurality Method January 18, / 13