Voting Theory Lecture Notes + Videos PDF

Voting Theory Introduction In this module, we are going to learn about voting theory. Voting theory is the mathematics used to determine in a fair way the winner of a vote or an election. There are several ways to take a vote and many ways to count the votes to determine the winner because no method...

Voting Theory Introduction In this module, we are going to learn about voting theory. Voting theory is the mathematics used to determine in a fair way the winner of a vote or an election. There are several ways to take a vote and many ways to count the votes to determine the winner because no method is ideal. This icon lets you know there is a video covering the Preference Schedule Voting associated material. Simply click on the icon to view the video. A traditional ballot allows you to choose one person to vote for. The votes are counted, or tallied up. There are very few ways to determine who the winner is for a traditional ballot vote. A majority winner has more than 50% of the votes. If no candidate has more than 50% of the vote, there is no majority winner. A plurality winner has more votes than any other candidate, regardless of percent. Ties are possible, and would have to be settled through some sort of run-off vote. This method is the popularity contest method. In order to allow for other methods of determining the winner of a vote, we need to introduce preference schedules, or ranked choice voting. A preference ballot is a ballot in which the voter ranks the choices in order of preference. Below is an example of a selection of preference votes for the destination of a vacation club. Hawaii (H), Orlando (O) and Anaheim (A) are the choices on the ballot. Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim 1 choice st A A O H A O H O H A 2nd choice O H H A H H A H A H 3rd choice H O A O O A O A O O These individual ballots are typically combined into one preference schedule, as follows: 1 3 3 3 1 choice st A A O H 2nd choice O H H A 3rd choice H O A O Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: 1 + 3 + 3 + 3 = 10 total votes. Also note that each column represents at least one vote, and often many more. Although this type of preference schedule allows for many more strategies for determining a winner, it can still be used to determine the plurality winner, or majority winner, if they exist. Example: To determine the plurality winner, we only count the 1st choice votes. A has 1 from the first column and 3 from the second column, so A has 1 + 3 = 4 votes. O has 3 votes, and H has 3 votes. The plurality winner is A. 4 A has the most votes, so they are the only possible majority candidate. However, = 40%, and since this is 10 less than 50%, there is no majority winner. Instant Runoff Voting (IRV)/Plurality with Elimination This method of counting votes is similar to the situations made popular on the Survivor TV series, as well as the Weakest Link gameshow. After counting votes, if there is no majority winner, the candidate with the least number of 1st place votes is eliminated from the ballot. Everyone below this candidate moves up on the preference schedule. The counting is repeated, and if there is a majority winner, they win the election. If not, the next weakest link is eliminated, and the process repeats. Eventually there are only two candidates left, and the majority winner wins the election. It is possible for there to be a tie, in which case this method is inconclusive, and another method is used. Example: Find the winner of the following election using Plurality-with-elimination (Instant Runoff Voting). Number of voters 12 13 21 7 48 20 By adding up the top row, we can see the number of votes is 12 + 13 + 21 + 7 + 48 + 20 = 121 votes. For a majority, we can 1st choice A A C B D C work out 50% of 121 to get (0.5)(121) = 60.5. So, 61 1st place 2nd choice D B B C A A votes are needed for a majority winner. 3rd choice B C A A B B Candidate A has 12 + 13 = 25 1st place votes. To see this, notice 4th choice C D D D C D A is on the top of the first and second columns. B is only on top of the 4th column, so B has 7 votes. Your turn: How many 1st place votes do the following have? A: ____ 25____ 7 B: _____ _____ C: __________ D: __________ As you can see, no candidate has the required 61 votes for a majority win in this election. So, we move to round 2: Throw B off the island!!! Your instructor might have some creative ways to notate this process, but to make this as clear as possible, we will show the entire resulting preference schedule: Number of voters 12 13 21 7 48 20 It is time for the recount. 1st choice A A C C D C A: ____ 25____ C: ____ 48____ D: __________ 2nd choice D C A A A A 3rd choice C D D D C D Once again, no candidate has the required 61 votes for a majority win in this election. So, we move to round 3: Throw A off the island!!! Number of voters 12 13 21 7 48 20 It is time for the recount. 1st choice D C C C D C C: ____ 61____ D: ____ 60______ 2nd choice C D D D C D At last, we have a candidate with the required 61 votes for the majority, so by IRV, C is the winner!!! Can you imagine being candidate D, and having the most 1st place votes, and losing? That must suck. Your turn: Find the winner of the following election using Plurality-with-elimination (Instant Runoff Voting). Number of voters 13 10 17 14 Total number of votes: ________________ 1st choice B D C A Number of votes needed for a majority: ________________ 2nd choice A B B D A: __________ B: __________ 3rd choice D C A C C: __________ D: __________ 4th choice C A D B Number of voters 13 10 17 14 1st choice A: __________ B: __________ 2nd choice C: __________ D: __________ 3rd choice Number of voters 13 10 17 14 1st choice A: __________ B: __________ 2nd choice C: __________ D: __________ Winner: _____ Borda Count The Borda Count method is a method of counting votes that takes into consideration all of the votes cast, including 2nd place, 3rd place, etc. This method is very common in selecting All Star participation in major league sports. This is a type of weighted voting where a certain number of points are given to different votes. The weights we will use will be 1 point for last place, 2 points for second to last place, etc. until the most points for 1st place. We then add up the points for each candidate and the candidate with the most points wins the election. Again, if there is a tie, another voting method is required to determine a winner. To remember this method, it takes the most amount of arithmetic, and some students get “borda” this method… Example: Find the winner of the following election using the Borda Count method. Number of voters 10 19 20 15 7 For our counting, we assign points as follows: 1st choice D A B C D 4th place votes are worth 1 point. 2nd choice C C A D A 3rd place votes are worth 2 points. 3rd choice A B D B B 2nd place votes are worth 3 points. 4th choice B D C A C 1st place votes are worth 4 points. Now we compute the Borda count for each candidate. We will do this column by column, though this is not the only way to organize the counting: For candidate A, the points are as follows: 10(2) + 19(4) + 20(3) + 15(1) + 7(3) = 𝟏𝟏𝟏𝟏𝟏𝟏 points. The first number represents the number of votes for the column, and the second number represents points per vote. For candidate B, the points are as follows: 10(1) + 19(2) + 20(4) + 15(2) + 7(2) = 𝟏𝟏𝟏𝟏𝟏𝟏 points. Notice that the first numbers (10, 19, 20, 15, 7) are the same, since the same voters are involved. The difference is in the value of the votes for candidate B. For candidate C, the points are as follows: 10(3) + 19(3) + 20(1) + 15(4) + 7(1) = 𝟏𝟏𝟏𝟏𝟏𝟏 points. For candidate D, the points are as follows: 10( ) + 19( ) + 20( ) + 15( ) + 7( ) = __________ points. As you can see, the candidate with the most points is Candidate A, the winner by the Borda Count method. Your turn: Determine the winner using the Borda Count method.: Number of voters 6 8 1 3 4 7 1st choice C A B D D C 2nd choice D B D C A B 3rd choice B C A A C A 4th choice A D C B B D Points for A: _________Points for B: ________ Points for C: ________ Points for D: ________ Winner: ____________ Pairwise Comparison/Copeland’s Method The pairwise comparison method, or Copeland’s method, is about comparing only one pair of candidates at a time to determine who is the most preferred. A point system is used to assign points to the winner of the pairwise comparison. 1 The winner gets 1 point. The loser gets 0 points. If there is a tie, each candidate gets point. After every possible pair is 2 considered and points are assigned, the winner is the one with the most points. A tie is possible, and another voting method would need to be used. Due to the small value of points assigned, you will have winners that have 2 points, unlike the Borda Count winners with large numbers of points. For example: Find the winner of the following election under Pairwise Comparison (Copeland's Method). Number of voters 13 10 17 14 1st choice B D C A With 4 candidates, there are 6 possible pairs to compare. They 2nd choice A B B D are as follows: (A, B), (A, C), (A, D), (B, C), (B, D), and (C, D). Since there are 6 comparisons, there will be a total of 6 points 3rd choice D C A C assigned in this voting method. 4th choice C A D B (A, B): To determine who wins this comparison, we need to determine which candidate is preferred among the voters in this election. We can proceed by column. For the 13 voters in the first column, they prefer B over A, since B is higher than A in that column. The 10 voters in the next column prefer B. The next 17 voters prefer B. Finally, the 14 voters in the last column prefer A. The total votes for A would be 14, and for B would be 13 + 10 + 17 = 40. (A, B): A: 14 votes B: 40 votes B wins: A gets 0 points. B gets 1 point. 1 1 (A, C): A gets 13 + 14 = 27 votes; C gets 10 + 17 = 27 votes. Tie vote: A gets point. C gets point. 2 2 (A, D): A gets 13 + 17 + 14 = 44 votes; D gets 10 votes. A wins: A gets 1 point. D gets 0 points. (B, C): B gets 13 + 10 = 23 votes; C gets 17 + 14 = 31 votes. C wins: B gets 0 points. C gets 1 point. (B, D): B gets _____________votes; D gets ___________ votes. ________: B gets ________ D gets _______ (C, D): C gets _____________votes; D gets ___________ votes. ________: C gets ________ D gets _______ Now, we can determine the total points for each candidate: 1 1 A has 0 + + 1 = 1 points. B has 1 + 0 + 1 = 2 points. C has __________ points. D has __________ points. 2 2 Under the Pairwise Comparison (Copeland's Method), the winner is candidate B. Notice that with 4 candidates, each candidate is in three comparisons, meaning the maximum points a candidate could have is 3. In order for a candidate to have 3 points, they would have to win all three of their pairwise comparisons. Such a candidate is called the Condorcet Candidate. Condorcet Candidate The Condorcet Candidate, or Condorcet Winner, is the candidate that is preferred in every one-to-one comparison with the other candidates. As we have seen, if there are four candidates, each candidate will have three comparisons. If a candidate wins all three comparisons, that is if they are preferred over the other three candidates in one-to-one comparisons, they are the Condorcet Candidate. They will also have the maximum of 3 points in the Pairwise Comparison method and would win under that voting method. The Condorcet Candidate ALWAYS wins the Pairwise Comparison voting method. Note that you don’t have to determine the number of points a candidate has in the Pairwise Comparison method to determine if that candidate is the Condorcet Winner. You just need to verify that they are preferred to each of the other candidates in one-to-one comparisons. Your Turn: If there are 5 candidates, how many one-to-one comparisons does each candidate have? ____________ How many of those comparisons does a candidate need to win in order to be the Condorcet Candidate? ____________ If using the Pairwise Comparison voting method, how many points are assigned in total? ____________ How many points would the Condorcet Candidate earn? ____________ Fairness Criteria The Fairness Criteria are statements that seem like they should be true in a fair election. Condorcet Criterion: The Condorcet Candidate, or Condorcet Winner, should win the election. In the following election, the Condorcet Candidate does not win under the plurality method. This doesn’t seem fair. 1 3 3 3 1 choice st A A O H 2nd choice O H H A 3rd choice H O A O Monotonicity Criterion: If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning. In the following situation, the results on the left schedule were announced but the physical votes were destroyed before the election was certified, and a second election was held. Those results are in the right schedule. Having seen the results of the left schedule, 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter. In other words, they changed their voting preference to favor the announced winner. Bandwagon fans, one and all… 37 22 12 29 47 22 2 29 1 choice st Adams Brown Brown Carter 1 choice st Adams Brown Brown Carter 2nd choice Brown Carter Adams Adams 2nd choice Brown Carter Adams Adams 3rd choice Carter Adams Carter Brown 3rd choice Carter Adams Carter Brown If you use the IRV voting method before and after, the winner changes from Adams to Carter, even though the change in the schedule was to favor Adams. This doesn’t seem fair. Majority Criterion: If a choice has a majority of first-place votes, that choice should be the winner. In the following election, the majority winner Seattle, loses to Tacoma under the Borda Count method. 51 25 10 14 1 choice st Seattle Tacoma Puyallup Olympia 2nd choice Tacoma Puyallup Tacoma Tacoma 3rd choice Olympia Olympia Olympia Puyallup 4th choice Puyallup Seattle Seattle Seattle  The Independence of Irrelevant Alternatives (IIA) Criterion: If a non-winning choice is removed from the ballot, it should not change the winner of the election. Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A. In the following election, the winner under the Pairwise Comparison method is Carlos, with last place going to Dimitry. Due to Dimitry being not eligible, he is removed, and the winner under Pairwise Comparison shifts to Anna. This doesn’t seem fair! Why would removing an irrelevant alternative (last place) affect the winner? 5 5 6 4 5 5 6 4 1 choice st Dimitry Anna Carlos Brian 1 choice st Anna Anna Carlos Brian 2nd choice Anna Carlos Brian Dimitry 2nd choice Carlos Carlos Brian Anna 3rd choice Carlos Brian Dimitry Anna 3rd choice Brian Brian Anna Carlos 4th choice Brian Dimitry Anna Carlos Arrow’s Impossibility Theorem: Arrow’s Impossibility Theorem states, roughly, that it is not possible for a voting method to satisfy every fairness criteria that we’ve discussed. Since there is no optimal voting method, there is usually a second and even third voting method used in case of ties. These alternate methods are voted on before the election is held. Approval Voting There are situations where instead of ranked voting, simple approval voting is done. For example, to determine which restaurant a family will go to, they might take a vote on who would approve of several different choices. Instead of 1st choice, 2nd choice, etc, just a simple yes or no can be tracked. Dimitry Brian Anna Carlos In this situation, we can see that Denny’s has 3 votes, while Denny’s X X X the other three choices each have 2 votes. Thus, Denny’s Pizza Hut X X would win this vote. KFC X X Wendy’s X X Approval Voting can easily violate the Majority Criterion You Try: Since the voting is given in a preference schedule, make Find the winner under approval voting: sure to count the total approvals along the top row. For example, A has 18 + 19 + 19 = 56 approval votes. Number of voters 18 17 17 19 18 19 A X X X B X X X C X X X D X X X