Lecture 1: Voting and Election Math Math for Liberal Arts Lecture 1: Voting and Election Math
Voting and Fairness The very essence of democracy? Majority Rule. In an election with two candidates, the candidate with more than half the votes wins. We call more than half the votes a majority. But what if there are more than two choices? What is a fair way to decide the outcome of an election?
MAC Election The 37 members of the Math Anxiety Club (MAC) cast ballots for President. There are four candidates: Alisha (A) Boris (B) Carmen (C) Dave (D)
Election Results Number of voters Favorite Candidate 14 11 8 4 A C D B Who has won?
Capture all the voters’ preferences with a preference ballot.
Plurality Method In the plurality method, the choice with the most first-place votes wins.
Winner: ______________ Number of voters 14 10 8 4 1 1st choice A C D B 2nd choice 3rd choice 4th choice Candidates A B C D 1st place votes Winner: ______________
Winner: ______________ Number of voters 14 10 8 4 1 1st choice A C D B 2nd choice 3rd choice 4th choice Candidates A B C D 1st place votes 14 4 11 8 Winner: ______________
Math Lovers Club Election The 11 members of the Math Lovers Club choose a president from among four candidates by preference ballot.
Number of voters 6 2 3 1st choice A B C 2nd choice D 3rd choice 4th choice Candidate A has 6 first place votes, a majority of the 11 voters. A majority (more than half the votes) is automatically a plurality (the most votes).
Majority Criterion If there is an alternative that is the first place choice of a majority of voters, then that alternative should be the winner of the election. The plurality method satisfies the majority criterion. However, there are some problems with the plurality method.
Problems with Plurality Method The Bowl Games The 100 members of the University Band must decide in which of five bowl games to march in: Rose Bowl (R) Hula Bowl (H) Cotton Bowl (C) Orange Bowl (O) Sugar Bowl (S)
Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Most of the band members list the Rose Bowl last, yet it wins by the plurality method.
One-to-One Comparison Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Comparison Result H versus R H versus C H versus O H versus S
One-to-One Comparison Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Comparison Result H versus R 51 – 49 H H versus C H versus O H versus S
One-to-One Comparison Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Comparison Result H versus R 51 – 49 H H versus C 97 – 3 H H versus O H versus S
One-to-One Comparison Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Comparison Result H versus R 51 – 49 H H versus C 97 – 3 H H versus O 100 – 0 H H versus S
One-to-One Comparison Number of voters 49 48 3 1st choice R H 2nd choice S C 3rd choice O 4th choice 5th choice Comparison Result H versus R 51 – 49 H H versus C 97 – 3 H H versus O 100 – 0 H H versus S
Note that the Hula Bowl wins all the one-to-one comparisons with other alternatives.
Insincere Voting (Strategic Voting) Actual Preferences Strategic Votes Number of voters 49 48 3 1st choice R H C 2nd choice S 3rd choice O 4th choice 5th choice Number of voters 49 48 3 1st choice R H 2nd choice S C 3rd choice O 4th choice 5th choice
Condorcet Candidate An alternative that wins on a one-to-one comparison with every other alternative is called the Condorcet candidate. In the Bowl Election, the Hula Bowl is the Condorcet candidate. An election can have at most one Condorcet candidate. An election may not have any Condorcet candidate.
Condorcet Criterion If there is an alternative that wins in a one-to-one comparison between it and every other alternative, then that alternative should be the winner of the election. The plurality method violates the Condorcet criterion.
Example with No Condorcet Candidate The mathematics faculty at Southside University hold an election to select a chair of the department from among three candidates (A, B, and C). The 9 preference ballots are tabulated below. Number of voters 4 3 2 1st choice A B C 2nd choice 3rd choice
Number of voters 4 3 2 1st choice A B C 2nd choice 3rd choice Comparison Result A versus B 6 – 3 A A versus C 4 – 5 C B versus C 7 – 2 B Since no candidate won all of her one-to-one comparisons, there is no Condorcet candidate.
Paradox of Voting Even though individual preferences are transitive, group preferences are not always transitive. As a group, the SU Math faculty prefer A to B and prefer B to C, but they do not prefer A to C! (Which they would if transitivity held.) This violation of transitivity for group preferences is often called the Paradox of Voting.
Borda Count Method Borda point assignment for n alternatives In the Borda Count Method each place on the ballot is assigned points. The alternative receiving the most points wins. Last place 1 point Next-to-last place 2 points Third place n-2 points Second place n-1 points First place n points Borda point assignment for n alternatives
MAC Election Borda Count Nbr of voters 14 10 8 4 1 1st choice:___ A C D B 2nd choice:___ 3rd choice:___ 4th choice:___
2015 MLB MVP Voting Based on 14-9-8-7-6-5-4-3-2-1 scoring system Player Team 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th *Points Bryce Harper Nationals 30 420 Paul Goldschmidt Diamondbacks 18 3 1 2 234 Joey Votto Reds 6 9 4 175 Anthony Rizzo Cubs 7 162 Andrew McCutchen Pirates 5 139 Jake Arrieta 134 Zack Greinke Dodgers 130 Nolan Arenado Rockies 102 Buster Posey Giants 84 Clayton Kershaw 49 Based on 14-9-8-7-6-5-4-3-2-1 scoring system
Problems with Borda Count Math Lovers Club Election Nbr of voters 6 2 3 1st choice: 4 A B C 2nd choice: 3 D 3rd choice: 2 4th choice: 1
Math Lovers Club Election Borda Counts A = 6(4) + 0(3) + 0(2) + 5(1) = 29 B = 2(4) + 6(3) + 3(2) + 0(1) = 32 C = 3(4) + 2(3) + 6(2) + 0(1) = 30 D = 0(4) + 3(3) + 2(2) + 6(1) = 19 Why might A have a problem with this outcome?
Observations Disadvantages of the Borda Count Method Borda count method violates the majority criterion. Consequently, Borda count method violates the Condorcet criterion. Advantages of the Borda Count Method Borda Count Method uses all the available voter preferences, not just first choices. Borda Count Method often produces the best compromise winner.
Summary So far, we have two voting methods applied to the MAC election and two different winners! Voting Method Winner Plurality Alisha Borda Count Boris
Plurality-with-Elimination Method Round 1 Count 1st place votes for each candidate. If a candidate has a majority, then that candidate is the winner. Otherwise, eliminate candidate(s) with fewest 1st place votes and simplify preference schedule.
Plurality-with-Elimination Method Round 2 Count 1st place votes for each candidate. If a candidate has a majority, then that candidate is the winner. Otherwise, eliminate candidate(s) with fewest 1st place votes and simplify preference schedule.
Plurality-with-Elimination Method Rounds 3, 4, etc.: Repeat above steps; eventually, some candidate will have a majority of 1st place votes.
MAC Election: Round One A = 14 B = 4 C = 11 D = 8
Round 2: B dropped C: D: Nbr of voters 14 10 8 4 1 1st choice:___ A: 2nd choice:___ 3rd choice:___
Round 2: Simplified more Nbr of voters 14 11 12 1st choice:___ A: C: D: 2nd choice:___ 3rd choice:___ A = 14 C = 11 D = 12
Round 3: C dropped D: Nbr of voters 14 11 12 1st choice:___ A: 2nd choice:___
Round 3: Simplified more Nbr of voters 14 23 1st choice:___ A: D: 2nd choice:___ A = 14 D = 23 D Wins
Example: Young Mathematicians Election The UAB Young Mathematicians elect a president of their club from among five candidates A, B, C, D, and E using the plurality-with-elimination method. There are 24 preference ballots. Number of voters 8 6 2 3 5 1st choice A B C D E 2nd choice 3rd choice 4th choice 5th choice
Number of voters 8 6 2 3 5 1st choice A B C D E 2nd choice 3rd choice 4th choice 5th choice What is the maximum number of rounds that might be needed to decide the winner of this election by the plurality-with-elimination method?
A = 8 B = 6 C = 2 D = 3 E = 5 Number of voters 8 6 2 3 5 1st choice A 2nd choice 3rd choice 4th choice 5th choice A = 8 B = 6 C = 2 D = 3 E = 5
A = 10 B = 6 D = 3 E = 5 Number of voters 8 6 2 3 5 1st choice A B D E 2nd choice 3rd choice 4th choice A = 10 B = 6 D = 3 E = 5
A = 10 B = 6 E = 8 Number of voters 8 6 2 3 5 1st choice A B E 2nd choice 3rd choice A = 10 B = 6 E = 8
A = 10 E = 14 E WINS Number of voters 8 6 2 3 5 1st choice A E 2nd choice A = 10 E = 14 E WINS
And the Best Actor Oscar goes to… To determine the magic number for nomination, take the total number of ballots received for a particular category and divide it by the total possible nominees plus one. Example: 600 potential ballots for the Best Actor category, divide that by 6 (5 possible nominees plus 1), thus making the magic number for the category 100 ballots to become an official nominee.
The starts based on a voter’s first choice selection until someone reaches the magic number. The ballots that named him as a first choice are then all set aside, and there are now four spots left for the Best Actor category. The actor with the fewest first-place votes is automatically knocked out, and those ballots are redistributed based on the voters' second place choices The counting continues, and actors or different categories rack up redistributed votes until all five spots are filled.
Problems with Plurality-with-Elimination Results of the Young Mathematicians Student of the Year Award Election Number of voters 7 8 10 4 1st choice A B C 2nd choice 3rd choice A = 11 B = 8 C = 10
Problems with Plurality-with-Elimination Number of voters 7 8 10 4 1st choice A C 2nd choice A = 11 C = 18
Because of election irregularities, the original election is declared void. Meanwhile, candidate C convinces the 4 voters represented by the last column of the preference schedule that she is better than candidate A. They switch their preference order to C, A, B. The reelection results are as follows.
Problems with Plurality-with-Elimination Results of the Young Mathematicians Student of the Year Award Election Number of voters 7 8 10 4 1st choice A B C 2nd choice 3rd choice A = 7 B = 8 C = 14
Problems with Plurality-with-Elimination Number of voters 7 8 10 4 1st choice B C 2nd choice B = 15 C = 14 C added first place votes, but lost the election
Monotonicity Criterion If an alternative X is the winner of an election, and, in a reelection, all the voters who change their preferences do so in a way that is favorable only to X, then X should still be the winner of the election. The plurality-with-elimination method violates the monotonicity criterion. The plurality-with-elimination method also violates the Condorcet criterion.
Fairness Criterion Voting Method Satisfied Violated Plurality Majority Monotonicity Condorcet Borda Count Monotonicity Majority Condorcet Plurality-with- Elimination Majority Monotonicity Condorcet
Plurality-with- Elimination Voting Method Winner Plurality Alisha Borda Count Boris Plurality-with- Elimination Dave In the MAC election, we have used three voting methods and have three different winners!
Method of Pairwise Comparisons Match each candidate on a one-to-one basis with every other candidate. Points Suppose X is compared with Y. If X wins the comparison, then X gets 1 point and Y gets 0. If Y wins the comparison, then Y gets 1 point and X gets 0. In case of a tie, both X and Y get ½ point. Winner. After all pairwise comparisons have been made, the candidate with the most points is the winner.
Pairwise Comparison This is similar to a round-robin tournament, where each player plays every other player to decide the winner. The method of pairwise comparisons is one of several so-called Condorcet methods. It is also called Copeland’s Method.
MAC Election Number of voters 14 10 8 4 1 1st choice A C D B 2nd choice 3rd choice 4th choice
MAC Election There are four candidates in the MAC election (A, B, C, D) There are 6 pairwise comparisons that must be made: A vs B B vs C C vs D A vs C B vs D A vs D
Nbr of voters 14 10 8 4 1 1st A C D B 2nd 3rd 4th Comparison Result Points A vs B A vs C A vs D B vs C B vs D C vs D
Nbr of voters 14 10 8 4 1 1st A C D B 2nd 3rd 4th Comparison Result Points A vs B 14 – 23 B = 1 A vs C C = 1 A vs D D = 1 B vs C 18 – 19 B vs D 28 – 9 C vs D 25 – 12 A = 0 B = 2 C = 3 D = 1 C Wins
Best won-lost-tied percentage in games played within the division. TO BREAK A TIE WITHIN A DIVISION If two or more clubs in the same division finish with identical won-lost-tied percentages, the following steps will be taken until a champion is determined. Head-to-head (best won-lost-tied percentage in games between the clubs). Best won-lost-tied percentage in games played within the division. Best won-lost-tied percentage in common games. Best won-lost-tied percentage in games played within the conference. Strength of victory. Strength of schedule. Best combined ranking among conference teams in points scored and points allowed. Best combined ranking among all teams in points scored and points allowed. Best net points in common games. Best net points in all games. Best net touchdowns in all games. Coin toss
Summary of MAC Election Results Voting Method Winner Plurality Alisha Borda Count Boris Plurality-with- Elimination Dave Pairwise Comparisons Carmen In the MAC election, we have used four voting methods and have four different results!
Problems with Pairwise Comparisons Number of voters 1 4 3 1st choice A C E 2nd choice B D 3rd choice 4th choice 5th choice
A Wins Comparison Result Points A vs B 5 – 4 A = 1 A vs C A vs D 5 – 4 A = 1 A vs C A vs D A vs E 1 – 8 E = 1 B vs C 7 – 2 B = 1 A = 3 B = 2 C = 2 D = 2 E = 1 B vs D 4 – 5 B = 1 B vs E 5 – 4 B = 1 C vs D C = 1 C vs E D vs E D = 1 A Wins
Because of an election irregularity, the votes have to be recounted. Meanwhile, B, C, and D drop out of the election. The new preference schedule is much simpler. Nbr of voters 1 8 1st A E 2nd E Wins
Independence of Irrelevant Alternatives Criterion If an alternative X is the winner of an election, and one or more of the other alternatives are removed and the ballots recounted, then X should still be the winner of the election. The method of pairwise comparisons violates the independence of irrelevant alternatives criterion. The method of pairwise comparisons satisfies the majority, Condorcet, and monotonicity criteria.
Conflict with Borda Count An even more serious problem with pairwise comparisons is revealed by the preceding example. Let us apply the Borda count method to it. Numberof voters 1 4 3 1st choice: 5 A C E 2nd choice: 4 B D 3rd choice: 3 4th choice: 2 5th choice: 1 A = 1(5)+4(4)+4(1) = 25 B = 1(4)+7(3)+1(2) = 27 C = 4(5)+1(3)+4(1) = 27 D = 4(4)+1(3)+4(2)= 27 E = 4(5)+4(2)+1(1)= 29
Observations A comes in last by Borda count, exactly the opposite order as with pairwise comparisons Pairwise comparisons: A, B-C-D tied, E. The Borda count winner E is preferred to the pairwise comparisons winner A by 8 of the 9 voters. This overwhelming margin of 8 to 1 is picked up by Borda count, but is missed by the method of pairwise comparisons. We call A a disliked winner since 8 of the 9 voters prefer E
More Problems with Pairwise Comparisons Example: Deadlock Number of voters 3 2 4 1st choice A B C 2nd choice 3rd choice Comparison Result Points A vs B 7 – 2 A = 1 A vs C 3 – 6 C = 1 B vs C 5 – 4 B = 1
Observations Advantages of the method of pairwise comparisons Pairwise comparisons satisfies The Condorcet criterion The majority criterion The monotonicity criterion Pairwise comparisons uses all the information provided by the voters.
Observations Disadvantages of the method of pairwise comparisons Violates the independence of irrelevant alternatives criterion. Often is indecisive, producing a deadlock. Can conflict spectacularly with Borda count, producing a disliked winner. The number of comparisons mounts fast as the number of candidates increases.
Fairness Criterion Voting Method Satisfied Violated Plurality Majority Monotonicity Condorcet Independence Irrelevant Alternatives Borda Count Monotonicity Majority Plurality-with- Elimination Pairwise Comparisons
Assumptions Voting is between two alternatives: “yes” or “no” The motion passes if the sum of the “yes” votes exceeds the quota. The motion fails if the sum of the “yes” votes does not exceed the quota. A motion cannot be deadlocked. It either passes or fails. Everyone must vote (no abstentions). There are 3 or more players.