Warm-Up Rank the following soft drinks according to your preference (1 being the soft drink you like best and 4 being the one you like least)  Dr. Pepper  Pepsi  Mt. Dew  Sprite

Election Theory Basics of Election Theory

How do we elect officials? Sometimes it is necessary to rank candidates instead of selecting a single candidate. We can summarize votes into a preference schedule.

Preference Ballot: a ballot in which voters are asked to rank the candidates in order There are 37 ballots, therefore 37 people voted

Preference Schedule: a table that organizes the ballots Number of voters st choiceACDBC 2 nd choiceBBCDD 3 rd choiceCDBCB 4 th choiceDAAAA

The Methods 1. Plurality 2. Borda Count 3. Pairwise Comparison (Copeland) 4. Plurality with Elimination (Hare) 5. Approval 6. Sequential Pairwise You will work with a group to prepare a lesson on your method. Must include:  Explanation  Example done for the class  Example for the class to do that you will go over.  What fairness criteria is broken?

7 The Mathematics of Voting Majority Majority The candidate with a more than half the votes should be the winner. Majority candidate Majority candidate The candidate with the majority of 1 st place votes.

The Plurality Method: if X has the most first-place votes, then X is the winner. X does not have to have a majority of 1 st place votes. A is the winner with 14 votes R is the winner with 49 votes

Example st ABBCD 2 nd CDCAC 3 rd BCADB 4 th DADBA 1) How many candidates? 4 2) How many people voted? 27 3) Which candidate has the most first- place votes? Is it a majority or plurality? B, Plurality

10 The Mathematics of Voting In the Borda Count Method Method each place on a ballot is assigned points. In an election with N candidates we give 1 point for last place, 2 points for second from last place, and so on.

11 The Mathematics of Voting Borda Count Method At the top of the ballot, a first-place vote is worth N points. The points are tallied for each candidate separately, and the candidate with the highest total is the winner. We call such a candidate the Borda winner.

12 The Mathematics of Voting Borda Count Method A gets 4(14)+1(10)+1(8)+1(4)+1(1) = 81 points B gets 3(14)+3(10)+2(8)+4(4)+2(1) = 106 points C gets 2(14)+4(10)+3(8)+2(4)+4(1) = 104 points D gets 1(14)+2(10)+4(8)+3(4)+3(1) = 81 points B is the winner!!!

The Plurality-with Elimination Method (Hare) Steps: 1) Count the first place votes for each candidate. If a candidate has a majority of the first-place votes, that candidate is the winner. 2) If there isn’t a candidate that has the majority of votes then, Cross out the candidate (or candidates if there is a tie) with the fewest first-place votes 3) Move other candidates up and count the number of the first-place votes again. If a candidate has a majority votes, that candidate is the winner. Otherwise, continue the process of crossing names and counting the first- place votes. 37 people voted so the majority would need 19 votes

Example 1: 37 VOTERS, need 19 votes for majority winner Number of voters st choiceACDBC 2 nd choiceBBCDD 3 rd choiceCDBCB 4 th choiceDAAAA Number of voters st choiceADDDD 2 nd choiceDAAAA Number of voters st choiceACDDC 2 nd choiceCDCCD 3 rd choiceDAAAA 4 th choice Step 1: No one receives 19 votes, so eliminate B and rewrite the table Step 2: No one with 19 votes yet, so eliminate C and re-write the table Step 3: D has 23 votes so D is the winner The Plurality-with Elimination Method

15 The Mathematics of Voting The Method of Pairwise Comparisons (Copeland) In a pairwise comparison between X and Y every vote is assigned to either X or Y,Y, the vote got in to whichever of the two candidates is listed higher on the ballot. The winner is the one with the most votes; if the two candidates split the votes equally, it ends in a tie.

16 The Mathematics of Voting The Method of Pairwise Comparisons The winner of the pairwise comparison gets 1 point and the loser gets none; in case of a tie each candidate gets ½ point. The winner of the election is the candidate with the most points after all the pairwise comparisons are tabulate.

17 The Mathematics of Voting The Method of Pairwise Comparisons There are 10 possible pairwise comparisons: A vs. B, A vs. C, A vs. D, A vs. E, B vs. C, B vs. D, B vs. E, C vs. D, C vs. E, D vs. E

18 The Mathematics of Voting The Method of Pairwise Comparisons A vs. B: B wins B gets 1 point. A vs. C: A wins A gets 1 point. etc. Final Tally: A-3, B-2.5, C-2, D-1.5, E-1. A wins.

Sequential Pairwise Voting  Sequential pairwise voting starts with an agenda and pits the first candidate against the second in a one-on-one contest.  The loser is deleted and the winner then moves on to confront the third candidate in the list, one on one.  This process continues throughout the entire agenda, and the one remaining at the end wins. Example: Who would be the winner using the agenda A, B, C, D for the following preference list ballots of three voters? 19 RankNumber of Voters (3) FirstACB SecondBAD ThirdDBC FourthCDA Using the agenda A, B, C, D, start with A vs. B and record (with tally marks) who is preferred for each ballot list (column). A vs. B II I A vs. C I II C vs. D I II A wins; B is deleted. C wins; A is deleted. D wins; C is deleted. Candidate D wins for this agenda.

Approval Voting  Under approval voting, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable.  No limit is set on the number of candidates for whom an individual can vote; however, preferences cannot be expressed.  Voters show disapproval of other candidates simply by not voting for them. 20

Approval Voting (cont)  The winner under approval voting is the candidate who receives the largest number of approval votes.  This approach is also appropriate in situations where more than one candidate can win,  EX: in electing new members to an exclusive society such as the National Academy of Sciences or the Baseball Hall of Fame.  Approval voting is also used to elect the secretary general of the United Nations.  Approval voting was proposed independently by several analysts in 1970s. 21

First basic fairness criterion The Majority Criterion: if X has the majority of the first-place votes (more than half), then X is the winner. The plurality method satisfies the majority criterion.

The Condorcet Criterion was introduced in 1785 by the French mathematician Le Marquis de Condorcet If candidate X is preferred over other candidates in a head-to-head comparison, then X is the winner If X is the winner under the Majority Criterion, then X is also the Condorcet winner. Second basic fairness criterion

Third basic fairness criterion 3. If a candidate is winning & votes are changed in FAVOR of the winner Monotonicity Criterion: If votes are changed in favor of the winning candidate, the winner should not change.

25 Monotonicity (The Hare system fails monotonicity.)  Monotonicity says that if a candidate is a winner and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain a winner.  In a new election, if a voter moves a winner higher up on his preference list, the outcome should still have the same winner. Number of Voters (13) Rank5431 FirstACBA SecondBBCB ThirdCAAC In this example, A won because A has the most 1 st place votes. Round 1: B is deleted with Hare method because B has the fewest 1 st place votes. Round 2: C moves up to replace B on the third column. However, C wins because now has the most 1 st place votes—this is a glaring defect! Number of Voters (13) Rank5431 FirstACCA SecondCAAC

Fourth basic fairness criterion What if a non-winning candidate drops out? Independence-of-Irrelevant Alternatives Criterion: If a non-winning candidate drops out, or is disqualified, the winner should not change. RankNumber of Voters (5) First (3 pts)AAABB Second (2pts)BBBCC Third (1 pt)CCCAA 26 RankNumber of Voters (5) First (3pts)AAACC Second (2 pts)BBBBB Third (1 pt)CCCAA Original Borda Score: A=11, B=10, C=9 Suppose the last two voters change their ballots (reverse the order of just the losers). This should not change the winner. New Borda Score: A= 11, B=12, C=8 B went from loser to winner and did not switch with A!

Summary MajorityCondercetMonotonicityIndependence Plurality YesNoYesNo Borda Count No YesNo Plurality Elimination YesNo Pairwise Comparison Yes No