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Chapter 1: The Mathematics of Voting

Chapter 1: The Mathematics of Voting. “It’s not the voting that’s democracy; it’s the counting.” -Tom Stoppard. Section 1.1: Preference Ballots and Preference Schedules. Example 1.1.1: The Math Club Election.

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Chapter 1: The Mathematics of Voting

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  1. Chapter 1: The Mathematics of Voting “It’s not the voting that’s democracy; it’s the counting.” -Tom Stoppard

  2. Section 1.1: Preference Ballots and Preference Schedules

  3. Example 1.1.1: The Math Club Election • The Math Club is holding its annual election for president. There are four candidates running for president: Alicia, Boris, Carmen and Dave (A, B, C, and D for short). Each of the 37 members of the club votes by means of a ballot indicating his or her first, second, third and fourth choice. Once the ballots are in, it’s decision time. Who should be the winner of the election? Why?

  4. Example 1.1.1: The Math Club Election (the ballots) 14 10 8 4 1

  5. Preference Ballots and Linear Ballots • Preference ballot: a ballot in which the voters are asked to rank the candidates in order of preference • Linear ballot: a ballot in which ties are not allowed

  6. Example 1.1.1: The Math Club Election (Preference Schedule)

  7. Section 1.2: The Plurality Method The candidate with the most first place votes wins!

  8. Example 1.2.1: The Math Club Election • Alicia clearly got the most first-place votes (she got 14). So, she’s the winner! • The candidate with a majority of the first-place votes (that’s Alicia) is called the majority candidate. • The Majority Criterion: A majority candidate should always win the election. We will define a majority to be more than half of the votes. • Disadvantage to the plurality method: When there are 3 or more candidates, there is no guarantee that there is going to be a majority candidate.

  9. The Condorcet Criterion • The Plurality Method fails to take into consideration a voter’s other preferences beyond first choice. • The Condorcet Criterion: If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.

  10. Example 1.2.2: The Marching Band Election The TSU Marching Band is so good that this coming bowl season they have invitations to perform at 5 different bowl games: The Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 members of the band to decide in which of the five bowl games they will perform. A preference schedule is given on the next slide.

  11. Example 1.2.2 Preference Schedule Which is the plurality candidate?

  12. Example 1.2.2: Marching Band Election • Under the Plurality Method, the Rose Bowl is the winner. Why would this be a bad choice? • Which bowl should be the winner? Look at the head-to-head comparison. • The Hula Bowl is called the Condorcet candidate and, hence, should be the winner.

  13. Section 1.3: The Borda Count Method Each place on a ballot is assigned points

  14. The Borda Count Method • If an election has N candidates, we give 1 point for last place, 2 points for second to last place, … and N points for first place. • The candidate with the highest total is the Borda winner.

  15. Example 1.3.1: The Math Club Election Again

  16. Example 1.3.1 Continued • Tally the points for each candidate. • A: 56+10+8+4+1 = 79 points • B: 42+30+16+16+2 = 106 points • C: 28+40+24+8+4 = 104 points • D: 14+20+32+12+3 = 81 points • Boris is the Borda winner! • Note: Alisha was the plurality winner.

  17. Issues with the Borda Count Method • Page 11 Example 1.6: The School Principal Election • Disadvantages: sometimes violates the Majority criterion and the Condorcet criterion • Advantages: takes into account all the information provided in the preference ballots

  18. Section 1.4: Instant Runoff Voting aka The Plurality-with-Elimination Method or the Hare Method

  19. Runoff Voting • In an election with three or more candidates, there is often no majority winner. • Runoff voting: eliminate the candidate(s) with the fewest first-place votes and hold another “runoff” election. • This is expensive and inefficient.

  20. Instant Runoff Voting (IRV) • Round 1: Count the first-place votes for each candidate. If a candidate has a majority of the first-place votes, then that candidate wins. Otherwise, eliminate the candidate with the fewest first-place votes. • Round 2: Cross out the name(s) of the candidates eliminated from the preference schedule and recount the first-place votes. If a candidate has a majority of the first-place votes, then that candidate win. Otherwise, eliminate the candidate with the fewest first-place votes. • Repeat this process until there is a winner.

  21. Example 1.4.1: Math Club Election Again

  22. Example 1.4.2: The School Principal Election again • There are 12 voters in this election. How many votes does a candidate need to win the majority? • Use IRV to find a winner. • Note that IRV satisfies the majority criterion.

  23. Issues with IRV • Pages 15-16 Example 1.10: There go the Olympics • The Monotonicity Criterion: If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election. • IRV does not satisfy this criterion. It also does not satisfy the Condorcet criterion.

  24. Uses of IRV • Municipal elections in places like San Francisco and Burlington, VT • In Australia to elect members of the House of Representatives • IRV is becoming increasingly popular, despite its discrepancies

  25. Section 1.5: The Method of Pairwise Comparisons Satisfies the Condorcet criterion

  26. Pairwise Comparisons • Pair the candidates, and compare them in a head-to-head match • The winner of the match gets 1 point and the loser gets 0. If there is a tie, each candidate gets 0.5 points. • The winner is the candidate with the most points at the end. • Note: This is how the FIRST Robotics Competitions are counted.

  27. Example 1.5.1: The Math Club Election Yet Again

  28. Issues with Pairwise Comparisons • Pages 18-19 Example 1.12: NFL Draft • Advantages: Satisfies the Majority criterion, Condorcet criterion and monotonicity criterion. • Disadvantages: • Often results in multiple ties for first place • If a non-winning candidate withdraws, it could change the winner of the election. (IIA Criterion)

  29. Independence-of-Irrelevant Alternatives (IIA) Criterion • If candidate X is a winner of an election and, in a recount, one of the non-winning candidates withdraws or is disqualified, then X should still be a winner of the election.

  30. Example 1.5.2: A Round-Robin Tournament • In a round-robin tournament, every team plays every other team once. Imagine you are in charge of scheduling a round-robin Ping-Pong tournament with 32 teams. The tournament organizers agree to pay you $1 per match for running the tournament. How much will you make? • Answer = 31 + 30 + 29 + … + 1 = too much work!

  31. Sum of Consecutive Integers • Try it: Add the first 20 positive integers. • Back to round-robin example.

  32. Example 1.5.3 • Consider an election with 10 candidates. How many pairwise comparisons are there?

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