1. Excursions in Modern Mathematics Sixth Edition
Peter Tannenbaum

2. Chapter 1 The Mathematics of Voting
The Paradoxes of Democracy

3. The Mathematics of Voting Outline/learning Objectives
Construct and interpret a preference schedule for an election involving preference ballots.
Implement the plurality, Borda count, plurality-with-elimination, and pairwise comparisons vote counting methods.
Rank candidates using recursive and extended methods.
Identify fairness criteria as they pertain to voting methods.
Understand the significance of Arrows’ impossibility theorem.

4. The Mathematics of Voting
1.1 Preference Ballots and Preference Schedules

5. The Mathematics of Voting
Preference ballots
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. The Mathematics of Voting

7. The Mathematics of Voting
A preference
schedule:

8. The Mathematics of Voting Important Facts
The first is that a voter’s preference are transitive, i.e., that a voter who prefers candidate A over candidate B and prefers candidate B over candidate C automatically prefers candidate A over C.
Secondly, that the relative preferences of a voter are not affected by the elimination of one or more of the candidates.

9. The Mathematics of Voting
Relative Preferences of a Voter

10. The Mathematics of Voting
Relative Preferences by elimination of one or more candidates

11. The Mathematics of Voting
1.2 The Plurality Method

12. The Mathematics of Voting
Plurality method – The candidate with the most 1st place votes wins the election
- most commonly used method for finding a winner
Plurality candidate – The candidate with the most 1st place votes. The plurality candidate is not necessarily a majority candidate.
Majority candidate - The candidate with more than half of the 1st place votes. A majority candidate is always the plurality candidate.

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

14. The Mathematics of Voting
The 1st of 4 “Fairness Criteria”
The Majority Criterion
If candidate X has a majority of the 1st place votes, then candidate X should be the winner of the election.
Good News: The plurality method satisfies the majority criterion!

15. The Mathematics of Voting
Bad News: The plurality method fails a different fairness criterion.
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.

16. The Mathematics of Voting
The plurality method fails to satisfy the Condorcet Criterion – H beats each other candidate head-to-head.

17. The Mathematics of Voting
Insincere Voting (or Strategic Voting)
If we know that the candidate we really want doesn’t have a chance of winning, then rather than “wasting our vote” on our favorite candidate we can cast it for a lesser choice that has a better chance of winning the election.

18. The Mathematics of Voting
Insincere Voting (or Strategic Voting)
Three voters decide not to “waste” their vote on F and swing the election over to H in doing so.

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1.3 The Borda Count Method

20. The Mathematics of Voting
In the Borda Count 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.

21. 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.

22. The Mathematics of Voting
Borda Count Method
A gets = 81 points
B gets = 106 points C gets = 104 points D gets = 81 points

23. The Mathematics of Voting
1.4 The Plurality-with-elimination Method

24. The Mathematics of Voting
Plurality-with-Elimination Method
Round 1. Count the first-place votes for each candidate, just as you would in the plurality method. If a candidate has a majority of first-place votes, that candidate is the winner. Otherwise, eliminate the candidate (or candidates if there is a tie) with the fewest first-place votes.

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Plurality-with-Elimination Method
Round 2. Cross out the name(s) of the candidates eliminated from the preference and recount the first-place votes. (Remember that when a candidate is eliminated from the preference schedule, in each column the candidates below it move up a spot.)

27. The Mathematics of Voting
Plurality-with-Elimination Method
Round 2 (continued). If a candidate has a majority of first-place votes, declare that candidate the winner. Otherwise, eliminate the candidate with the fewest first-place votes.

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29. The Mathematics of Voting
Plurality-with-Elimination Method
Round 3, 4, etc. Repeat the process, each time eliminating one or more candidates until there is a candidate with a majority of first-place votes. That candidate is the winner of the election.

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31. The Mathematics of Voting
So what is wrong with the plurality-with-elimination method?
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.

32. The Mathematics of Voting
1.5 The Method of Pairwise Comparisons

33. The Mathematics of Voting
The Method of Pairwise Comparisons
In a pairwise comparison between between X and Y every vote is assigned to either X or 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.

34. 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.

35. 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

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

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So what is wrong with the method of pairwise comparisons?
The Independence-of-Irrelevant-Alternatives Criterion (IIA)
If candidate X is a winner of an election and in a recount one of the non-winning candidates is removed from the ballots, then X should still be a winner of the election.

38. The Mathematics of Voting
Eliminate C (an irrelevant alternative) from this election and B wins (rather than A).

39. The Mathematics of Voting
How Many Pairwise Comparisons?
In an election between 5 candidates, there were 10 pairwise comparisons. How many comparisons will be needed for an election having 6 candidates?
Ans = 15

40. The Mathematics of Voting
The Number of Pairwise Comparisons
In an election with N candidates the total number of pairwise comparisons between candidates is

41. The Mathematics of Voting Rankings
Extended Ranking
Extended Plurality
Extended Borda Count
Extended Plurality with Elimination
Extended Pairwise Comparisons
Recursive Ranking
Recursive Plurality
Recursive Plurality with Elimination

42. The Mathematics of Voting Rankings
Recursive Ranking
Step 1: [Determine first place] Choose winner using method and remove that candidate.
Step 2: [Determine second place] Choose winner of new election (without candidate removed in step 1) and remove that candidate.
Steps 3, 4, etc.: [Determine third, fourth, etc. places] Continue in same manner using method on remaining candidates yet to be ranked.

43. The Mathematics of Voting Rankings- Recursive Plurality
First-place: A
Second-place: B
Third-place: C
Fourth-place: D

44. The Mathematics of Voting Conclusion
Methods of Vote Counting
Fairness Criteria
Arrow’s Impossibility Theorem
It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria.