Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern Mathematics Sixth Edition. Peter Tannenbaum. Chapter 1 The Mathematics of Voting. The Paradoxes of Democracy. The Mathematics of Voting Outline/learning Objectives. Construct and interpret a preference schedule for an election involving preference ballots.

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Peter Tannenbaum

Chapter 1The Mathematics of Voting

The Mathematics of VotingOutline/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.

The Mathematics of Voting

1.1 Preference Ballots and Preference Schedules

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.

The Mathematics of Voting

A preference

schedule:

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.

The Mathematics of VotingImportant Facts
The Mathematics of Voting

Relative Preferences of a Voter

The Mathematics of Voting

Relative Preferences by elimination of one or more candidates

The Mathematics of Voting

1.2 The Plurality Method

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.
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 .

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!

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.

The Mathematics of Voting

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

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.

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.

The Mathematics of Voting

1.3 The Borda Count Method

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.

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.

The Mathematics of Voting

Borda Count Method

A gets 56 + 10 + 8 + 4 + 1 = 81 points

B gets 42 + 30 + 16 + 16 + 2 = 106 pointsC gets 28 + 40 + 24 + 8 + 4 = 104 pointsD gets 14 + 20 + 32 + 12 + 3 = 81 points

The Mathematics of Voting

1.4 The Plurality-with-elimination Method

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.
The Mathematics of Voting
• 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.)
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.
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.
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.

The Mathematics of Voting

1.5 The Method of Pairwise Comparisons

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.

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.

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

The Mathematics of Voting

The Method of Pairwise Comparisons

A vs. B: B wins 15-7. B gets 1 point. A vs. C: A wins 16-6. C gets 1 point. etc.

Final Tally: A-3, B-2.5, C-2, D-1.5, E-1. A wins.

The Mathematics of Voting

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.

The Mathematics of Voting

Eliminate C (an irrelevant alternative) from this election and B wins (rather than A).

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. 5 + 4 + 3 + 2 + 1 = 15

The Mathematics of Voting

The Number of Pairwise Comparisons

In an election with N candidates the total number of pairwise comparisons between candidates is

Extended Ranking

Extended Plurality

Extended Borda Count

Extended Plurality with Elimination

Extended Pairwise Comparisons

Recursive Ranking

Recursive Plurality

Recursive Plurality with Elimination

The Mathematics of VotingRankings
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.
The Mathematics of Voting Rankings- Recursive Plurality

First-place: A

Second-place: B

Third-place: C

Fourth-place: D

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.