excursions in modern mathematics sixth edition

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Excursions in Modern Mathematics

Sixth Edition

Peter Tannenbaum

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Chapter 1The Mathematics of Voting

The Paradoxes of Democracy

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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.

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The Mathematics of Voting

1.1 Preference Ballots and Preference Schedules

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Preference ballotsPreference ballots

A ballot in which the voters are asked to rank the candidates in order of preference.

Linear ballotLinear ballot

A ballot in which ties are not allowed.

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schedule:schedule:A preferenceA preference

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The Mathematics of VotingImportant Facts

The first is that a voter’s preference are transitivetransitive, 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.

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Relative Preferences of a Voter

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Relative Preferences by elimination of one or more candidates

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1.2 The Plurality Method

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Plurality method – Plurality method – The candidate with the The candidate with the mostmost 1 1stst place votes wins the electionplace votes wins the election

- most commonly used method for finding a - most commonly used method for finding a winnerwinner

Plurality candidate – Plurality candidate – The candidate with the most 1st place votes. The plurality candidate is not necessarily a majority candidate.

Majority candidate - Majority candidate - TThe candidate with more than half of the 1st place votes. A majority candidate is always the plurality candidate.

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Majority ruleMajority rule

The candidate with a more than half the votes should be the winner.

Majority candidateMajority candidate

The candidate with the majority of 1st place votes .

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The 1The 1stst of 4 “Fairness Criteria” of 4 “Fairness Criteria”

The Majority CriterionThe 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!

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Bad News: The plurality method fails a different fairness criterion.

The Condorcet CriterionThe Condorcet CriterionIf 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.

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

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Insincere Voting (or Strategic 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.

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Insincere Voting (or Strategic Voting)Insincere Voting (or Strategic Voting)

Three voters decide not to “waste” their vote on Three voters decide not to “waste” their vote on F and swing the election over to H in doing so.F and swing the election over to H in doing so.

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1.3 The Borda Count Method

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In the Borda Count MethodBorda 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.

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Borda Count MethodBorda 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.

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Borda Count MethodBorda Count Method

A gets 56 + 10 + 8 + 4 + 1 = 81 pointsB gets 42 + 30 + 16 + 16 + 2 = 106 pointsC gets 28 + 40 + 24 + 8 + 4 = 104 pointsD gets 14 + 20 + 32 + 12 + 3 = 81 points

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1.4 The Plurality-with-elimination Method

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– Plurality-with-Elimination MethodPlurality-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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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.)

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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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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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So what is wrong with the plurality-with-So what is wrong with the plurality-with-elimination method?elimination method?

The Monotonicity CriterionThe Monotonicity CriterionIf 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.

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1.5 The Method of Pairwise Comparisons

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The Method of Pairwise ComparisonsThe 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.

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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.

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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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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.

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So what is wrong with the method of pairwise So what is wrong with the method of pairwise comparisons?comparisons?

The Independence-of-Irrelevant-Alternatives The Independence-of-Irrelevant-Alternatives Criterion (IIA)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.

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

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How Many Pairwise Comparisons?

In an election between 5 candidates, there were In an election between 5 candidates, there were 10 pairwise comparisons. 10 pairwise comparisons.

How many comparisons will be needed for an How many comparisons will be needed for an election having 6 candidates?election having 6 candidates?

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

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The Number of Pairwise ComparisonsThe Number of Pairwise Comparisons

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

2(N - 1)N

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The Mathematics of VotingRankings

Extended Ranking

Extended Plurality Extended Borda

Count Extended Plurality

with Elimination Extended Pairwise

Comparisons

Recursive Ranking

Recursive Plurality Recursive Plurality

with Elimination

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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.

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The Mathematics of Voting Rankings- Recursive Plurality

First-place: A

Second-place: B

Third-place: C

Fourth-place: D

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The Mathematics of Voting Conclusion

Methods of Vote CountingMethods of Vote CountingFairness CriteriaFairness CriteriaArrow’s Impossibility TheoremArrow’s Impossibility Theorem

It is mathematically impossible for a It is mathematically impossible for a democratic voting method to satisfy all of democratic voting method to satisfy all of the fairness criteria.the fairness criteria.

excursions in modern mathematics sixth edition

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favorite candidate

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