one more voting method: plurality with eliminationpkoester/teaching/ma111/slides/voting/vote08...one...

One More Voting Method: Plurality with Elimination

Our final voting method is an elimination method: we start with a large number ofcandidates, then successively remove “weak” candidates until very few candidatesremain.

The most common elimination method is the Plurality with EliminationMethod:

Step 1: If a candidate has a MAJORITY of first place votes, declare thatcandidate the winner.

Step 2: If no candidate has a majority, then remove the candidate with theFEWEST number of first place votes. Repeat Step 1.

Paul Koester () MA 111, Plurality With Elimination Method, And Fairness CriteriaJanuary 30 and February 1 2012 1

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Plurality with Elimination in Action

Determine the winner of this election using Plurality with Elimination:

7 7 8 5

A C D B

B B C A

C A B C

D D A D

A candidate requires at more than7 + 7 + 8 + 5

2= 13.5 first place votes for a

majority. Since no candidate has a majority of first place votes, we proceed to Step 2.Who has the fewest first place votes?


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B has the fewest first place votes, so we remove B.

7 7 8 5

A C D //B

//B //B C A

C A //B C

D D A D

Pay attention! We are NOT removing the ballots that favored B. We are removing Bfrom all ballots. Any candidate that was previously ranked below B will move up aspot.


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We rewrite the preference schedule with the three remaining candidates:

7 7 8 5

A C D A

C A C C

D D A D

Notice that, upon removing B, the first and last columns involve the same ballottype. It will simplify things if we combine those columns:

12 7 8

A C D

C A C

D D A

Now we repeat the Plurality with Elimination algorithm for this 3 candidate election.


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12 7 8

A C D

C A C

D D A

Step 1: Is there a Majority candidate? A majority requires at least 14 first placevotes. There is no majority candidate, so go to step 2.

Step 2: Eliminate candidate with fewest first place votes. In this case, C will beeliminated.


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12 7 8

A //C D

//C A //C

D D A

Rewriting the table without C:

12 7 8

A A D

D D A

Combining similar columns:

19 8

A D

D A


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19 8

A D

D A

Now repeat the Plurality with Elimination algorithm:

Step 1: Is there a Majority candidate? Yes! A is the majority candidate, so we canfinally declare a winner. A wins using Plurality with elimination.


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Re-examining the Election

7 7 8 5

A C D B

B B C A

C A B C

D D A D

We just showed that A is the Plurality with Elimination Winner of this election.

On a recent worksheet you looked at this same election and found:

B is the Borda Count Winner

C is the Pairwise Comparison Winner

D is the Plurality Winner.

So, WHO SHOULD WIN?


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The Remainder of the Voting Theory Chapter

We have now seen four different methods that can be used to determine the winnerof an election.

There are certainly many other methods used in practice, but these are four of themost common.

Other common methods are combinations of these methods (For example, acandidate might be awarded 3 points for each first place, two points for each secondplace, 1 point for each third, and no points if below third, then the candidates withthe lowest point totals are eliminated and the process is repeated until a majoritycandidate is found. This could be desribed as a “Borda Count with Elimination”)

Our focus for the remaining 4 class periods will be to understand the strengths andweaknesses of the four main methods. In particular, the previous example shows thatit is possible for these four methods to all produce different winners for the sameelection.

“It’s not the voting that’s democracy; it’s the counting.” Tom Stoppard.Before determining which methods are actually “fair” voting methods, we look at afew more technical issues involving Plurality with Elimination.


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A Quick Plurality with Elimination Example

An election involves 25 voters and 6 candidates.

2 7 6 1 1 4 4

A B B C C D E

D F A B D A C

F C D A A E F

B D F D F C B

E E C E E B A

C A E F B F D

Explain why you can determine the Plurality with Elimination winner in this electionwith almost no work.

Solution: Step 1 of the elimination algorithm says that a Majority candidate shouldbe selected as the winner. The elimination steps are only required IF the electiondoes not already have a majority candidate.

B is a majority candidate, so B wins using Plurality with Elimination.


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Plurality with Elimination: What to do in the case of a tie?

Determine the winner, using Plurality with Elimination.

5 4 4 6

A B C D

D C A A

B A B C

C D D B

There is no Majority Candidate, so we should eliminate the candidate with thefewest first place votes.B and C are tied for least first place votes.

There are several different logical tie breaking rules we could use. The mostcommonly used rule is that if there is a tie for who has the fewest first place votes,then both (or all) of the tie-ing candidates should be eliminated. In this case, weremove B and C.


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Plurality with Elimination: What to do in the case of a tie?

5 4 4 6

A //B //C D

D //C A A

//B A //B //C

//C D D //B

After removing B and C (moving A or D up if necessary) and combining similarballots, we obtain

13 6

A D

D A

A now has a Majority, so A is the Plurality with Elimination Winner.


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Plurality with Elimination: Other tie breaking rules

Return to the original election

5 4 4 6

A B C D

D C A A

B A B C

C D D B

B and C are tied for least first place votes.A different tie-breaking rule that is sometimes used in practice is to randomly chooseone of the candidates with the fewest first place votes to be eliminated.Suppose we choose to eliminate B first.

5 4 4 6

A //B C D

D C A A

//B A //B C

C D D //B


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Remove B, move other candidates up where necessary, and combine similar columns:

5 8 6

A C D

D A A

C D C

No candidate has a majority, so we eliminate the candidate with the fewest firstplace votes. In this case, we eliminate..... A.Removing A and simplifying the schedule:

11 8

D C

C D

Thus, D is the Plurality with Elimination Winner.


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This last example shows how delicate these voting methods can be.

Not only does the winner of the election depend on which voting method is used, evena small change in a technical tie breaking rule can affect the winner of the election.

We accept the first tie breaking rule as being the “correct” one. i.e., if multiplecandidates receive the fewest number of first place votes, we eliminate all of thesecandidates.


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Measuring Fairness:

Recall:

A candidate is a Majority Candidate if this candidate receives more than half of thefirst place votes.

A candidate is a Condorcet Candidate if this candidate wins all of the one-on-onecomparisons involving this candidate.

Typically, we think of Majority Candidates and Condorcet Candidates as being verystrong candidates, and these types of candidates lead to our first two measures offairness.


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Majority Criterion

The Majority Criterion: If a candidate X receives a majority of first place votes,then candidate X should be declared the winner of the election.

Please note that this is not a RULE. This is a notion of fairness.We believe that majority candidates are very strong candidates, and the MajorityCriterion essentially says that we would think a voting method was unfair if acandidate had a majority of first place votes and did not win the election.

A given voting method may satisfy or violate a given Fairness Criterion. Failing afairness criterion simply acts as a strike against that particular voting method.


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Proofs that Some Methods Satisfy the Majority Criterion

PLURALITY: Suppose candidate X is a Majority candidate. Since X receives morethan half of the first place votes, X automatically receives more first place votes thanany other candidate, so X will be declared the winner in the Plurality Method. Thisshows that the Plurality Method satisfies the Majority Criterion.

PLURALITY WITH ELIMINATION: Suppose candidate X is a Majority candidate.By step 1 of the Plurality with Elimination algorithm, X is declared the winner usingPlurality with Elimination. Therefore, the Plurality with Elimination Methodsatisfies the Majority Criterion.

PAIRWISE COMPARISON: Suppose candidate X is a Majority candidate. Since Xreceives more than half of the first place votes, X will receive more votes than theother candidate in any one-on-one comparison. In other words, X will win all of herone-on-one comparisons, and so X wins more one-on-one comparisons than any othercandidate, so X wins using Pairwise Comparisons. This shows that the PairwiseComparison Method satisfies the Majority Criterion.


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Borda Count FAILS the Majority Criterion

BORDA COUNT: Recall the example from the January 23 (Pairwise Comparisonand Borda Count) set of slides.

21 26 3

A B C

C C A

B A B

We saw that C won using Borda Count. Thus, B loses using Borda Count, despitethe fact that B was a Majority Candidate.In fact, that set of slides contains several examples of elections in which the MajorityCandidate LOSES using Borda. That set of slides even had an example of acandidate that received 79% of the first place votes and still lost using the BordaCount Method.


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Borda Count FAILS the Majority Criterion

On an exam, you could be asked to construct an example which shows that a givencriterion fails a given Fairness Criterion.

If you have to construct an example on the fly, it is usually easiest to try to create asmall example that does the trick.

This is perhaps the “smallest” example of an election in which a Majority Candidateloses using Borda Count Method.

3 2

X Y

Y Z

Z X

BORDA COUNT: Borda Count FAILS the Majority Criterion.


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What does it mean to SATISFY a Fairness Criterion?

First, Fairness Criteria are applied to Voting Methods. They are not applied to theindividual elections.

To say that a voting method, VM, satisfies a given fairness criterion, FC, means thatin EVERY ELECTION, whenever voting method VM is used to determine thewinner of the election, AND the hypotheses of the fairness criterion are met, then theconclusion of the fairness criterion is guaranteed to hold.

In order to show that a given voting method SATISFIES a given fairness criterion,you need to give a logical argument in which you assume the hypotheses of theFairness Criterion, then, ONLY using those hypotheses and the rules for that votingmethod, demonstrate that the conclusion of the fairness criterion are true.

Chances are that you have not had a lot of practice providing formal logical proofsand counterexamples, so we discuss a few ptifalls to avoid.


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Incorrect Arguments for Plurality Method and Majority Criterion

Consider the election

21 26 3

A B C

C C A

B A B

B receives more votes than any other candidate, so B is the winner using Plurality.Also, B is a Majority Candidate. Therefore, the Plurality Method satisfies theMajority Criterion.

This is NOT a valid proof that the Plurality Method satisfies the Majority Criterion.All this shows is that, in this one example, the winner of the plurality methodhappened to be a majority candidate.

A valid proof has to explain why, in ANY ELECTION WITH A MAJORITYCANDIDATE, said majority candidate MUST win using Plurality. To say thatPlurality Method satisfies the Majority Criterion means that NO election, usingPlurality, would ever produce a violation of the Majority Criterion. The above onlyshows that no violation occured in this particular example.


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Incorrect Arguments for Plurality Method and Majority Criterion

2 2 3

A B C

C C A

B A B

C receives more votes than any other candidate, so C is the winner using thePlurality Method. C is not a Majority candidate (since a majority would require 4 ormore first place votes.)

The Plurality Method fails the Majority Criterion because a non-majority candidatewon using Plurality.

This is NOT a valid argument. The Majority Criterion does not say that the winnerof the election must be a Majority Candidate. Rather, it says that IF the electionhas a Majority Candidate, then that candidate must win. Since this election did nothave a Majority candidate, there is nothing to violate.If it helps, think of the Majority Criterion as saying “A majority candidate shouldnot lose” instead of “A majority candidate should win.”


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What does it mean to FAIL a Fairness Criterion?

To say that a voting method, VM, SATISFIES a given fairness criterion, FC, meansthat in EVERY ELECTION, whenever voting method VM is used to determine thewinner of the election, AND the hypotheses of the fairness criterion are met, then theconclusion of the fairness criterion is guaranteed to hold.

In order to show that a VOTING METHOD fails a fairness criterion, it is sufficientto find a single example of an election in which that fairness criterion is not met.

For example, we showed that the Borda Count Method FAILS the Majority Criterionby finding a single example in which there was a Majority Candidate AND thatMajority Candidate lost the election using the Borda Count Method.


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Incorrect Arguments for Borda Count Method and Majority Criterion

Consider the election

2 1

A B

C A

B C

A receives 2 · 3 + 1 · 2 = 8 Borda points, B receives 2 · 1 + 1 · 3 = 5 Borda points, Creceives 2 · 2 + 1 · 1 = 5 Borda points.So A wins using Borda Method. Also, A is a Majority Candidate. Therefore, BordaMethod satisfies the Majority Criterion.

This is NOT a valid proof that the Plurality Method satisfies the Majority Criterion.All this shows is that, in this one example, the winner of the plurality methodhappened to be a majority candidate.A valid proof has to explain why, in ANY ELECTION WITH A MAJORITYCANDIDATE, said majority candidate MUST win using Borda Count. To say thatBorda Cound satisfies the Majority Criterion means that NO election, using BordaCount, would ever produce a violation of the Majority Criterion. The above onlyshows that no violation occured in this particular example.


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Incorrect Arguments for Borda Count Method and Majority Criterion

Suppose that candidate A is a Majority Candidate. Thus, A receives more than halfof the first place votes. Since A has more votes than anyone else, A will have thehighest Borda point total, so A will win using Borda Count Method. This shows thatBorda Count satisfies the Majority Criterion.

This is NOT a valid proof. In spirit, it is better than the previous invalid proof, sincethis TRIES to provide a logical argument that applies to all elections, instead oftrying to draw general conclusions from one example. However, the implication“Since A has more votes than anyone else, A will have the highest Borda point total”is incorrect. (We have seen examples in which the candidate with the highest Bordapoint total had NO first place votes.)


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Recap

To show that a voting method SATISFIES a fairness criterion, you need to provide alogical argument in which you assume the hypothesis of the voting method and,using ONLY those hypotheses and the rules for the given voting method, you showthe conclusion of the voting method must be true.

To show that a voting method FAILS a fairness criterion, it is enough to find aSINGLE election showing a violation. A violation consists of an election in which thehypotheses of that voting method are met, but the conclusion is not.


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Recap

Majority Criterion

Plurality Satisfies

Borda Count Fails

Pairwise Comp Satisfies

Plur. with Elim Satisfies

On an exam, you could be asked which methods satisfy the Majority Criterion andwhich fail it.

You could also be asked to explain why a given method satisfies the MajorityCriterion. In that case you would need to provide a proof, like we did a few slides ago.

You could also be asked to construct an example showing that a given method failsthe Majority Criterion.

It is probably a good idea to commit the above proofs and countereamples to memoryuntil you can reconstruct these proofs or counterexamples in your own words.


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Condorcet Criterion

The Condorcet Criterion: If a candidate X is a Condorcet Candidate, then candidateX should be declared the winner of the election.

Please note that this is not a RULE. This is a notion of fairness. We believe thatCondorcet candidates are very strong candidates, and the Condorcet Criterionessentially says that we would think a voting method was unfair if a candidate was aCondorcet Candidate and did not win the election.


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Proof that Pairwise Comparisons Satisfies Condorcet Criterion

PAIRWISE COMPARISONS: Suppose candidate X is a Condorcet candidate. Xwins all of her one-on-one comparisons, and so X wins more one-on-one comparisonsthan any other candidate, so X wins using Pairwise Comparisons. This shows thatthe Pairwise Comparison Method satisfies the Condorcet Criterion.

It turns out that the other three methods fail the Condorcet Criterion.


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The Other Methods Fail the Condorcet Criterion

Recall the example from the January 25 Workhseet.

7 7 8 5

A C D B

B B C A

C A B C

D D A D

C is a Condorcet Candidate.

We have seen that A is the Plurality with Elimination Winner of this election, B isthe Borda Count Winner, and D is the Plurality Winner.

Thus, the Condorcet Candidate C LOST in each of Plurality with Elimination,Borda Count, and Plurality, thus showing that each of these methods violates theCondorcet Criterion.


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Recap

Majority Criterion Condorcet Criterion

Plurality Satisfies Fails

Borda Count Fails Fails

Pairwise Comp Satisfies Satisfies

Plur. with Elim Satisfies Fails

On an exam, you could be asked which methods satisfy a given criterion and whichmethods fail it.

You could also be asked to explain why a given method satisfies a given criterion. Inthat case you would need to provide a proof, like we did a few slides ago.

You could also be asked to construct an example showing that a given method fails agiven criterion.

It is probably a good idea to commit the above proofs and countereamples to memoryuntil you can reconstruct these proofs or counterexamples in your own words.


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