ma 111, fairness criteria - university of...

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.

Paul Koester () MA 111, Fairness Criteria February 1 2012 1 / 35

Majority Criterion: Reveiew from last class

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.


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.


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.


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.


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.


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.


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.”


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.


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.


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.)


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.


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.


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.


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.


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.


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.


Monotonicity Criterion

The Monotonicity Criterion: Suppose that a candidate X wins an election. Nowsuppose a similar preference schedule is used in a re-election, with the only changesbeing changes that favor X. (i.e., the only changes are changes in which X is placedHIGHER than before). The X should win the re-election as well.

Again, this is not a RULE. This is a notion of fairness.

If a candidate’s position only improves, they couldn’t possibly lose the re-election,could they?


Plurality with Elimination Violates Monotonicity

Determine the winner of this election using Plurality with Elimination.

7 6 5 2

A B C C

B A B A

C C A B

No candidate receives a majority of first place votes, so we eliminate B (thecandidate with fewest first place votes)After eliminating B and simplifying the schedule

13 7

A C

C A

So A wins using Plurality with Elimination.



Now suppose there is a re-election, and the only change is on the last ballot.Both of these voters now favor A over C.

7 6 5 2

A B C A

B A B C

C C A B

No candidate receives a majority of first place votes, so we eliminate C (thecandidate with fewest first place votes).After eliminating C and simplifying the schedule

9 11

A B

B A

So B wins using Plurality with Elimination in this re-election.



The previous example shows that Plurality with Elimination VIOLATES theMonotonicity Criterion:A won the original election, in a re-election, A received even more first place votes,but managed to lose the re-election to B.

How did this happen? It helps to think about from a different point of view: Insteadof thinking “A’s position improved in the re-election” think “C’s position worsened inthe re-election”. In fact, C’s position worsened enough so that in the second round ofthe re-election, A was put against B instead of C.


Which Methods Satisfy Monotonicity?

PLURALITY: Suppose candidate X receives more first place votes than any othercandidate. In a re-election, the only changes on ballots are changes in which X’sposition improves. In particular, in the re-election X receives at least as many firstplace votes as he did in the original election, AND the other candidates receive nomore first place votes than they did in the original election, so X still receives morefirst place votes than any other candidate, so X still wins using Plurality. Therefore,Plurality SATISFIES the Monotonicity Criterion.

BORDA COUNT: Suppose candidate X receives the highest Borda point total in theoriginal election. In a re-election, the only changes on ballots are changes in whichX’s position improves. In particular, in the re-election, X’s Borda point total is atleast as high as in the original election. The remaining candidates either stay in thesame position as before, or drop a position due to being surpassed by X. Thus, in there-election, the remaining candidates’ point total are no greater than in the originalelection. Therefore, X still has the highest Borda point total, so X wins there-election. This verfies that Borda Count satisies the Monotonicty Criterion.


Which Methods Satisfy Monotonicity?

PAIRWISE COMPARISONS: Suppose candidate X wins using PairwiseComparisons. Therefore, candidate X wins more one-on-one comparisons than anyother candidate. In a re-election, the only changes on ballots are changes in whichX’s position improves. Now, if X beat Y one-on-one in the original election, then Xstill beats Y one-on-one in the re-election. Therefore, X wins at least as manyone-on-one comparisons in the re-election as in the original election. Furthermore,since the only changes made are changes in which X’s position improves, theone-on-one comparisons between the remaining candidates are unchanged. (In otherwords, the other candidates win no more one-on-one comparisons in the re-electionthan they did in the original election.) Therefore, X still wins the re-election. Thisproves that Pairwise Comparisons satisfies the Monotonicity Criterion.


Invalid Argument for Plurality and Monotonocity

Suppose candidate X receives more first place votes than any other candidate. In are-election, X’s position improves on some of the ballots. In particular, in there-election X receives at least as many first place votes as he did in the originalelection so X still receives more first place votes than any other candidate, so X stillwins using Plurality.

Why is this not a valid proof? The above argument actually claims to prove that ifX’s position improves on some ballots, then X will still have more first place votesthan any other candidate. This implication is false! For example, X and Y couldboth improve on some number of ballots. If Y improved on more ballots than X, Ycould end up stealing the victory from X. The trouble is that this argument does nottake into account the full assumptions of Monotonicity: the ONLY changes arechanges in which the original winner’s position improves.


Recapping

Majority Criterion Condorcet Criterion Monotonicity

Plurality Satisfies Fails Satisifies

Borda Count Fails Fails Satisfies

Pairwise Comp Satisfies Satisfies Satisfies

Plur. with Elim Satisfies Fails Fails

Pairwise Comparisons appears to be the only fair method.


Pairwise Comparisons Also has a Defect

Determine the winner of this election, using Pairwise Comparisons:

3 1 5 3

B A C A

A D D D

C C B B

D B A C

Here are the results of the comparisons:

B vs A B A vs C A

B vs C Tie: B, C A vs D A

B vs D D C vs D C

A wins two, B wins one and ties one, C wins one and ties one, and D ties one.Therefore, A wins using Pairwise Comparisons.



Suppose that candidate D realizes he has no chance of winning the election, so Ddrops out of the race right before the election takes place. What is the outcome ofthe new election?

3 1 5 3

B A C A

A //D //D //D

C C B B

//D B A C

Here are the results of the comparisons:

B vs A B A vs C A

B vs C Tie: B, C //A////vs///D //A

//B////vs///D //D //C////vs///D //C

A wins one, B wins one and ties one, C ties one. Therefore, B wins the new electionusing Pairwise Comparisons.



What just happened? A losing candidate, D, decided to drop out of the election.Even though D had no chance of winning, D’s dropping out or staying in the electionCHANGED the outcome!

How did it happen? Originally, A won the election since A won more one-on-onecomparisons than any other candidate. However, B barely lost against A.Furthemore, one of A’s one-on-one victories was against D, but B had lost to D.Thus, when D dropped out, one of A’s one-on-one victories vanished, but none of B’sone-on-one victories vanished.

This example leads us to our final fairness criterion.


The Independence of Irrelavent Alternatives Criterion

IIA: Suppose that candidate X wins an election. Now suppose a NON-winningcandidate, Y, is removed from the preference schedule. Then X should win the newelection as well.

The previous example shows that Pairwise Comparisons VIOLATES the IIACriterion.


Plurality Violates IIA

The “Burger-Pizza-Thai” example from a few classes ago shows that Pluralityviolates IIA.

2 2 3

T P B

P T P

B B T

The Plurality Method declares B as the winner. Now suppose T is removed justbefore the election.

2 2 3

//T P B

P //T P

B B //T

P now inherits the two first place votes that originally went to T, and so P wins thenew election.

The outcome of the election changed after removing a non-winning candidate, so thisshows Plurality violates the IIA Criterion.


Plurality Violates IIA

The “Burger-Pizza-Thai” example also shows that Plurality with Eliminationviolates IIA.

2 2 3

T P B

P T P

B B T

P and T are tied for fewest first place votes, so both are eliminated, so B wins usingPlurality with Elimination.

Now suppose T is removed just before the election.

2 2 3

//T P B

P //T P

B B //T

In the new election, P has a majority of the first place votes, so P is declared thewinner of the new election using Plurality with Elimination.

The outcome of the election changed afer removing a non-winning candidate, so thisshows Plurality with Elimination violates the IIA Criterion.


Borda Fails IIA

The following example shows that Borda Count violates the IIA Criterion.

4 2 3 2

A B C C

B C B A

C A A B

A receives 21 Borda points, B receives 22 Borda points, and C receives 23 Bordapoints, so C is the Borda Count winner.

Suppose A decides to drop out of the race. Removing A from the election results inthe preference schedule:

6 5

B C

C B

In the new election, B receives 17 Borda points and C receives 16 Borda points, so Bwins the re-election.

The outcome of the election changed after removing a non-winning candidate, so thisshows Borda Count violates the IIA Criterion.


Recapping

Majority Condorcet Monotonicity IIA

Plurality S F S F

Borda Count F F S F

Pairwise Comp S S S F

Plur. with Elim S F F F

We have discussed FOUR common VOTING METHODS. We have discussed FOURmeasures of FAIRNESS. We have seen that every one of our voting methodsVIOLATES at least one Fairness Criterion.

Social Scientients spent a lot of time trying to devise better (or “fair-er”) votingmethods, but every method they came up with violated at least one of these fourcriteria..... Then the mathematical economist Kenneth Arrow made a remarkablediscovery:


Arrow’s Impossibility Theorem

The precise version of Arrow’s Theorem is too technical to state in this course, butwe will give a semi-precise statement of this result, as well as a colloquial statementof the result:

ARROW’S IMPOSSIBILITY THEOREM: If at least two voters and at least threecandidates are involved, then there is NO preferential voting method which satisfiesall four of the fairness criterion.

Informal Restatement: There does not exist a “fair” preferential voting method.

The restatement is the way people typically desribe Arrow’s Theorem, although therestatement is rather vague and imprecise.

The more precise version of the theorem suggests that “fair” voting methods mayexist for two candidate elections. (We’ll look into this a little more later.)

Also notice the imprecise version essentially says “all voting systems are unfair,” but“unfair” is left as a vague, undefined term. The more precise statement quantifies“unfairness:” How are the voting systems unfair? At least one of the MajorityCriterion, the Condorcet Criterion, the Monotonicity Criterion, or the Independenceof Irrelevant Alternatives Criterion will be violated by the voting method.


Arrow’s Impossibility Theorem

Notice that Arrow’s Theorem is a statement about all VOTING METHODS beingunfair. It does not say that all ELECTIONS are unfair.

What’s the difference?Saying that a voting method violates a fairness criteria means that are elections inwhich violations of said fairness criterion will be observed; it does not say that theviolations will be observed in every election.

For example, saying that the Borda Count Method violates the Majority FairnessCriterion means that a Majority candidate COULD lose if the election is decidedusing Borda Count; this does not imply that Majority candidates must lose usingBorda Count.


ma 111, fairness criteria - university of...

Documents