Fair Elections

Are they possible?

Acknowledgment

Many of the examples are taken from Excursions in Modern Mathematics by Peter Tannenbaum and Robert Arnold

Presidential Election

• Not decided by majority vote

• Decided by electoral college

• In 2000, Bush won without receiving a majority of votes

Senate and House Elections

• Majority wins

• If no candidate wins a majority, then the outcome depends on the state’s rules

Today’s Assumptions

• At least two choices

• Voters give preference list:A>B means A preferred over B

• Vote ABCD

Means A>B>C>D

Who Wins?

• Given everyone’s preference vote, what method should be used to determine winner?

• Some bad methods:– My vote wins: Called dictatorship– Ignore the votes and pick one at random:

Decision should be deterministic– Always pick A as the winner regardless of

vote: Imposition, Method should be non-impositional

Vote for Class President

No. Votes

14 10 8 4 1

1st A C D B C

2nd B B C D D

3rd C D B C B

4th D A A A A

Plurality Method

• Candidate with most first place votes wins

• In case of class president, A wins• We will not worry about ties. (Ways to

resolve ties, or just have more than one winner!)

Plurality Method

• AdvantageIf majority of voters place a choice as their first preference, then that choice wins.

• DisadvantageIgnores the lower preference choices

Majority Criterion:

If a choice receives a majority of the top preference votes, then that choice should win

Plurality Method

No. votes

23 21 3

1st A B C

2nd B C B

3rd C A A

A wins plurality, but if B goes head to head with either A or C, B wins!

Plurality Method

• Used for many political elections (common in England, India, US, Canada)

• Often used to pick corporate executive officers

Condercet Criterion

• If any choice wins in a head-to-head comparison over every other choice, then that choice should win.

• Plurality does not satisfy Condercet criterion– Means that plurality will not satisfy Condercet

criterion for some voting outcome, but for some voting outcomes it might

Insincere Voting

• Sometimes in order to avoid undesirable winner, voter may put first choice second

• Other schemes, may move second choice to last

No. votes

23 21 3

1st A B C

2nd B C B

3rd C A A

No. votes

23 21 3

1st A B B

2nd B C C

3rd C A A

Borda Count

• Assign points for each vote base on preference

• Example A 3 pointsD 2 points B 1 pointC 0 points

Borda Count

• Who wins class president?

• A 79

• B 106

• C 104

• D 81

•B wins!

Borda Count

No. Voters

6 2 3

1st A B C

2nd B C D

3rd C D B

4th D A A

A 6 x 3 = 18

B 6 x 2 + 2 x 3 + 3 x 1 = 21

C 6 x 1 + 2 x 2 + 3 x 3 = 19

D 2 x 1 + 3 x 2 = 8

Violates Majority Criterion and Condercet

Borda Count

• Used in some political elections (Slovenia and Micronesia)

• Baseball MPV in AL and NL

• Heisman Trophy

• Universities often use it for hiring

Plurality with Elimination

1. If a choice has a majority of first place votes, that choice wins

2. Eliminate choice with fewest first place votes and pretend the election was only among the other choices

3. Back to 1.

Plurality with Elimination

• Who wins class president?

•D

Plurality with Elimination

• Three candidates in an election. Word gets out who is voting for who. Consequently, a few people change their votes so they vote for a winner.

No. Votes

7 8 10 4

1st A B C A

2nd B C A C

3rd C A B B

No. Votes

7 8 14

1st A B C

2nd B C A

3rd C A B

Plurality with Elimination

• Advantage– Majority Criterion Satisfied

• Disadvantage– Violates Monotonicity

Criterion:

If votes are changed that only improves the winners position, the winner should not change

Plurality with Elimination

• Variations used in political elections (France)

• Often used in hiring

• Olympic Committee uses to determine location of Olympics

Pairwise Comparison

• For each pair of choices, x and y, count the number of votes that places x before y and the number that places y before x.

• One point to the winner, ½ point if tie

• Tally all the points and the most points wins

Pairwise Comparison

• Who wins class president?• A 14, B 23 … B gets 1 point• A 14, C 23 … C gets 1 point• A 14, D 23 … D gets 1 point• B 18, C 19 … C gets 1 point• B 28, D 9 … B gets 1 point• C 25, D 12 … C gets 1 point

• C Wins with 3 points!

Pairwise ComparisonNo. Votes

2 6 4 1 1 4 4

1st A B B C C D E

2nd D A A B D A C

3rd C C D A A E D

4th B D E D B C B

5th E E C E E B A

A vrs B B 1A vrs C A 1 A vrs D A 1 A vrs E A 1B vrs C C 1 B vrs D B .5 D .5 B vrs E B 1C vrs D C 1C vrs E E 1D vrs E D 1

A: 3 B: 2.5 C: 2 D: 1.5 E: 1 Winner is A But what if C withdraws just before vote?

Pairwise Comparison

• Advantage– Satisfies Majority– Satisfies Condercet– Satisfies Monotonicity

• Disadvantage– Does not satisfy

Independence-of-Irrelevant-Alternative Criterion:

If x is the winner and a choice other than x is removed, x should still be the winner.

Pairwise Comparison

• Not currently used in any political election – variation last used in Marquette Michigan in the 1920’s

• Variation used by Wikimedia Foundation for election of Trustees

• Various other private organizations

Who Should be Class President?

• Four reasonable methods yielded four different answers!

• Critical that before voting takes place, method of determining winner is well established!

• But, is there some method that satisfies all the fairness criteria?

Is Fairness an Illusion?

No. Votes

5 4 3

1st A B C

2nd B C A

3rd C A B

Theorem: If a voting scheme satisfies the Majority Criterion, it cannot satisfy the Independence-of-Alternative Criterion.Proof:

Suppose A wins. If B withdraws, then C wins.

Suppose B wins. If C withdraws, then A wins.

Suppose C wins. If A withdraws, then B wins.

Arrows Theorem

Theorem: If the method to determine the winner is deterministic, non-impositional, and it satisfies both Monotonicity and Independence-of-Alternative criteria, then it is a dictatorship.

Sometimes people say there is no fair way to determine who wins!

Summary

Majority Condercet Monotonicity Independence

Plurality Yes No Yes No

Borda Count No No Yes No

Plurality Elimination

Yes No No No

Pairwise

ComparisonYes Yes Yes No

Best Method

• Let’s vote on which method we think is best.

• Before we vote, we have to decide which method to use to determine winner.

• Let’s vote on which method to use to determine the best method.

• Before we vote on which method to use, we have to …