Excursions in Modern Mathematics Peter Tannenbaum Chapter 10
Fair Elections Are they possible?. Acknowledgment Many of the examples are taken from Excursions in...
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![Page 1: Fair Elections Are they possible?. Acknowledgment Many of the examples are taken from Excursions in Modern Mathematics by Peter Tannenbaum and Robert.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649cf65503460f949c538f/html5/thumbnails/1.jpg)
Fair Elections
Are they possible?
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Acknowledgment
Many of the examples are taken from Excursions in Modern Mathematics by Peter Tannenbaum and Robert Arnold
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Presidential Election
• Not decided by majority vote
• Decided by electoral college
• In 2000, Bush won without receiving a majority of votes
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Senate and House Elections
• Majority wins
• If no candidate wins a majority, then the outcome depends on the state’s rules
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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
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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
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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
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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!)
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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
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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!
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Plurality Method
• Used for many political elections (common in England, India, US, Canada)
• Often used to pick corporate executive officers
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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
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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
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Borda Count
• Assign points for each vote base on preference
• Example A 3 pointsD 2 points B 1 pointC 0 points
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Borda Count
• Who wins class president?
• A 79
• B 106
• C 104
• D 81
•B wins!
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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
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Borda Count
• Used in some political elections (Slovenia and Micronesia)
• Baseball MPV in AL and NL
• Heisman Trophy
• Universities often use it for hiring
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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.
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Plurality with Elimination
• Who wins class president?
•D
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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
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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
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Plurality with Elimination
• Variations used in political elections (France)
• Often used in hiring
• Olympic Committee uses to determine location of Olympics
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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
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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!
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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?
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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.
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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
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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?
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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.
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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!
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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
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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 …
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