1. Fairness Criteria Fairness Criteria: properties we expect a good voting system to satisfy.Fairness Criteria: properties we expect a good voting system to satisfy. Three fairness criteria will be studied:Three fairness criteria will be studied: The majority criterion The majority criterion The head-to-head criterion The head-to-head criterion The irrelevant alternatives criterion The irrelevant alternatives criterion

2. Majority Criterion Majority Criterion If a candidate is the first choice of a majority of voters, then that candidate should win.If a candidate is the first choice of a majority of voters, then that candidate should win.

3. Majority Criterion Majority Criterion For the majority criterion to be violated:For the majority criterion to be violated: A candidate ha s more than half of the votes. A candidate ha s more than half of the votes. The same candidate does not win the election. The same candidate does not win the election. Note:Note: D oes not apply if no candidate receives a majority. D oes not apply if no candidate receives a majority. D oes not say that the winner of an election must win by a majority. D oes not say that the winner of an election must win by a majority.

4. The Majority Criterion If a candidate is the first choice of a majority of voters, that candidate will always win using:If a candidate is the first choice of a majority of voters, that candidate will always win using: The plurality method. The plurality method. The plurality with elimination method. The plurality with elimination method. The pairwise comparison method. The pairwise comparison method.

5. The Majority Criterion If a candidate is the first choice of a majority of voters, then that candidate might not win using:If a candidate is the first choice of a majority of voters, then that candidate might not win using: The Borda count method. The Borda count method. The candidate with the most points may not be the candidate with the most first-place votes.The candidate with the most points may not be the candidate with the most first-place votes.

6. Example 1 Four cities are being considered for an annual trade show. The preferences of the organizers are given in the table.Four cities are being considered for an annual trade show. The preferences of the organizers are given in the table.

7. Example 1, cont’d a) Which site has a majority of first-place votes? b) Which site wins using Borda count?

8. Example 1, cont’d Point totals for Borda count:Point totals for Borda count: Chicago: 5(4) + 0(3) + 2(2) + 2(1) = 26Chicago: 5(4) + 0(3) + 2(2) + 2(1) = 26 Seattle: 3(4) + 4(3) + 2(2) + 0(1) = 28Seattle: 3(4) + 4(3) + 2(2) + 0(1) = 28 Phoenix: 1(4) + 2(3) + 2(2) + 4(1) = 18Phoenix: 1(4) + 2(3) + 2(2) + 4(1) = 18 Boston: 0(4) + 3(3) + 3(2) + 3(1) = 18Boston: 0(4) + 3(3) + 3(2) + 3(1) = 18

9. Example 1, cont’d This is an example of the Borda count method failing the majority criterion. In this case, some would say that the Borda count method was unfair.This is an example of the Borda count method failing the majority criterion. In this case, some would say that the Borda count method was unfair.

10. Head-to-Head Criterion If a candidate is favored when compared with each of the other candidates, then that candidate should win the election.If a candidate is favored when compared with each of the other candidates, then that candidate should win the election. This is also called the Condorcet criterion. This is also called the Condorcet criterion.

11. Head-to-Head Criterion If the head-to-head criterion is violated:If the head-to-head criterion is violated: A candidate is preferred pairwise to every other candidate. A candidate is preferred pairwise to every other candidate. This same candidate does not win the election. This same candidate does not win the election.

12. Head-to-Head Criterion If a candidate is favored pairwise to every other candidate, then that candidate will always win using:If a candidate is favored pairwise to every other candidate, then that candidate will always win using: The pairwise comparison method. The pairwise comparison method. This candidate will earn the most points from the pairwise comparisons.This candidate will earn the most points from the pairwise comparisons.

13. Head-to-Head Criterion If a candidate is favored pairwise to every other candidate, then that candidate might not win using:If a candidate is favored pairwise to every other candidate, then that candidate might not win using: The plurality method. The plurality method. The plurality with elimination method. The plurality with elimination method. The Borda count method. The Borda count method.

14. Example 2 7 people are choosing what to do for a party: catering, picnic, or restaurant.7 people are choosing what to do for a party: catering, picnic, or restaurant. a) Which site wins using the plurality method? b) Show that the head-to-head criterion is violated.

15. Example 2, cont’d Solution:Solution: b) The pairwise comparisons are made: R is preferred to P 4 to 3.R is preferred to P 4 to 3. R is preferred to C 5 to 2.R is preferred to C 5 to 2. R is preferred to every other candidate, but R did not win the election. This is a violation of the head-to-head criterion. R is preferred to every other candidate, but R did not win the election. This is a violation of the head-to-head criterion.

16. Irrelevant Alternatives Criterion Candidate X is selected in an election.Candidate X is selected in an election. If this election were redone with one or more of the unselected candidates removed from the vote, then X should still win.If this election were redone with one or more of the unselected candidates removed from the vote, then X should still win.

17. Irrelevant Alternatives Criterion The irrelevant alternatives criterion is not always satisfied by any of the 4 voting methods studied.The irrelevant alternatives criterion is not always satisfied by any of the 4 voting methods studied.

18. Example 4 5 members of a book club vote on what book to read: mystery, historical novel, or science fiction.5 members of a book club vote on what book to read: mystery, historical novel, or science fiction.

19. Example 4, cont’d a)Which book wins using plurality with elimination? No book has a majority, so H is eliminated. No book has a majority, so H is eliminated.

20. Example 4, cont’d Book M has 3 first-place votes, so it wins. The book club will read the mystery. Book M has 3 first-place votes, so it wins. The book club will read the mystery.

21. Example 4, cont’d If the science fiction book is removed, is the irrelevant alternatives criterion violated?

22. Example 4, cont’d In this table, M has 2 votes and H has 3. In this table, M has 2 votes and H has 3. Book H is the new winner Book H is the new winner

23. Arrow Impossibility Theorem Arrow Impossibility Theorem: no system of voting will always satisfy all of the 4 fairness criteria.Arrow Impossibility Theorem: no system of voting will always satisfy all of the 4 fairness criteria. This fact was proven by Kenneth Arrow in 1951. This fact was proven by Kenneth Arrow in 1951.

24. Approval Voting No voting system is always fair, but some systems are unfair less often than others. One such system is called approval voting.No voting system is always fair, but some systems are unfair less often than others. One such system is called approval voting. In approval voting:In approval voting: Each voter votes for all candidates he/she considers acceptable. Each voter votes for all candidates he/she considers acceptable. The candidate with the most votes is selected. The candidate with the most votes is selected.

25. Example 5 Three candidates are running for two positions. There are 9 voters.Three candidates are running for two positions. There are 9 voters. Who wins under approval voting?Who wins under approval voting?

26. Example 5, cont’d The vote totals are:The vote totals are: Ammee: 6 Ammee: 6 Bonnie: 7 Bonnie: 7 Celeste: 5 Celeste: 5 Bonnie and Ammee are selected for the two positions.Bonnie and Ammee are selected for the two positions.

27. Weighted Voting Systems In a weighted voting system, an individual voter may have more than one vote.In a weighted voting system, an individual voter may have more than one vote. The number of votes a voter controls is called the weight of the voter.The number of votes a voter controls is called the weight of the voter. Example: the election of the U.S. President by the Electoral College. Example: the election of the U.S. President by the Electoral College.

28. Weighted Voting Systems, cont’d Weights of voters are listed as a sequence of numbers in brackets.Weights of voters are listed as a sequence of numbers in brackets. For example, If Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11, it’s shown as [12, 11, 9, 8].For example, If Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11, it’s shown as [12, 11, 9, 8].

29. Weighted Voting Systems, cont’d The voter with the most weight is called the “first voter”, or P 1.The voter with the most weight is called the “first voter”, or P 1. The weight of the first voter is written as W 1.The weight of the first voter is written as W 1. The rest of the voters and their weights are represented similarly, in order of weight.The rest of the voters and their weights are represented similarly, in order of weight.

30. Example 1 If Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11, it looks like this: [12, 11, 9, 8].If Angie has a weight of 9, Roberta has a weight of 12, Carlos has a weight of 8, and Darrell has a weight of 11, it looks like this: [12, 11, 9, 8]. P 1 = Roberta, P 2 = Darrell, P 3 = Angie, andP 1 = Roberta, P 2 = Darrell, P 3 = Angie, and P 4 = Carlos.P 4 = Carlos. W 1 = 12, W 2 = 11, W 3 = 9, and W 4 = 8.W 1 = 12, W 2 = 11, W 3 = 9, and W 4 = 8.

31. Weighted Voting Systems, cont’d Simple majority: a motion must get more than half of the votes to pass.Simple majority: a motion must get more than half of the votes to pass. Supermajority: the number of votes required to pass a motion is higher than half of the total weight.Supermajority: the number of votes required to pass a motion is higher than half of the total weight. A common supermajority is two-thirds of the total weight. A common supermajority is two-thirds of the total weight.

32. Weighted Voting Systems, cont’d The weight required to pass a motion is called the quota.The weight required to pass a motion is called the quota. Example: A simple majority quota for the weighted voting system [12, 11, 9, 8] would be ___.Example: A simple majority quota for the weighted voting system [12, 11, 9, 8] would be ___.

33. Question: Given the weighted voting system [10, 9, 8, 8, 5], find the quota for a supermajority requirement of two-thirds of the total weight. a. 27 b. 21 c. 26 d. 20

34. Weighted Voting Systems, cont’d The quota can be added to the front of the list of weights.The quota can be added to the front of the list of weights. Example: The weighted voting system [12, 11, 9, 8] with a quota of 21 is [21: 12, 11, 9, 8].Example: The weighted voting system [12, 11, 9, 8] with a quota of 21 is [21: 12, 11, 9, 8].

35. Example 2 Given the weighted voting systemGiven the weighted voting system [21: 10, 8, 7, 7, 4, 4] suppose P 1, P 3, and P 5 vote ‘Yes’. Is the motion passed or defeated?Is the motion passed or defeated?

36. Example 2, cont’d Solution:Solution: The given voters have a combined weight of 10 + 7 + 4 = 21. The given voters have a combined weight of 10 + 7 + 4 = 21. The quota is met, so the motion passes. The quota is met, so the motion passes.

37. Example 3 Given the weighted voting systemGiven the weighted voting system [21: 10, 8, 7, 7, 4, 4] suppose P 1, P 5, and P 6 vote ‘Yes’. Is the motion passed or defeated?Is the motion passed or defeated?

38. Example 3, cont’d Solution:Solution: The given voters have a combined weight of 10 + 4 + 4 = 18. The given voters have a combined weight of 10 + 4 + 4 = 18. The quota is not met, so the motion is defeated. The quota is not met, so the motion is defeated.

39. Homework Pg. 172 Pg. 172 2, 4, 12, 24a, 30 2, 4, 12, 24a, 30 Pg. 193 Pg. 193 2, 6(a&b), 12 2, 6(a&b), 12

40. Dictators and Dummies Dummy: A voter whose presence makes no difference in the outcome of a voteDummy: A voter whose presence makes no difference in the outcome of a vote Dictator: A voter whose presence completely determines the outcome of a voteDictator: A voter whose presence completely determines the outcome of a vote When a weighted voting system has a dictator, the other voters in the system are automatically dummies. When a weighted voting system has a dictator, the other voters in the system are automatically dummies.

41. Veto Power A voter with veto power can defeat a motion by voting ‘No’ but cannot necessarily pass a motion by voting ‘Yes’.A voter with veto power can defeat a motion by voting ‘No’ but cannot necessarily pass a motion by voting ‘Yes’. Any dictator has veto power, but a voter with veto power is not necessarily a dictator. Any dictator has veto power, but a voter with veto power is not necessarily a dictator.

42. Example 7 Consider the weighted voting system [10: 10, 5, 4].Consider the weighted voting system [10: 10, 5, 4]. Are there any dummies, dictators, or voters with veto power?Are there any dummies, dictators, or voters with veto power?

43. Example 7, cont’d Solution:Solution: P 1 has enough weight to pass a motion by voting ‘Yes’ no matter how anyone else votes. P 1 has enough weight to pass a motion by voting ‘Yes’ no matter how anyone else votes. If P 1 votes ‘No’, the motion will not pass no matter how anyone else votes. If P 1 votes ‘No’, the motion will not pass no matter how anyone else votes. P 1 is a dictator, so all other voters are dummies. P 1 is a dictator, so all other voters are dummies.

44. Example 6 Consider the weighted voting systemConsider the weighted voting system [12: 7, 6, 4]. a) Are there any dummies or dictators? b) Are there any voters with veto power?