Optimal Voting Rules. Alexander Scheer. November 14, 2012

Transcription

1 Optimal Voting Rules Alexander Scheer November 14,

2 Introduction What we have seen in the last weeks: Borda's Count Condorcet's Paradox 2

3 Introduction What we have seen in the last weeks: Independence of Irrelevant Alternatives (IIA) Social choices should be independent of irrelevant alternatives. Pareto axiom If everybody prefers x to y, then x should be socially preferred to y. 3

4 Introduction What we have seen in the last weeks: Impossibility theorems Arrow's theorem Any Social Welfare Function for at least 3 alternatives that satisfies Pareto and IIA must be a dictatorship. 4

5 Introduction But today: Different point of view Assumption: There is an objectively best alternative! We want to find the right one with highest probability. 5

6 Outline 0. Introduction 1. Two Alternatives Condorcet's Majority Rule 2. Condorcet's Rule of Three Borda's Rule and Condorcet's Response Local Independence of Irrelevant Alternatives A Compromise: Kemeny's Approach 3. Conclusion 6

7 Two Alternatives 7

8 Condorcet's Majority Rule Problem: Group of voters 2 alternatives Assumptions: either a or b is objectively best each voter is more likely to make the right choice than the wrong one (p > ½) Two Alternatives 8

9 Condorcet's Majority Rule Majority Rule: If the voters choose independently, then the choice with the most votes is most likely to be correct. Two Alternatives 9

10 Condorcet's Majority Rule Example: 100 individuals, alternatives a, b Assumption: individuals are right 60% of time Two Alternatives 10

11 Condorcet's Majority Rule Example: 100 individuals, alternatives a, b Assumption: individuals are right 60% of time a: 55 votes b: 45 votes 2 cases Two Alternatives 11

12 Condorcet's Majority Rule Case 1: a is the best choice p a = probability for voting pattern a: 55, b:45 p a = 100! 55! 45! Two Alternatives 12

13 Condorcet's Majority Rule Case 2: b is the best choice p b = probability for voting pattern a: 55, b:45 p b = 100! 45! 55! Two Alternatives 13

14 Condorcet's Majority Rule p a p b = a is 58 times more likely to be correct than b Maximum Likelihood Estimation Two Alternatives 14

15 Condorcet's Majority Rule Maximum Likelihood Estimation: consider every possible outcome calculate probability for voting pattern outcome with highest probability most likely correct Two Alternatives 15

16 Condorcet's Majority Rule majority rule is a statistically optimal method (when assumptions are fulfilled) larger group: probability of majority decision to be correct reaches unity Two Alternatives 16

17 Three or More Alternatives 17

18 Ideally: Choose alternative with majority over every other! 18

19 But: Alternative with simple majority may not exist: 60 voters, ranking over 3 alternatives abc: 23 bca: 17 bac: 2 cab: 10 cba: 8 a > b (33 : 27) b > c (42 : 18) c > a (35 : 25) Cyclic! 19

20 But: Alternative with simple majority may not exist: 60 voters, ranking over 3 alternatives abc: 23 bca: 17 bac: 2 cab: 10 cba: 8 a > b (33 : 27) b > c (42 : 18) c > a (35 : 25) Condorcet's Paradox 20

21 Condorcet's Rule of Three 21

22 Condorcet's Rule of Three Visualization of the problem: Vote graph 22

23 Condorcet's Rule of Three Example: Probability of ranking abc a > b b > c a > c 23

24 Condorcet's Rule of Three Example: Probability of ranking abc a b b c a c 24

25 Condorcet's Rule of Three a b b c a c Total pairwise support: =

26 Condorcet's Rule of Three Total pairwise support: abc: 100 bca: 104 acb: 76 cab: 86 bac: 94 cba: 80 Condorcet's Rule of Three 26

27 Condorcet's Rule of Three Again: Maximum Likelihood Estimation: Probability that a certain voting pattern occurs given that the ranking is correct Assumption: p > ½, voters choose independently 27

28 Condorcet's Rule of Three Ex.: Case abc is correct a b b c a c (33) (42) (25) p 33 (1 p) 27 p 42 (1 p) 18 p 25 (1 p) 35 28

29 Condorcet's Rule of Three Ex.: Case abc is correct a b b c a c (33) (42) (25) p 33 (1 p) 27 p 42 (1 p) 18 p 25 (1 p) 35 = p 100 (1 p) 80 29

30 Condorcet's Rule of Three Ex.: Case abc is correct a b b c a c (33) (42) (25) p 33 (1 p) 27 p 42 (1 p) 18 p 25 (1 p) 35 = p 100 (1 p) 80 other cases are calculated similarly since p > ½, maximum likelihood ranking is the one with greatest total support can be applied to more than 3 alternatives 30

31 Borda's Rule 31

32 Borda's Rule Borda's observation: The voting with the most first places is not necessarily the one that has the highest standing overall. 32

33 Borda's Rule Example: 21 voters, 3 alternatives bca: 7 acb: 7 cba: 6 abc: 1 First places: a 8 b 7 c 6 33

34 Borda's Rule Example: 21 voters, 3 alternatives But: bca: 7 acb: 7 cba: 6 abc: 1 First places: a 8 b 7 c 6 7 who prefer a like c better than b 7 who prefer b like c better than a c should be the compromise candidate (even though it has fewest first places) 34

35 Borda's Rule each voter ranks the candidates and scores the alternatives b c a

36 Borda's Rule each voter ranks the candidates and scores the alternatives b c a calculate Borda score for every candidate and order them accordingly c b a exactly the opposite of first-place vote 36

37 Condorcet's Response Counterexample: (81 voters) Peter Peter Paul Paul Jack Jack Paul Jack Peter Jack Peter Paul Jack Paul Jack Peter Paul Peter 37

38 Condorcet's Response Counterexample: (81 voters) Peter Peter Paul Paul Jack Jack Paul Jack Peter Jack Peter Paul Jack Paul Jack Peter Paul Peter Borda's Rule: Paul > Peter > Jack 38

39 Condorcet's Response Counterexample: (81 voters) Peter Peter Paul Paul Jack Jack Paul Jack Peter Jack Peter Paul Jack Paul Jack Peter Paul Peter But: Peter has a simple majority over both Peter and Jack Peter > Paul (41:40), Peter > Jack (60:21) Peter should be ranked first! 39

40 Condorcet's Response More generally... Majority principle: If there is a majority alternative, it should be ranked first! 40

41 Condorcet's Response comparison between Peter and Paul should not depend on their relation to Jack Arrow: Independence of Irrelevant Alternatives (IIA) 41

42 Condorcet's Response Condorcet: Any scoring system yields outcomes that are based on irrelevant factors and violates the majority principle. 42

43 Recall: more than 2 alternatives violation of IIA (Arrow's theorem) But why is IIA so important? Irrelevant factors (e.g. Jack) affect the outcome We need a weaker version! 43

44 Local Independence of Irrelevant Alternatives (LIIA) 44

45 LIIA Ordering: a b c d e f 45

46 LIIA Ordering: a b c d e f 46

47 LIIA Ordering: a b c d e f Removing candidates outside of the interval may not affect the ordering inside of the interval. 47

48 LIIA a b c d e f Sum a b c d e f voters, 6 alternatives each cell: number of votes by row candidate over column candidate (pairwise comparison) 48

49 LIIA (100 voters) a b c d e f Sum a b c d e f Sum = Borda score Borda's rule cabdef 49

50 LIIA (100 voters) a b c d e f Sum a b c d e f a, b, c only choices under discussion d, e, f included for manipulation? 50

51 LIIA (100 voters) a b c d e f Sum a b c d e f a, b, c only choices under discussion d, e, f included for manipulation? 51

52 LIIA (100 voters) a b c Sum a b c Borda's rule bac exactly the opposite 52

53 LIIA And again: Maximum Likelihood Solution: If an alternative has a simple majority over every other, it must be ranked first. 53

54 LIIA (100 voters) a b c d e f Sum a b c d e f a wins every pairwise comparison a 54

55 LIIA (100 voters) a b c d e f Sum a b c d e f a wins every pairwise comparison a b wins every pairwise comparison ab 55

56 LIIA (100 voters) a b c d e f Sum a b c d e f a wins every pairwise comparison a b wins every pairwise comparison ab c wins every pairwise comparison abc 56

57 LIIA (100 voters) a b c d e f Sum a b c d e f a wins every pairwise comparison a b wins every pairwise comparison ab c wins every pairwise comparison abc abcdfe 57

58 LIIA We ranked every alternative as it would be in the absence of the better ones. Maximum Likelihood Solution is the only reasonable ranking method that satisfies LIIA 58

59 Previous presumption: there is a best ordering Not always useful in reality! Now: compromise between conflicting opinions 59

60 Kemeny's Approach

61 Kemeny's Approach John Kemeny (1959) view voters' opinions as data find an ordering that averages the data 61

62 Kemeny's Approach John Kemeny (1959) view voters' opinions as data find an ordering that averages the data We need: metric defined on the set of rankings 62

63 Kemeny's Approach Kemeny's proposal: d(r, R') = number of pairs of alternatives on which they differ 63

64 Kemeny's Approach Kemeny's proposal: d(r, R') = number of pairs of alternatives on which they differ Example: R = abcd, R'= dabc d(r, R') = 3 (a,d), (b,d), (c,d) 64

65 Kemeny's Approach Best definition of compromise ordering? mean: minimizes sum of squares of distances from a given set of n rankings median: minimizes sum of absolute distances from the given n rankings 65

66 Kemeny's Approach Best definition of compromise ordering? mean: minimizes sum of squares of distances from a given set of n rankings median: minimizes sum of absolute distances from the given n rankings 66

67 Kemeny's Approach Example: 41 voters abc: 21 bca: 5 cab: 4 cba: 11 median: abc maximum likelihood: abc (most first-places) mean: bac? 67

68 Kemeny's Approach Example: 41 voters abc: 21 bca: 5 cab: 4 cba: 11 median: abc maximum likelihood: abc (most first-places) mean: bac? median seems to be most appropriate solution 68

69 Conclusion Maximum Likelihood Method can be justified from several points of view arguably best method to estimate what decision most likely represents the common objective resistant to strategic manipulation (LIIA) represents the median opinion 69

70 Thank you for your attention! 70

71 References: Young, P. (1995) Optimal Voting Rules. Journal of Economic Perspectives. 9 (1), Images: Young, P. (1995) Optimal Voting Rules. Journal of Economic Perspectives. 9 (1), Page 54 References and Images 71