Changing Perspectives

Transcription

1 Harvey Mudd College

2 Applied Representation Theory Group at HMC Thesis Students Zajj Daugherty 05 Eric Malm 05 Melissa Banister 04 Will Chang 04 Nate Eldredge 03 Matthew Macauley 03 David Uminsky 03 Elizabeth Norton 01

3 Research Students Gregory Minton 08 Alex Eustis 06 Masanori Koyama 07 Zajj Daugherty 05 Mike Hansen 06 Marshall Pierce 06 Moana Evans 06 Julijana Gjorgjieva 06 Grant Clifford 04 Ruben Arenas 05 Nate Eldredge 03 David Uminsky 03

4 Connections

5 Connections Linear Algebra linear transformations, inner products, projections, norms

6 Connections Linear Algebra linear transformations, inner products, projections, norms Abstract Algebra symmetry, representation theory, module theory

7 Connections Linear Algebra linear transformations, inner products, projections, norms Abstract Algebra symmetry, representation theory, module theory Graph Theory adjacency matrices, distance transitve graphs, linear complexity

8 Connections Linear Algebra linear transformations, inner products, projections, norms Abstract Algebra symmetry, representation theory, module theory Graph Theory adjacency matrices, distance transitve graphs, linear complexity Numerical Analysis Lanczos Algorithm

9 New Views

10 Sum Fun

11 Sum Fun Adding

12 Sum Fun Adding

13 Sum Fun Adding

14 Sum Fun Adding

15 Sum Fun Adding ( ) + (2 + 99) + (3 + 98) + (4 + 97) + = 5050

16 Sum Fun Adding ( ) + (2 + 99) + (3 + 98) + (4 + 97) + = 5050 Child s Play 10 year old Carl Friederich Gauss ( )

17 Ants on a Stick

18 Ants on a Stick Situation 100 ants dropped on meter stick

19 Ants on a Stick Situation 100 ants dropped on meter stick each ant moving either left or right at 1 meter per min

20 Ants on a Stick Situation 100 ants dropped on meter stick each ant moving either left or right at 1 meter per min if two meet, bounce off and reverse direction

21 Ants on a Stick Situation 100 ants dropped on meter stick each ant moving either left or right at 1 meter per min if two meet, bounce off and reverse direction if ant reaches end, it falls off

22 Ants on a Stick Situation 100 ants dropped on meter stick each ant moving either left or right at 1 meter per min if two meet, bounce off and reverse direction if ant reaches end, it falls off Question How long do we have to wait for all of the ants to fall off?

23 Ants on a Stick Situation 100 ants dropped on meter stick each ant moving either left or right at 1 meter per min if two meet, bounce off and reverse direction if ant reaches end, it falls off Question How long do we have to wait for all of the ants to fall off? Answer No more than a minute!

24 Change of Perspective is Everywhere

25 Change of Perspective is Everywhere Mathematics

26 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus)

27 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus) Integration by Parts (Calculus, Differential Equations)

28 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus) Integration by Parts (Calculus, Differential Equations) Bayes Theorem (Probability, Statistics)

29 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus) Integration by Parts (Calculus, Differential Equations) Bayes Theorem (Probability, Statistics) Change of Basis (Linear Algebra, Signal Processing)

30 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus) Integration by Parts (Calculus, Differential Equations) Bayes Theorem (Probability, Statistics) Change of Basis (Linear Algebra, Signal Processing) Life

31 Change of Perspective is Everywhere Mathematics Substitution (Algebra, Calculus) Integration by Parts (Calculus, Differential Equations) Bayes Theorem (Probability, Statistics) Change of Basis (Linear Algebra, Signal Processing) Life insert just about anything here

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33 Voting

34 Voting Example Eleven voters have the following preferences: 2 ABC 3 ACB 4 BCA 2 CBA.

35 Voting Example Eleven voters have the following preferences: 2 ABC 3 ACB 4 BCA 2 CBA. Change of Perspective Focus on the procedure, not the preferences.

36 The Will of the People?

37 The Will of the People?...rather than reflecting the views of the voters, it is entirely possible for an election outcome to more accurately reflect the choice of an election procedure. D. Saari

38 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA

39 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Favorite

40 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Favorite A: 5 points

41 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Favorite A: 5 points B: 4 points

42 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Favorite A: 5 points B: 4 points C: 2 points

43 Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Favorite A: 5 points B: 4 points C: 2 points Outcome A > B > C

44 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA

45 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Top Two Favorites

46 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Top Two Favorites A: 5 points

47 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Top Two Favorites A: 5 points B: 8 points

48 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Top Two Favorites A: 5 points B: 8 points C: 9 points

49 Anti-Plurality Preferences 2 ABC 3 ACB 4 BCA 2 CBA Vote for Top Two Favorites A: 5 points B: 8 points C: 9 points Outcome C > B > A

50 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA

51 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA 1 Point for First, 1 2 Point for Second

52 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA 1 Point for First, 1 2 A: 5 points Point for Second

53 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA 1 Point for First, 1 Point for Second 2 A: 5 points B: 6 points

54 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA 1 Point for First, 1 Point for Second 2 A: 5 points B: 6 points C: points

55 Borda Count Preferences 2 ABC 3 ACB 4 BCA 2 CBA 1 Point for First, 1 Point for Second 2 A: 5 points B: 6 points C: points Outcome B > C > A

56 Common Thread

57 Common Thread Positional Voting

58 Common Thread Positional Voting 1st: 1 point

59 Common Thread Positional Voting 1st: 1 point 2nd: t points, 0 t 1

60 Common Thread Positional Voting 1st: 1 point 2nd: t points, 0 t 1 3rd: 0 points

61 Common Thread Positional Voting 1st: 1 point 2nd: t points, 0 t 1 3rd: 0 points Tally Matrix

62 Common Thread Positional Voting 1st: 1 point 2nd: t points, 0 t 1 3rd: 0 points Tally Matrix 2 ABC 1 1 t 0 t 0 3 ACB t t 0 BAC 0 t 0 t BCA 0 CAB 2 CBA

63 Common Thread Positional Voting 1st: 1 point 2nd: t points, 0 t 1 3rd: 0 points Tally Matrix 2 ABC 1 1 t 0 t 0 3 ACB t t 0 5 A BAC 0 t 0 t = 4 + 4t B BCA t C CAB 2 CBA

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66 Main Idea

67 Main Idea Decompose the voting profile into symmetry-invariant pieces.

68 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic.

69 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic. Use the pieces to tell a story or build paradoxes.

70 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic. Use the pieces to tell a story or build paradoxes. Symmetry

71 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic. Use the pieces to tell a story or build paradoxes. Symmetry Based on permuting the labels of the candidates.

72 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic. Use the pieces to tell a story or build paradoxes. Symmetry Based on permuting the labels of the candidates. Commutative gives classical harmonic analysis.

73 Main Idea Decompose the voting profile into symmetry-invariant pieces. Each piece should capture some fundamental characteristic. Use the pieces to tell a story or build paradoxes. Symmetry Based on permuting the labels of the candidates. Commutative gives classical harmonic analysis. Noncommutative gives noncommutative harmonic analysis.

74 Framework

75 Framework Situation

76 Framework Situation Voters are given list of n candidates.

77 Framework Situation Voters are given list of n candidates. They return their votes.

78 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied.

79 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced.

80 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced. Algebraic Insights

81 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced. Algebraic Insights There is no set order on the candidates.

82 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced. Algebraic Insights There is no set order on the candidates. The votes form a vector.

83 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced. Algebraic Insights There is no set order on the candidates. The votes form a vector. Tallying is a linear transformation.

84 Framework Situation Voters are given list of n candidates. They return their votes. The votes are tallied. The winner(s) are announced. Algebraic Insights There is no set order on the candidates. The votes form a vector. Tallying is a linear transformation. Winner depends on tallying method.

85 Big Idea

86 Big Idea Decompositions Use symmetry to decompose the profile space V = V 0 V 1 V k+1 V k.

87 Big Idea Decompositions Use symmetry to decompose the profile space V = V 0 V 1 V k+1 V k. Analysis Use the projections of the profile into these subspaces to explain the profile.

88 Big Idea Decompositions Use symmetry to decompose the profile space V = V 0 V 1 V k+1 V k. Analysis Use the projections of the profile into these subspaces to explain the profile. Paradoxes Compare the impact that different voting procedures have on these subspaces to construct paradoxical profiles.

89 Example

90 Example If the profile is p = [ ] 6 18 A B

91 Example If the profile is p = then it decomposes as [ ] 6 = 18 [ ] 6 18 [ ] A B [ ] 6. 6

92 Example If the profile is 4 p = A B C

93 Example If the profile is 4 p = A B C then it decomposes as =

94 Bigger Example 2 AB 12 AC 11 AD 6 AE 17 BC p = 8 BD 4 BE 24 CD 20 CE 6 DE

95 Bigger Example p = AB AC AD AE BC BD BE CD CE DE = /3 16/3 8/3 7 16/3 8/3 7 34/ /3 13/3 8/3 2 2/3 1/3 0 5/3 2 4

96 Bigger Example (cont.)

97 Bigger Example (cont.) Analysis p 0 2 = 1210, p , and p

98 Bigger Example (cont.) Analysis p 0 2 = 1210, p , and p Inner products for first order effects: A B C D E

99 Bigger Example (cont.) Analysis p 0 2 = 1210, p , and p Inner products for first order effects: A B C D E Inner products for second order effects: AB AC AD AE BC BD BE CD CE DE

100 Bigger Example (cont.) Analysis p 0 2 = 1210, p , and p Inner products for first order effects: A B C D E Inner products for second order effects: AB AC AD AE BC BD BE CD CE DE Story People seem to really like C.

101 Committee Voting Data

102 Committee Voting Data Situation Given n-member committee.

103 Committee Voting Data Situation Given n-member committee. Each vote yields winners and losers.

104 Committee Voting Data Situation Given n-member committee. Each vote yields winners and losers. New Perspective

105 Committee Voting Data Situation Given n-member committee. Each vote yields winners and losers. New Perspective Let the issues do all of the voting!

106 Committee Voting Data Situation Given n-member committee. Each vote yields winners and losers. New Perspective Let the issues do all of the voting! Look for coalitions.

107 Supreme Court Table: Rehnquist Court , 192 non-unanimous cases split subspace norm 2 four largest (using absolute value) inner products 8-1 M Sv.996 By Gi Sc M ThSc.732 SvGi.476 SvBy.354 SvSc M 1 59 Sv.695 O So Ke M 3 72 ReThSc.647 O ThSc.345 ByThSc ReO Sc M ThSc.626 SvGi.345 GiTh ReTh M 1 16 Th.656 Sc.540 Ke By M SvGiBySo.954 KeReThSc.344 O ReThSc.341 SvBySoSc M SvBySo.379 SvGiSo.368 GiBySo.293 SvGiBy M SvSo.315 SvBy.301 ThSc.282 SvGi M 1 22 Ke So.418 O Gi.380 Sv Stevens Gi Ginsburg By Breyer So Souter Ke Kennedy O O Connor Re Rehnquist Th Thomas Sc Scalia

108 Paradoxes

109 Paradoxes Positional vs Pairwise

110 Paradoxes Positional vs Pairwise D. Saari Borda Count is best (geometry).

111 Paradoxes Positional vs Pairwise D. Saari Borda Count is best (geometry). Z. Daugherty Borda-like Count is best (algebra).

112 Paradoxes Positional vs Pairwise D. Saari Borda Count is best (geometry). Z. Daugherty Borda-like Count is best (algebra). Positional vs Approval

113 Paradoxes Positional vs Pairwise D. Saari Borda Count is best (geometry). Z. Daugherty Borda-like Count is best (algebra). Positional vs Approval G. Minton and A. Eustis Borda Count is close, but not close enough! (algebra)

114 Thanks!

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