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CMU 15-896 Social choice 1: The basics Teacher: Ariel Procaccia Social choice theory A mathematical theory that deals with aggregation of individual preferences Origins in ancient Greece Formal foundations: 18 th Century

  1. CMU 15-896 Social choice 1: The basics Teacher: Ariel Procaccia

  2. Social choice theory • A mathematical theory that deals with aggregation of individual preferences • Origins in ancient Greece • Formal foundations: 18 th Century (Condorcet and Borda) • 19 th Century: Charles Dodgson • 20 th Century: Nobel prizes to Arrow and Sen 15896 Spring 2016: Lecture 1 2

  3. The voting model • Set of voters • Set of alternatives 1 2 3 • Each voter has a ranking over a c b the alternatives b a c • Preference profile = collection c b a of all voters’ rankings 15896 Spring 2016: Lecture 1 3

  4. Voting rules • Voting rule = function from preference profiles to alternatives that specifies the winner of the election • Plurality Each voter awards one point to top o alternative Alternative with most points wins o Used in almost all political elections o 15896 Spring 2016: Lecture 1 4

  5. More voting rules • Borda count Each voter awards � � � o points to alternative ranked � ’th Alternative with most points o wins Proposed in the 18 th Century o by the chevalier de Borda Used for elections to the o national assembly of Slovenia Lordi, Eurovision 2006 winners Similar to rule used in the o Eurovision song contest 15896 Spring 2016: Lecture 1 5

  6. More voting rules • Positional scoring rules Defined by vector �� � , … , � � � o Plurality = �1,0, … , 0� , Borda = �� � 1, � � 2, … , 0� o beats in a pairwise election if the majority of • voters prefer to • Plurality with runoff First round: two alternatives with highest plurality o scores survive Second round: pairwise election between these two o alternatives 15896 Spring 2016: Lecture 1 6

  7. More voting rules • Single Transferable vote (STV) rounds o In each round, alternative with least o plurality votes is eliminated Alternative left standing is the winner o Used in Ireland, Malta, Australia, and New o Zealand (and Cambridge, MA) 15896 Spring 2016: Lecture 1 7

  8. STV: example 2 2 1 2 2 1 voters voters voter voters voters voter a b c a b c b a d b a b c d b c c a d c a 2 2 1 2 2 1 voters voters voter voters voters voter a b b b b b b a a 15896 Spring 2016: Lecture 1 8

  9. Social choice axioms • How do we choose among the different voting rules? Via desirable properties! • Majority consistency = if a majority of voters rank alternative first, then should be the winner Which of the rules we talked about is not majority consistent? 15896 Spring 2016: Lecture 1 9

  10. Marquis de Condorcet • 18 th Century French Mathematician, philosopher, political scientist • One of the leaders of the French revolution • After the revolution became a fugitive • His cover was blown and he died mysteriously in prison 15896 Spring 2016: Lecture 1 10

  11. Condorcet winner • Recall: beats in a pairwise election if a majority of voters 1 2 3 rank above a c b • Condorcet winner beats every b a c other alternative in pairwise c b a election • Condorcet paradox = cycle in majority preferences 15896 Spring 2016: Lecture 1 11

  12. Condorcet consistency • Condorcet consistency = select a Condorcet winner if one exists Which of the rules we talked about is Condorcet consistent? 15896 Spring 2016: Lecture 1 12

  13. Condorcet consistency Poll: What is the relation between majority consistency and Condorcet consistency? Majority cons. Condorcet cons. 1. Condorcet cons. Majority cons. 2. Equivalent 3. Incomparable 4. 15896 Spring 2016: Lecture 1 13

  14. More voting rules • Copeland Alternative’s score is #alternatives it beats o in pairwise elections Why does Copeland satisfy the Condorcet o criterion? • Maximin Score of is min � � o Why does Maximin satisfy the Condorcet o criterion? 15896 Spring 2016: Lecture 1 14

  15. Application: web search • Generalized Condorcet: if there is a partition of such that a majority prefers every to every , then is ranked above • Assumption: spam website identified by a majority of search engines • When aggregating results from different search engines, spam websites will be ranked last [Dwork et al., WWW 2001] 15896 Spring 2016: Lecture 1 15

  16. Application: Web Search x a b a a y z b b b a z z z x x y x y y overall 15896 Spring 2016: Lecture 1 16

  17. Dodgson’s Rule • Distance function between profiles: #swaps between adjacent candidates • Dodgson score of = the min distance from a profile where is a Condorcet winner • Dodgson’s rule: select candidate that minimizes Dodgson score • The problem of computing the Dodgson score is NP-complete! 15896 Spring 2016: Lecture 1 18

  18. Dodgson Unleashed a b e e b b a b c e c c c d d d d a a a e e d b c Voter 1 Voter 2 Voter 3 Voter 4 Voter 5 15896 Spring 2016: Lecture 1 19

  19. Awesome example • Plurality: • Borda: 33 16 3 8 18 22 voters voters voters voters voters voters • Condorcet a b c c d e winner: b d d e e c c c b b c b • STV: d e a d b d • Plurality e a e a a a with runoff: 15896 Spring 2016: Lecture 1 20

  20. Is social choice practical? • UK referendum: Choose between plurality and STV as a method for electing MPs • Academics agreed STV is better... • ... but STV seen as beneficial to the hated Nick Clegg • Hard to change political elections! 15896 Spring 2016: Lecture 1 21

  21. Computational social choice • However: in human computation o systems... in multiagent systems... o the designer is free to employ any voting rule! • Computational social choice focuses on positive results through computational thinking 15896 Spring 2016: Lecture 1 22

  22. Example: Robobees • Robobees need to decide on a joint plan (alternative) • Many possible plans • Each robobee (agent) has a numerical evluation (utility) for each alternative • Want to maximize sum of utilities = social welfare • Communication is restricted 15896 Spring 2016: Lecture 1 23

  23. 15896 Spring 2016: Lecture 1 24

  24. Example: Robobees • Approach 1: communicate utilities a May be infeasible o n/2  1 agents • Approach 2: each agent votes for favorite alternative (plurality) a b c d e f g h log bits per agent o n/2 + 1 agents May select a bad o alternative 15896 Spring 2016: Lecture 1 25

  25. Example: Robobees • Approach 3: each agent votes for an alternative with probability proportional to its utility • Theorem [Caragiannis & P 2011]: if then this approach gives almost optimal social welfare in expectation 15896 Spring 2016: Lecture 1 26

  26. Example: Pnyx 15896 Spring 2016: Lecture 1 27

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