CMU Social choice 2: Manipulation. Teacher: Ariel Procaccia

Transcription

1 CMU Social choice 2: Manipulation Teacher: Ariel Procaccia

2 Reminder: Voting Set of voters Set of alternatives Each voter has a ranking over the alternatives means that voter prefers to Preference profile = collection of all voters rankings Voting rule = function from preference profiles to alternatives Important: so far voters were honest! 2

3 Manipulation Using Borda count Top profile: b wins Bottom profile: a wins By changing his vote, voter 3 achieves a better outcome! b b a a a b c c c d d d b b a a a c c c d d d b 3

4 Borda responds to critics My scheme is intended only for honest men! Random 18 th Century French Dude 4

5 Strategyproofness A voting rule is strategyproof (SP) if a voter can never benefit from lying about his preferences: Maximum value of which plurality is SP? for 5

6 Strategyproofness A voting rule is dictatorial if there is a voter who always gets his most preferred alternative A voting rule is constant if the same alternative is always chosen Constant functions and dictatorships are SP Dictatorship Constant function 6

7 Gibbard-Satterthwaite A voting rule is onto if any alternative can win Theorem (Gibbard-Satterthwaite): If then any voting rule that is SP and onto is dictatorial In other words, any voting rule that is onto and nondictatorial is manipulable Gibbard Satterthwaite 7

8 Proof sketch of G-S Lemmas (prove in HW1): o Strong monotonicity: is SP rule, profile,. Then for all profiles s.t. o Pareto optimality: is SP+onto rule, profile. If for all then Let us assume that, and neutrality: for all 8

9 Proof sketch of G-S Say and Consider the following profile a b c d b c d a c d a b d a b c e e e e Pareto optimality is not the winner Suppose 9

10 Proof sketch of G-S a b c d b c d a c d a b d a b c e e e e a d d d d a a a b b b b c c c c e e e e Strong monotonicity 10

11 a d d d d a a a b b b b c c c c e e e e a d d d d b a a b c b b c e c c e a e e Poll 1: How many options are there for?

12 a d d d d b a a b c b b c e c c e a e e a d d d d b b a b c c b c e e c e a a e Pareto optimality,, [SP ] Strong monotonicity for every where ranks first Neutrality is a dictator a d d d d b b b b c c c c e e e e a a a 12

13 Circumventing G-S Restricted preferences (this lecture) Money mechanism design (later) Computational complexity (this lecture) 13

14 Single peaked preferences We want to choose a location for a public good (e.g., library) on a street Alternatives = possible locations Each voter has an ideal location (peak) The closer the library is to a voter s peak, the happier he is 14

15 Single peaked preferences Leftmost point mechanism: return the leftmost point Midpoint mechanism: return the average of leftmost and rightmost points Which of the two mechanisms is SP? 15

16 The median Select the median peak The median is a Condorcet winner! The median is onto The median is nondictatorial 16

17 The median is SP 17

18 Complexity of manipulation Manipulation is always possible in theory But can we design voting rules where it is difficult in practice? Are there reasonable voting rules where manipulation is a hard computational problem? [Bartholdi et al., SC&W 1989] 18

19 The computational problem -MANIPULATION problem: o o Given votes of nonmanipulators and a preferred candidate Can manipulator cast vote that makes uniquely win under? Example: Borda, b b a a c c d d b b a a a c c c d d d b 19

20 A greedy algorithm Rank in first place While there are unranked alternatives: o o If there is an alternative that can be placed in next spot without preventing from winning, place this alternative Otherwise return false 20

21 Example: Borda b b a a a c c d d b b a a a c c c b d d b b a a a b c c d d b b a a a c c c d d d b b a a a c c c d d b b a a a c c c d d d b 21

22 Example: Copeland a b e e a b a c c c d b b d e a a e c d d Preference profile a b c d e a b c d e Pairwise elections 22

23 Example: Copeland a b e e a b a c c c c d b b d e a a e c d d Preference profile a b c d e a b c d e Pairwise elections 23

24 Example: Copeland a b e e a b a c c c c d b b d d e a a e c d d Preference profile a b c d e a b c d e Pairwise elections 24

25 Example: Copeland a b e e a b a c c c c d b b d d e a a e e c d d Preference profile a b c d e a b c d e Pairwise elections 25

26 Example: Copeland a b e e a b a c c c c d b b d d e a a e e c d d b Preference profile a b c d e a b c d e Pairwise elections 26

27 When does the alg work? Theorem [Bartholdi et al., SCW 89]: Fix and the votes of other voters. Let be a rule s.t. function such that: 1. For every, chooses a candidate that uniquely maximizes, 2. : :,, Then the algorithm always decides -MANIPULATION correctly What is for plurality? 27

28 Proof of Theorem Suppose the algorithm failed, producing a partial ranking Assume for contradiction makes win alternatives not ranked in highest ranked alternative in according to Complete by adding others arbitrarily first, then Output of alg, 28

29 Proof of Theorem Property 2 Property 1 and makes winner the Property 2 Conclusion:, so the alg could have inserted next Output of alg, 29

30 Voting rules that are hard to manipulate Natural rules o Copeland with second order tie breaking [Bartholdi et al., SCW 89] o STV [Bartholdi&Orlin, SCW 91] o Ranked Pairs [Xia et al., IJCAI 09] Order pairwise elections by decreasing strength of victory Successively lock in results of pairwise elections unless it leads to cycle Winner is the top ranked candidate in final order Can also tweak easy to manipulate voting rules [Conitzer&Sandholm, IJCAI 03]

31 Example: ranked pairs a 6 b d 2 c 31

32 Example: ranked pairs a 6 b d 2 c 32

33 Example: ranked pairs a 6 b 8 4 d 2 c 33

34 Example: ranked pairs a 6 b 4 d 2 c 34

35 Example: ranked pairs a b 4 d 2 c 35

36 Example: ranked pairs a b d 2 c 36

37 Example: ranked pairs a b d c 37