Efficient Algorithms for Hard Problems on Structured Electorates

Transcription

1 Aspects of Computation 2017 Efficient Algorithms for Hard Problems on Structured Electorates Neeldhara Misra

2 The standard Voting Setup and some problems that we will encounter.

3 The standard Voting Setup and some problems that we will encounter. Single Peaked Preferences definition, recognition, strategy-proofness, elicitation

4 The standard Voting Setup and some problems that we will encounter. Single Peaked Preferences definition, recognition, strategy-proofness, elicitation Single Crossing preferences definition, recognition and Condorcet winners

5 The standard Voting Setup and some problems that we will encounter. Single Peaked Preferences definition, recognition, strategy-proofness, elicitation Single Crossing preferences definition, recognition and Condorcet winners Concluding remarks nearly structured preferences, dichotomous preferences

6 The standard Voting Setup and some problems that we will encounter. The Handbook of Computational Social Choice, Brandt, Conitzer, Endriss, Lang and Procaccia; 2016

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8 A typical voting scenario involves a set of alternatives, and a set of voters. Voters have preferences over the alternatives, which can be modelled in various ways. In this talk, we ll think of preferences as linear orders over the rankings.

9 A typical voting scenario involves a set of alternatives, and a set of voters. Voters have preferences over the alternatives, which can be modelled in various ways. In this talk, we ll think of preferences as linear orders over the rankings.

10 A typical voting scenario involves a set of alternatives, and a set of voters. Voters have preferences over the alternatives, which can be modelled in various ways. In this talk, we ll think of preferences as linear orders over the rankings.

11 A group of friends have an evening to spend.

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14 (P lurality)

15 (P lurality)

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17 We say that a voter (or a group of voters) can manipulate if they can obtain a more desirable outcome by misreporting their preferences. We ll see that plurality is vulnerable to this behaviour.

18 We say that a voter (or a group of voters) can manipulate if they can obtain a more desirable outcome by misreporting their preferences. We ll see that plurality is vulnerable to this behaviour.

19 (P lurality)

20 (P lurality)

21 (P lurality)

22 (P lurality)

23 Manipulation (P lurality)

24 Borda, 1770 This scheme is intended only for honest men.

25 Elimination over multiple rounds

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29 (ST V)

30 Pairwise Elections

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36 An alternative that beats all the others in pairwise comparisons.

37 An alternative that beats all the others in pairwise comparisons. (C ondorcet)

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39 An alternative that beats all the others in pairwise comparisons.

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41 An alternative that beats all the others in pairwise comparisons. may not exist!

42 Desirable properties of voting rules.

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44 We ll now pause to formally define social choice functions and social welfare functions.

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46 Social Welfare Functions (SWF )

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48 Social Choice Functions (SCF )

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50 Unanimity

51 Unanimity

52 Unanimity If everyone prefers A to B then the consensus ranking also prefers A over B.

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54 Independence of Irrelevant Alternatives

55 Independence of Irrelevant Alternatives

56 Independence of Irrelevant Alternatives

57 Independence of Irrelevant Alternatives Suppose a SWF prefers A over B in the consensus ranking for a profile P. Let Q be a profile that is the same as P when projected on the candidates A and B. Then the SWF must prefer A over B in the consensus ranking that it determines for Q as well.

58 Arrow (1949)

59 Arrow (1949) When voters have three or more alternatives, any social welfare function which respects unanimity and independence of irrelevant alternatives is a dictatorship.

60 Gibbard-Satterthwaite (1973/75)

61 Gibbard-Satterthwaite (1973/75) Any strategy- proof SCF where at least three candidates have some chance of being selected must be a dictatorship.

62 Some Popular Workarounds

63 Some Popular Workarounds Domain Restrictions

64 Some Popular Workarounds Domain Restrictions Randomised Voting Rules

65 Some Popular Workarounds Domain Restrictions Randomised Voting Rules Computational Hardness

66 Some Popular Workarounds Domain Restrictions Randomised Voting Rules Computational Hardness

67 Single Peaked Preferences definition, recognition, strategy-proofness, elicitation

68 Single Peaked Preferences Definition The Theory of Committees and Elections. Black, D.,New York: Cambridge University Press, 1958

69 A B C D E F G Left Center Right

70 A B C D E F G Left Center Right

71 A B C D E F G Left Center Right E D C F G B A

72 A B C D E F G Left Center Right E D C F G B A

73 A B C D E F G Left Center Right E D C F G B A

74 A B C D E F G Left Center Right E D C F G B A

75 A B C D E F G Left Center Right E D C F G B A E D F C B G A

76 A B C D E F G Left Center Right

77 A B C D E F G Left Center Right

78 A B C D E F G Left Center Right If an agent with single-peaked preferences prefers x to y, one of the following must be true: - x is the agent s peak, - x and y are on opposite sides of the agent s peak, or - x is closer to the peak than y.

79 A B C D E F G Left Center Right The notion is popular for several reasons: - No Condorcet Cycles. - No incentive for an agent to misreport its preferences. - Identifiable in polynomial time. - Reasonable (?) model of actual elections.

80 Single Peaked Preferences Recognition Stable Matching with Preferences Derived from a Psychological Model Bartholdi and Trick, Operations Research Letters, 1986

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82 Recognising if a given profile is single-peaked with respect to some preference ordering. Checking if a 0/1 matrix has the consecutive-ones property.

83 A B C D E F G

84 A B C D E F G E D F C B A G

85 A B C D E F G E D F C B A G A fixed permutation

86 A B C D E F G E D F C B A G A fixed permutation

87 A B C D E F G E D F C B A G A fixed permutation

88 A B C D E F G E D F C B A G A fixed permutation

89 A B C D E F G E D F C B A G A fixed permutation

90 A B C D E F G E D F C B A G A fixed permutation

91 A B C D E F G E D F C B A G A fixed permutation

92 A B C D E F G E D F C B A G A fixed permutation

93 A B C D E F G E D F C B A G A fixed permutation

94 A B C D E F G E D F C B A G A fixed permutation

95 A B C D E F G E D F C B A G A fixed permutation

96 A B C D E F G E D F C B A G A fixed permutation

97 A B C D E F G E D F C B A G A fixed permutation

98 A B C D E F G E D F C B A G A fixed permutation

99 Single Peaked Preferences Strategyproofness The Theory of Committees and Elections. Black, D.,New York: Cambridge University Press, 1958

100 A B C D E F G

101 A B C D E F G

102 A B C D E F G

103 A B C D E F G Claim: D beats all other candidates in pairwise elections.

104 A B C D E F G Claim: D beats all other candidates in pairwise elections.

105 A B C D E F G Claim: D beats all other candidates in pairwise elections.

106 A B C D E F G (Peak to the left of D, F further than D) Claim: D beats all other candidates in pairwise elections.

107 A B C D E F G Claim: D beats all other candidates in pairwise elections.

108 A B C D E F G Claim: D beats all other candidates in pairwise elections.

109 A B C D E F G Claim: D beats all other candidates in pairwise elections.

110 A B C D E F G (Peak to the right of D, B further than D) Claim: D beats all other candidates in pairwise elections.

111 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

112 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

113 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

114 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

115 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

116 A B C D E F G Claim: Choosing D also leaves nobody with any incentive to manipulate.

117 Single Peaked Preferences Preference Elicitation Eliciting single-peaked preferences using comparison queries, Conitzer, J. Artif. Intell. Res.; 2016

118 When the number of candidates is large, soliciting a full ranking can be a little unmanageable.

119 When the number of candidates is large, soliciting a full ranking can be a little unmanageable. Voters typically find it easier to answer comparison queries : Would you rather hang out over coffee or join me for a concert?

120 How many such queries to do we need to make to be able to reconstruct the full preference?

121 How many such queries to do we need to make to be able to reconstruct the full preference? Just like the weighing scale puzzles, except you can only compare two single options at a time.

122 Using a merge sort like idea, O(m log m) comparisons are enough.

123 Using a merge sort like idea, O(m log m) comparisons are enough. Recursively order half the alternatives.

124 Using a merge sort like idea, O(m log m) comparisons are enough. Recursively order half the alternatives.

125 Using a merge sort like idea, O(m log m) comparisons are enough. Merge the two lists with a linear number of queries.

126 A B C D E F G Left Center Right

127 and we can do better if the preferences are single-peaked! A B C D E F G Left Center Right

128 and we can do better if the preferences are single-peaked! (i) Identify the peak: use binary search. A B C D E F G Left Center Right

129 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. A B C D E F G Left Center Right

130 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. E A B C D F G Left Center Right

131 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. this signals that the peak is to the right. E A B C D F G Left Center Right

132 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. A B C D E F G Left Center Right

133 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. D A B C E F G Left Center Right

134 [Better query complexities for elicitation.] (i) Identify the peak: use binary search. D this signals that the peak is to the left. A B C E F G Left Center Right

135 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. A B C D E F G Left Center Right

136 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. A B C D E F G Left Right

137 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C A B C D E F G Left Right

138 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C A B C D E F G Left Right

139 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C A B C D E F G Left Right

140 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C B A C D E F G Left Right

141 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C B A C D E F G Left Right

142 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C A B C D E F G Left Right

143 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C D A B C E F G Left Right

144 [Better query complexities for elicitation.] (ii) Once we know the peak, the rest is O(m) queries. C D A B C E F G Left Right

145 [Better query complexities for elicitation.] A B C D E F G Left Center Right

146 [Better query complexities for elicitation.] (i) Identify the peak: use binary search (O(log m) queries). (ii) Once we know the peak, the rest is O(m) queries. A B C D E F G Left Center Right

147 Single Crossing Preferences definition, recognition and Condorcet winners

148 Single Crossing Preferences Definition

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150 A profile is single-crossing if it admits an ordering of the voters such that for every pair of candidates (a,b), either: a) all voters who prefer a over b appear before all voters who prefer b over a, or, b) all voters who prefer a over b appear after all voters who prefer b over a, or,

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152 The notion is popular for several reasons: - No Condorcet Cycles. - Identifiable in polynomial time. - Reasonable (?) model of actual elections.

153 Single Crossing Preferences Recognition A Characterization of the Single-Crossing Domain Bredereck, Chen and Woeginger, Social Choice and Welfare, 2013

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155 Recognising if a given profile is single-crossing with respect to some preference ordering. Checking if a 0/1 matrix has the consecutive-ones property.

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157 Single Crossing Preferences No Condorcet Cycles

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160 For any pair of candidates, the opinion of the midd le voter is also the majority opinion.

161 Single Crossing Preferences Chamberlin-Courant The Complexity of Fully Proportional Representation for Single-Crossing Electorates Skowron, SAGT, 2013

162 Voters Candidates

163 Voters Candidates

164 Voters Candidates

165 Voters Candidates

166 Voters Candidates dissatisfaction of voter v = rank of best candidate in the committee in his vote

167 Voters Candidates dissatisfaction of voter v = rank of best candidate in the committee in his vote

168 On single-crossing profiles, optimal CC solutions exhibit a contiguous blocks property. Voters Candidates dissatisfaction of voter v = rank of best candidate in the committee in his vote

169 On single-crossing profiles, optimal CC solutions exhibit a contiguous blocks property. Voters Candidates dissatisfaction of voter v = rank of best candidate in the committee in his vote

170 On single-crossing profiles, optimal CC solutions exhibit a contiguous blocks property. Voters Candidates dissatisfaction of voter v = rank of best candidate in the committee in his vote

171 Concluding Remarks nearly structured preferences, dichotomous preferences

172 The single-peaked and single-crossing domains have been generalised to notions of single-peaked and singlecrossing on trees. The generalised domains continue to exhibit many of the nice properties we saw today. Single-peaked orders on a tree, Gabrielle Demange, Math. Soc. Sci, 3(4), Generalizing the Single-Crossing Property on Lines and Trees to Intermediate Preferences on Median Graphs, Clearwater, Puppe, and Slinko, IJCAI 2015

173 The single-peaked and single-crossing domains have been generalised to notions of single-peaked-width and singlecrossing-width. Here, it is common that algorithms that work in the singlepeaked or single-crossing settings can be generalised to profiles of width w at an expense that is exponential in w. Kemeny Elections with Bounded Single-peaked or Single-crossing Width, Cornaz, Galand, and Spanjaard, IJCAI 2013

174 Profiles that are close to being single-peaked or singlecrossing (closeness measured usually in terms of candidate or voter deletion) have also been studied. It s typically NP-complete to determine the optimal distance, but FPT and approximation algorithms are known. On Detecting Nearly Structured Preference Profiles Elkind and Lackner, AAAI 2014 Computational aspects of nearly single-peaked electorates, Erdélyi, Lackner, and Pfandler, AAAI 2013 Are There Any Nicely Structured Preference Profiles Nearby? Bredereck, Chen, and Woeginger, AAAI 2013

175 Summary from: On Detecting Nearly Structured Preference Profiles Elkind and Lackner, AAAI 2014

176 While the domain restrictions we saw today apply to votes given as linear orders, similar restrictions have also been studied in the context of votes that are given by approval ballots. As was the case here, there are close connections with COP and many hard problems become easy on these structured (dichotomous) profiles. Structure in Dichotomous Preferences, Elikind and Lackner; IJCAI 2015.

177 Euclidean preferences capture settings where voters and alternatives can be identified with points in the Euclidean space, with voters preferences driven by distances to alternatives. Psychological scaling without a unit of measurement, Coombs; Psychological review, 1950

178 The Dark Side: Domain restrictions also have some sideeffects: problems like manipulation, bribery, and so forth also become easy! The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control, Faliszewski et al; TARK 2009 Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates, Brandt et al; AAAI 2010

179 Thank you! The Handbook of Computational Social Choice, Brandt, Conitzer, Endriss, Lang and Procaccia; 2016 Preference Restrictions in Computational Social Choice: Recent Progress, Elikind, Lackner, and Peters; IJCAI 2016.