computational social choice

Computational Social Choice

IJCAI-13 Tutorial

Aug 4, 2013

Lirong Xia

• About RPI

– The first technological institutes in English-speaking

countries

– Graduate school rankings in US

• 47th Computer Science

• 28th Computer Engineering

• 17th Applied Math

• I am generally interested in both theory and

application of economics and computation.

– Hiring highly motivated Ph.D. students

– If interested, please send me an email ([email protected])2

A shameless advertisement…

• The second nationwide

referendum in UK

– 1st was in 1975

• Member of Parliament election:

Plurality rule Alternative vote rule?

• 68% No vs. 32% Yes

3

2011 UK Referendum

Ordinal Preference Aggregation: Social Choice

> > social choice

mechanism

> >

> >

4

A profile

Carol

Alice

Bob

A B C

A B C

ACB

A

5

A B C

A B C

Turker 1 Turker 2 Turker n

…

> >

Ranking pictures [PGM+ AAAI-12]

...

.

.

.

....

. ....

. . .

.. .

. .

. .. . .

> > AB > B C>

6

Social choice

R1 R1*

Outcome

R2 R2*

Rn Rn*

social choice mechanism

… …

Profile

Ri, Ri*: full rankings over a set A of alternatives

Social Choice

Computational thinking + optimization algorithms

CSSocial

Choice

7PLATO

4thC. B.C.LULL

13thC.BORDA

18thC.

CONDORCET

18thC.

ARROW

20thC.TURING et al.

20thC.

21th Century

and Computer Science

PLATO et al.4thC. B.C.---20thC.

Strategic thinking + methods/principles of aggregation

Applications: real world

• People/agents often have conflicting

preferences, yet they have to make a

joint decision

8

• Multi-agent systems [Ephrati and Rosenschein 91]

• Recommendation systems [Ghosh et al. 99]

• Meta-search engines [Dwork et al. 01]

• Belief merging [Everaere et al. 07]

• Human computation (crowdsourcing) [Mao et al.

AAAI-13]

• etc.9

Applications: academic world

A burgeoning area

• Recently has been drawing a lot of attention– IJCAI-11: 15 papers, best paper

– AAAI-11: 6 papers, best paper

– AAMAS-11: 10 full papers, best paper runner-up

– AAMAS-12 9 full papers, best student paper

– EC-12: 3 papers

• Workshop: COMSOC Workshop 06, 08, 10, 12, 14

• Courses: – Technical University Munich (Felix Brandt)

– Harvard (Yiling Chen)

– U. of Amsterdam (Ulle Endriss)

– RPI (2013 fall Lirong Xia)

• Book in progress: Handbook of Computational Social Choice 10

Flavor of this tutorial

• High-level objectives for– design

– evaluation

– logic flow among research topics

“Give a man a fish and you feed him for a day.

Teach a man to fish and you feed him for a lifetime.”

-----Chinese proverb

• Plus some concrete results

11

How to design a good social choice mechanism?

12

What is being “good”?

Two goals for social choice mechanisms

GOAL1: democracy

13

GOAL2: truth

1. Classical Social Choice3. Statistical approaches

2. Computational aspects

14

Outline

1. Classical Social Choice

2.1 Computational aspectsPart 1

3. Statistical approaches

45 min

55 min

75 min


5 min

5 min

15 min

30 minNP-Hard

NP-Hard

NP-Hard

Common voting rules(what has been done in the past two centuries)

• Mathematically, a social choice mechanism (voting rule) is a mapping from {All profiles} to {outcomes}– an outcome is usually a winner, a set of winners, or a ranking

– m : number of alternatives (candidates)

– n : number of agents (voters)

– D=(P1,…,Pn) a profile

• Positional scoring rules• A score vector s1,...,sm

– For each vote V, the alternative ranked in the i-th position gets si points

– The alternative with the most total points is the winner

– Special cases

• Borda, with score vector (m-1, m-2, …,0)

• Plurality, with score vector (1,0,…,0) [Used in the US]

An example

• Three alternatives {c1, c2, c3}

• Score vector (2,1,0) (=Borda)

• 3 votes,

• c1 gets 2+1+1=4, c2 gets 1+2+0=3,

c3 gets 0+0+2=2

• The winner is c1

1 2 3c c c 2 1 3c c c 3 1 2c c c

2 1 0 2 1 0 2 1 0

• The election has two rounds

– In the first round, all alternatives except the two

with the highest plurality score drop out

– In the second round, the alternative that is

preferred by more voters wins

• [used in Iran, North Carolina State]

17

Plurality with runoff

10 7 6 3

a > b > c > da > d d > a > b > c d > a

c > d > a >bd > a b > c > d >ad >a

d

• Also called instant run-off voting or

alternative vote

• The election has m-1 rounds, in each round,

– The alternative with the lowest plurality score

drops out, and is removed from all votes

– The last-remaining alternative is the winner

• [used in Australia and Ireland]

18

Single transferable vote (STV)

10 7 6 3

a > b > c > da > c > d d > a > b > c d > a > c

c > d > a >bc > d > a b > c > d >a

a

c > d >aa > c a > c c > a c > a

• Kendall tau distance

– K(V,W)= # {different pairwise comparisons}

• Kemeny(D)=argminW K(D,W)=argminW ΣP∈DK(D,W)

• For single winner, choose the top-ranked alternative

in Kemeny(D)

• [Has a statistical interpretation] 19

The Kemeny rule

K( b ≻ c ≻ a , a ≻ b ≻ c ) =112

• Approval, Baldwin, Black, Bucklin,

Coombs, Copeland, Dodgson, maximin,

Nanson, Range voting, Schulze, Slater,

ranked pairs, etc…

20

…and many others

21

• Q: How to evaluate rules in terms of

achieving democracy?

• A: Axiomatic approach

22

Axiomatic approach(what has been done in the past 50 years)

• Anonymity: names of the voters do not matter– Fairness for the voters

• Non-dictatorship: there is no dictator, whose top-ranked alternative is always the winner– Fairness for the voters

• Neutrality: names of the alternatives do not matter– Fairness for the alternatives

• Consistency: if r(D1)∩r(D2)≠ϕ, then r(D1∪D2)=r(D1)∩r(D2)

• Condorcet consistency: if there exists a Condorcet winner, then it must win– A Condorcet winner beats all other alternatives in pairwise elections

• Easy to compute: winner determination is in P– Computational efficiency of preference aggregation

• Hard to manipulate: computing a beneficial false vote is hard

23

Which axiom is more important?

• Some of these axiomatic properties are not compatible with others

• Food for thought: how to evaluate partial satisfaction of axioms?

Condorcet consistency

Consistency Easy to compute

Positional scoring rules

N Y Y

Kemeny Y N N

Ranked pairs Y N Y

24

An easy fact

• Theorem. For voting rules that selects a single winner, anonymity is not compatible with neutrality– proof:

>

>

>

>

≠W.O.L.G.

NeutralityAnonymity

Alice

Bob

• Thm. No positional scoring rule is

Condorcet consistent:

– suppose s1 > s2 > s3

25

Another easy fact [Fishburn APSR-74]

> >

> >

> >

> >

3 Voters

2 Voters

1 Voter

1 Voter

is the Condorcet winner

: 3s1 + 2s2 + 2s3

: 3s1 + 3s2 + 1s3

<

CONTRADICTION

26

Not-So-Easy facts• Arrow’s impossibility theorem

– Google it!

• Gibbard-Satterthwaite theorem

– Next section

• Axiomatic characterization

– Template: A voting rule satisfies axioms A1, A2, A2 if it is rule X

– If you believe in A1 A2 A3 are the most desirable properties then X is

optimal

– (anonymity+neutrality+consistency+continuity) positional scoring

rules [Young SIAMAM-75]

– (neutrality+consistency+Condorcet consistency) Kemeny [Young&Levenglick SIAMAM-78]

• Can we quantify a voting rule’s satisfiability

of these axiomatic properties?

– Tradeoffs between satisfiability of axioms

– Use computational techniques to design new

voting rules

• use AI techniques to automatically prove or

discover new impossibility theorems [Tang&Lin AIJ-09]

27

Food for thought

28

Outline




45 min

55 min

75 min


5 min

5 min

15 min

30 min

• Easy to compute: – the winner can be computed in polynomial time

• Hard to manipulate: – computing a beneficial false vote is hard

29

Computational axioms



30


• Almost all common voting rules, except

– Kemeny: NP-hard [Bartholdi et al. 89], Θ2p-complete [Hemaspaandra et

al. TCS-05]

– Young: Θ2p-complete [Rothe et al. TCS-03]

– Dodgson: Θ2p-complete [Hemaspaandra et al. JACM-97]

– Slater: NP-complete [Hurdy EJOR-10]

• Practical algorithms for Kemeny (also for others)

– ILP [Conitzer, Davenport, & Kalagnanam AAAI-06]

– Approximation [Ailon, Charikar, & Newman STOC-05]

– PTAS [Kenyon-Mathieu and W. Schudy STOC-07]

– Fixed-parameter analysis [Betzler et al. TCS-09]31

Which rule is easy to compute?

• Easy to compute axiom: computing the

winner takes polynomial time in the input

size

– input size: nmlog m

• What if m is extremely large?

32

Really easy to compute?

Combinatorial domains(Multi-issue domains)

• The set of alternatives can be uniquely

characterized by multiple issues

• Let I={x1,...,xp} be the set of p issues

• Let Di be the set of values that the i-th issue

can take, then A=D1×... ×Dp

• Example:

– Issues={ Main course, Wine }

– Alternatives={ } ×{ }33

• In California, voters voted on 11 binary issues

( / )

– 211=2048 combinations in total

– 5/11 are about budget and taxes

34

Multiple referenda

• Prop.30 Increase sales

and some income tax

for education• Prop.38 Increase

income tax on almost

everyone for education

35

Overview

Combinatorial voting

Preference representation

New voting rule

Evaluation

Preference representation: CP-nets[Boutilier et al. JAIR-04]

Variables: x,y,z.

Graph CPTs

This CP-net encodes the following partial order:

{ , },xD x x { , },yD y y { , }.zD z z

x

zy

36

Sequential voting rules [Lang IJCAI-07]

• Issues: main course, wine

• Order: main course > wine

• Local rules are majority rules

• V1: > , : > , : >

• V2: > , : > , : >

• V3: > , : > , : >

• Step 1:

• Step 2: given , is the winner for wine

• Winner: ( , )37

• How can we say that sequential voting is

good?

– computationally efficient

– satisfies good axioms [Lang and Xia MSS-09]

– need to worry about manipulation in the worst

case [Xia, Conitzer &Lang EC-11]

• Other compact languages

– GAI network [Gonzales et al. AIJ-11]

– TCP-net [Li et al. AAMAS-11]

– Soft constraints [Pozza et al. IJCAI-11] 38

Research topics

• Belief merging [Gabbay et al. JLC-09]

• Judgment aggregation [List and Pettit EP-02]

39

Other combinatorial domains

K1 merging operator

K2

Kn

…

Action P Action Q Liable? (P∧Q)

Judge 1 Y Y Y

Judge 2 Y N N

Judge 3 N Y N

Majority Y Y N



40


Strategic behavior (of the agents)

• Manipulation: an agent (manipulator) casts a

vote that does not represent her true

preferences, to make herself better off

• A voting rule is strategy-proof if there is never a

(beneficial) manipulation under this rule

• How important strategy-proofness is as an

desired axiomatic property?

– compared to other axiomatic properties

Manipulation under plurality rule (ties are broken in favor of )

> >

> >

> >

>>

Plurality ruleAlice

Bob

Carol

Any strategy-proof voting rule?

• No reasonable voting rule is strategyproof

• Gibbard-Satterthwaite Theorem [Gibbard Econometrica-73,

Satterthwaite JET-75]: When there are at least three

alternatives, no voting rules except dictatorships satisfy

– non-imposition: every alternative wins for some

profile

– unrestricted domain: voters can use any linear order

as their votes

– strategy-proofness

• Axiomatic characterization for dictatorships!

• Relax non-dictatorship: use a dictatorship

• Restrict the number of alternatives to 2

• Relax unrestricted domain: mainly pursued

by economists

– Single-peaked preferences:

– Range voting: A voter submit any natural

number between 0 and 10 for each alternative

– Approval voting: A voter submit 0 or 1 for each

alternative 44

A few ways out

• Use a voting rule that is too complicated so that

nobody can easily predict the winner

– Dodgson

– Kemeny

– The randomized voting rule used in Venice Republic for

more than 500 years [Walsh&Xia AAMAS-12]

• We want a voting rule where

– Winner determination is easy

– Manipulation is hard45

Computational thinking

46

Overview

Manipulation is inevitable(Gibbard-Satterthwaite Theorem)

Yes

No

Limited information

Limited communication

Can we use computational complexity as a barrier?

Is it a strong barrier?

Other barriers?

May lead to very undesirable outcomes

Seems not very often

Why prevent manipulation?

How often?

If it is computationally too hard for a

manipulator to compute a manipulation,

she is best off voting truthfully

– Similar as in cryptography

For which common voting rules

manipulation is computationally hard?

47

Manipulation: A computational complexity perspective

NP-Hard

• Initiated by [Bartholdi, Tovey, &Trick SCW-89b]

• Votes are weighted or unweighted

• Bounded number of alternatives [Conitzer, Sandholm, &Lang JACM-

07]

– Unweighted manipulation: easy for most common rules

– Weighted manipulation: depends on the number of

manipulators

• Unbounded number of alternatives (next few slides)

• Assuming the manipulators have complete

information! 48

Computing a manipulation

Unweighted coalitional manipulation (UCM) problem

• Given

– The voting rule r

– The non-manipulators’ profile PNM

– The number of manipulators n’

– The alternative c preferred by the manipulators

• We are asked whether or not there exists a profile PM (of the manipulators) such that c is the winner of PNM∪PM under r

49

50

The stunningly big table for UCM

#manipulators One manipulator At least two

Copeland P [BTT SCW-89b] NPC [FHS AAMAS-08,10]

STV NPC [BO SCW-91] NPC [BO SCW-91]

Veto P [ZPR AIJ-09] P [ZPR AIJ-09]

Plurality with runoff P [ZPR AIJ-09] P [ZPR AIJ-09]

Cup P [CSL JACM-07] P [CSL JACM-07]

Borda P [BTT SCW-89b] NPC[DKN+ AAAI-11][BNW IJCAI-11]

Maximin P [BTT SCW-89b] NPC [XZP+ IJCAI-09]

Ranked pairs NPC [XZP+ IJCAI-09] NPC [XZP+ IJCAI-09]

Bucklin P [XZP+ IJCAI-09] P [XZP+ IJCAI-09]

Nanson’s rule NPC [NWX AAA-11] NPC [NWX AAA-11]

Baldwin’s rule NPC [NWX AAA-11] NPC [NWX AAA-11]

• For some common voting rules,

computational complexity provides some

protection against manipulation

• Is computational complexity a strong

barrier?

– NP-hardness is a worst-case concept

51

What can we conclude?

52

Probably NOT a strong barrier

1. Frequency of

manipulability

2. Easiness of

Approximation3. Quantitative G-S

• Non-manipulators’ votes are drawn i.i.d.

– E.g. i.i.d. uniformly over all linear orders (the

impartial culture assumption)

• How often can the manipulators make c

win?– Specific voting rules [Peleg T&D-79, Baharad&Neeman

RED-02, Slinko T&D-02, Slinko MSS-04, Procaccia and

Rosenschein AAMAS-07]53

A first angle: frequency of manipulability

• Theorem. For any generalized scoring rule

– Including many common voting rules

• Computational complexity is not a strong barrier against

manipulation

– UCM as a decision problem is easy to compute in most cases

– The case of Θ(√n) has been studied experimentally in [Walsh

IJCAI-09]54

A general result [Xia&Conitzer EC-08a]

# manipulatorsAll-powerful

No powerΘ(√n)

• Unweighted coalitional optimization

(UCO): compute the smallest number of

manipulators that can make c win

– A greedy algorithm has additive error no more

than 1 for Borda [Zuckerman, Procaccia,

&Rosenschein AIJ-09]

55

A second angle: approximation

• A polynomial-time approximation algorithm that

works for all positional scoring rules

– Additive error is no more than m-2

– Based on a new connection between UCO for positional

scoring rules and a class of scheduling problems

• Computational complexity is not a strong barrier

against manipulation

– The cost of successful manipulation can be easily

approximated (for positional scoring rules)

56

An approximation algorithm for positional scoring rules[Xia,Conitzer,& Procaccia EC-10]

The scheduling problems Q|pmtn|Cmax

• m* parallel uniform machines M1,…,Mm*

– Machine i’s speed is si (the amount of work done in unit time)

• n* jobs J1,…,Jn*

• preemption: jobs are allowed to be interrupted (and resume later maybe on another machine)

• We are asked to compute the minimum makespan– the minimum time to complete all jobs 57

s2=s1-s3

s3=s1-s4

p1p

p2

p3

Thinking about UCOpos

• Let p,p1,…,pm-1 be the total points that c,c1,…,cm-1

obtain in the non-manipulators’ profile

p c

c1

c3

c2

∨

∨

∨

PNM V1

=

c

c1

c2

c3

p1 -pp1 –p-(s1-s2)

p p2 -pp2 –p-(s1-s4)

p p3 -pp3 –p-(s1-s3)

s1-s3

s1-s4

s1-s2

∪{V1=[c>c1>c2>c3]}

s1=s1-s2(J1)

(J2)

(J3)

58

59

The approximation algorithm

Original UCOScheduling

problem

Solution to the

scheduling problem

Solution to the

UCO

[Gonzalez&Sahni

JACM 78]

Rounding

No more than

OPT+m-2

• Manipulation of positional scoring rules =

scheduling (preemptions at integer time points)

– Borda manipulation corresponds to scheduling

where the machines speeds are m-1, m-2, …, 0

• NP-hard [Yu, Hoogeveen, & Lenstra J.Scheduling 2004]

– UCM for Borda is NP-C for two manipulators

• [Davies et al. AAAI-11 best paper]

• [Betzler, Niedermeier, & Woeginger IJCAI-11 best paper]

60

Complexity of UCM for Borda

• G-S theorem: for any reasonable voting rule there

exists a manipulation

• Quantitative G-S: for any voting rule that is “far

away” from dictatorships, the number of

manipulable situations is non-negligible

– First work: 3 alternatives, neutral rule [Friedgut, Kalai,

&Nisan FOCS-08]

– Extensions: [Dobzinski&Procaccia WINE-08, Xia&Conitzer EC-08b,

Isaksson,Kindler,&Mossel FOCS-10]

– Finally proved: [Mossel&Racz STOC-12]61

A third angle: quantitative G-S

• The first attempt seems to fail

• Can we obtain positive results for a

restricted setting?

– The manipulators has complete information

about the non-manipulators’ votes

– The manipulators can perfectly discuss their

strategies

62

Next steps

• Limiting the manipulator’s information can

make dominating manipulation computationally

harder, or even impossible [Conitzer,Walsh,&Xia

AAAI-11]

• Bayesian information [Lu et al. UAI-12]63

Limited information

• The leader-follower model

– The leader broadcast a vote W, and the potential

followers decide whether to cast W or not

• The leader and followers have the same preferences

– Safe manipulation [Slinko&White COMSOC-08]: a vote W

that

• No matter how many followers there are, the leader/potential

followers are not worse off

• Sometimes they are better off

– Complexity: [Hazon&Elkind SAGT-10, Ianovski et al. IJCAI-11]64

Limited communication among manipulators

65

OverviewManipulation is inevitable

(Gibbard-Satterthwaite Theorem)

Yes

No

Limited information

Limited communication

Can we use computational complexity as a barrier?

Is it a strong barrier?

Other barriers?

May lead to very undesirable outcomes

Seems not very often

Why prevent manipulation?

How often?

• How to predict the outcome?

– Game theory

• How to evaluate the outcome?

• Price of anarchy [Koutsoupias&Papadimitriou STACS-99]

–

– Not very applicable in the social choice setting

• Equilibrium selection problem

• Social welfare is not well defined

• Use best-response game to select an equilibrium and use scores

as social welfare [Brânzei et al. AAAI-13]66

Research questions

Worst welfare when agents are fully strategic

Optimal welfare when agents are truthful

67

Simultaneous-move voting games

• Players: Voters 1,…,n

• Strategies / reports: Linear orders over

alternatives

• Preferences: Linear orders over alternatives

• Rule: r(P’), where P’ is the reported profile

68

Equilibrium selection problem

> >

>>

Plurality rule

> >

>>

> >

>>

Alice

Bob

Carol

69

Stackelberg voting games[Xia&Conitzer AAAI-10]

• Voters vote sequentially and strategically– voter 1 → voter 2 → voter 3 → … → voter n

– any terminal state is associated with the winner under rule r

• At any stage, the current voter knows– the order of voters

– previous voters’ votes

– true preferences of the later voters (complete information)

– rule r used in the end to select the winner

• Called a Stackelberg voting game– Unique winner in SPNE (not unique SPNE)

– Similar setting in [Desmedt&Elkind EC-10]

70

General paradoxes (ordinal PoA)

• Theorem. For any voting rule r that satisfies majority

consistency and any n, there exists an n-profile P

such that:

– (many voters are miserable) SGr(P) is ranked somewhere

in the bottom two positions in the true preferences of n-2 voters

– (almost Condorcet loser) SGr(P) loses to all but one

alternative in pairwise elections

• Strategic behavior of the voters is extremely harmful

in the worst case

Simulation results

• Simulations for the plurality rule (25000 profiles uniformly at random)

– x: #voters, y: percentage of voters

– (a) percentage of voters who prefer SPNE winner to the truthful winner minus those who prefer truthful winner to the SPNE winner

– (b) percentage of profiles where SPNE winner is the truthful winner

• SPNE winner is preferred to the truthful r winner by more voters than vice versa

(a) (b)

71

• Procedure control by

– {adding, deleting} × {voters, alternatives}

– partitioning voters/alternatives

– introducing clones of alternatives

– changing the agenda of voting– [Bartholdi, Tovey, &Trick MCM-92, Tideman SCW-07, Conitzer,Lang,&Xia IJCAI-09]

• Bribery [Faliszewski, Hemaspaandra, &Hemaspaandra JAIR-09]

• See [Faliszewski, Hemaspaandra, &Hemaspaandra CACM-10] for a

survey on their computational complexity

• See [Xia Axriv-12] for a framework for studying many of these

for generalized scoring rules 72

Other types of strategic behavior

(of the chairperson)

73

Food for thought

• The problem is still very open!

– Shown to be connected to integer factorization [Hemaspaandra, Hemaspaandra, & Menton STACS-13]

• What is the role of computational complexity in

analyzing human/self-interested agents’ behavior?

– Explore information/communication assumptions

– In general, why do we want to prevent strategic

behavior?

• Practical ways to protect elections

74

Outline




45 min

55 min

75 min


5 min

5 min

15 min

30 min

75

A B C

A B C

Turker 1 Turker 2 Turker n

…

> >

Ranking pictures [PGM+ AAAI-12]

...

.

.

.

....

. ....

. . .

.. .

. .

. .. . .

> > AB > B C>

Two goals for social choice mechanisms

GOAL1: democracy

76

GOAL2: truth

1. Classical Social Choice3. Statistical approaches

2. Computational aspects

Outline: statistical approaches

77

Condorcet’s MLE model(history)

A General framework

Why MLE? Why Condorcet’s

model?

Random Utility Models

Model selection

The Condorcet Jury theorem.

• Given

– two alternatives {a,b}.

– 0.5<p<1,

• Suppose

– each agent’s preferences is generated i.i.d., such that

– w/p p, the same as the ground truth

– w/p 1-p, different from the ground truth

• Then, as n→∞, the majority of agents’ preferences

converges in probability to the ground truth78

The Condorcet Jury theorem [Condorcet 1785]

Condorcet’s MLE approach• Parametric ranking model Mr: given a “ground truth”

parameter Θ

– each vote P is drawn i.i.d. conditioned on Θ, according to Pr(P|Θ)

– Each P is a ranking

• For any profile D=(P1,…,Pn),

– The likelihood of Θ is L(Θ|D)=Pr(D|Θ)=∏P∈D Pr(P|Θ)

– The MLE mechanism

MLE(D)=argmaxΘ L(Θ|D)

– Break ties randomly

“Ground truth” Θ

P1 P2 Pn…

79

• Parameterized by a ranking

• Given a “ground truth” ranking W and p>1/2, generate each

pairwise comparison in V independently as follows (suppose c

≻ d in W)

• MLE ranking is the Kemeny rule [Young JEP-95]

– Pr(P|W) = pnm(m-1)/2-K(P,W) (1-p) K(P,W) =

– The winning rankings are insensitive to the choice of p (>1/2) 80

Condorcet’s model [Condorcet 1785]

Pr( b ≻ c ≻ a | a ≻ b ≻ c )

=

(1-p)p (1-p)p (1-p)2

Constant<1

c≻d in W

c≻d in Vp

d≻c in V1-p

• Learning [Lu and Boutilier ICML-11]

• Approximation by common voting rules [Caragiannis, Procaccia & Shah EC-13]

81

Recent studies on Condorcet’s model


82



model?

A General framework

83

Statistical decision framework[Azari, Parkes, and Xia draft 13]

ground truth Θ

P1 Pn……

Mr

Decision (winner, ranking, etc)

Information about theground truth

P1 P2 Pn…

Step 1: statistical inference

Data D

Given MrStep 2: decision making

Mr = Condorcet’ model

Step 1: MLE

Step 2: top-alternative

84

Example: Kemeny

Winner

The most probable ranking

P1 P2 Pn…

Step 1: MLE

Data D

Step 2: top-1 alternative

• Frequentist

– there is an unknown

but fixed ground truth

– p = 10/14=0.714

– Pr(2heads|p=0.714)

=(0.714)2=0.51>0.5

– Yes!85

Frequentist vs. Bayesian in general

• Bayesian

– the ground truth is

captured by a belief

distribution

– Compute Pr(p|Data)

assuming uniform prior

– Compute Pr(2heads|

Data)=0.485<0.5

– No!

Credit: Panos Ipeirotis

& Roy Radner

• You have a biased coin: head w/p p

– You observe 10 heads, 4 tails

– Do you think the next two tosses will be two heads in a row?


86

Kemeny = Frequentist approach

Winner

The most probable ranking

P1 P2 Pn…

Step 1: MLE

Data D

Step 2: top-1 alternative

This is the Kemeny rule

(for single winner)!

87

Example: Bayesian


Winner

Posterior over rankings

P1 P2 Pn…

Step 1: Bayesian update

Data D

Step 2: mostly likely top-1

This is a new rule!

Anonymity, neutrality,

monotonicityConsistency Condorcet

Easy to compute

Frequentist(Kemeny)

Y NY N

Bayesian N Y

88

Frequentist vs. Bayesian[Azari, Parkes, and Xia draft 13]


89



model?

A General framework

• When the outcomes are winning alternatives

– MLE rules must satisfy consistency: if r(D1)∩r(D2)≠ϕ,

then r(D1∪D2)=r(D1)∩r(D2)

– All classical voting rules except positional scoring

rules are NOT MLEs

• Positional scoring rules are MLEs

• This is NOT a coincidence!

– All MLE rules that outputs winners satisfy anonymity and

consistency

– Positional scoring rules are the only voting rules that satisfy

anonymity, neutrality, and consistency! [Young SIAMAM-75]

90

Classical voting rules as MLEs [Conitzer&Sandholm UAI-05]

• When the outcomes are winning rankings

– MLE rules must satisfy reinforcement (the

counterpart of consistency for rankings)

– All classical voting rules except positional scoring

rules and Kemeny are NOT MLEs

• This is not (completely) a coincidence!

– Kemeny is the only preference function (that outputs

rankings) that satisfies neutrality, reinforcement, and

Condorcet consistency [Young&Levenglick SIAMAM-78]91

Classical voting rules as MLEs [Conitzer&Sandholm UAI-05]

• Condorcet’s model

– not very natural

– computationally hard

• Other classic voting rules

– Most are not MLEs

– Models are not very natural either

92

Are we happy?

New mechanisms via the statistical decision framework

Model selection

– How can we evaluate fitness?

• Frequentist or Bayesian?

– Focus on frequentist

• Computation

– How can we compute MLE efficiently?

93

Decision

Information about theground truth

Data D

decision making

inference

• Closely related, but

– We need economic insight to build the model

– We care about satisfaction of traditional social

choice criteria

• Also want to reach a compromise (achieve

democracy)

94

Why not just a problem of machine learning or statistics?


95


A General framework


model?


• Continuous parameters: Θ=(θ1,…, θm)

– m: number of alternatives

– Each alternative is modeled by a utility distribution μi

– θi: a vector that parameterizes μi

• An agent’s perceived utility Ui for alternative ci is generated

independently according to μi(Ui)

• Agents rank alternatives according to their perceived utilities

– Pr(c2≻c1≻c3|θ1, θ2, θ3) = PrUi ∼ μi (U2>U1>U3)

96

Random utility model (RUM)[Thurstone 27]

U1 U2U3

θ3 θ2 θ1

• Pr(Data |θ1, θ2, θ3) = ∏R∈Data Pr(R |θ1, θ2, θ3)

97

Generating a preference-profile

Parameters

P1= c2≻c1≻c3Pn= c1≻c2≻c3…

Agent 1 Agent n

θ3 θ2 θ1

• μi’s are Gumbel distributions

– A.k.a. the Plackett-Luce (P-L) model [BM 60, Yellott 77]

• Equivalently, there exist positive numbers λ1,…,λm

• Pros:

– Computationally tractable

• Analytical solution to the likelihood function

– The only RUM that was known to be tractable

• Widely applied in Economics [McFadden 74], learning to rank [Liu 11], and

analyzing elections [GM 06,07,08,09]

• Cons: does not seem to fit very well98

RUMs with Gumbel distributions

c1 is the top choice in { c1,…,cm }c2 is the top choice in { c2,…,cm }cm-1 is preferred to cm

• μi’s are normal distributions

– Thurstone’s Case V [Thurstone 27]

• Pros:

– Intuitive

– Flexible

• Cons: believed to be computationally intractable

– No analytical solution for the likelihood function Pr(P |

Θ) is known

99

RUM with normal distributions

Um: from -∞ to ∞ Um-1: from Um to ∞ … U1: from U2 to ∞

• Location family: RUMs where each μi is parameterized

by its mean θi

– Normal distributions with fixed variance

– P-L

• Theorem. For any RUM in the location family, if the PDF

of each μi is log-concave, then for any preference-profile

D, the likelihood function Pr(D|Θ) is log-concave

– Local optimality = global optimality

– The set of global maxima solutions is convex100

Unimodality of likelihood[APX. NIPS-12]

• Utility distributions μl’s belong to the exponential family

(EF)

– Includes normal, Gamma, exponential, Binomial, Gumbel, etc.

• In each iteration t

• E-step, for any set of parameters Θ

– Computes the expected log likelihood (ELL)

ELL(Θ| Data, Θt) = f (Θ, g(Data, Θt))

• M-step

– Choose Θt+1 = argmaxΘ ELL(Θ| Data, Θt)

• Until |Pr(D|Θt)-Pr(D|Θt+1)|< ε101

Approximately computed

by Gibbs sampling

MC-EM algorithm for RUMs [APX NIPS-12]


102


A General framework


model?


Model selection

103

Model selection

Value(Normal)- Value(PL)

LL Pred. LL AIC BIC

44.8(15.8) 87.4(30.5) -79.6(31.6) -50.5(31.6)

• Compare RUMs with Normal distributions and PL for

– log-likelihood

– predictive log-likelihood,

– Akaike information criterion (AIC),

– Bayesian information criterion (BIC)

• Tested on an election dataset

– 9 alternatives, randomly chosen 50 voters

Red: statistically significant with 95% confidence

• Generalized RUM [APX UAI-13]

– Learn the relationship between agent features

and alternative features

• Preference elicitation based on experimental

design [APX UAI-13]

– c.f. active learning

• Faster algorithms [ACPX, ACP in submission]

– Generalized Method of Moments (GMM)104

Recent progress

Computational thinking + optimization algorithms

CSSocial

Choice

Strategic thinking + methods/principles of aggregation

2. Computational aspects 3. Statistical approaches

• Easy-to-compute axiom• Hard-to-manipulate axiom

• Computational thinking +

game-theoretic analysis

• Framework based on

statistical decision theory• Model selection

• Condorcet vs. RUM

Thank you!

computational social choice

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