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Computational social choice

Computational Social Choice

Lirong Xia

IJCAI-13 Tutorial

Aug 4, 2013


A shameless advertisement

A shameless advertisement…

  • About RPI

    • The first technological institutes in English-speaking countries

    • Graduate school rankings in US

      • 47thComputer 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])


2011 uk referendum

2011 UK Referendum

  • The second nationwide referendum in UK

    • 1stwas in 1975

  • Member of Parliament election:

    Plurality rule  Alternative vote rule?

  • 68% No vs. 32% Yes


Ordinal preference aggregation social choice

Ordinal Preference Aggregation: Social Choice

A profile

> >

A

B

C

Alice

social choice mechanism

> >

A

C

A

B

Bob

> >

C

B

A

Carol


Ranking pictures pgm aaai 12

Ranking pictures [PGM+ AAAI-12]

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>

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A

C

B

C

C

B

A

B

>

>

A

B

>

>

Turker1

Turker2

Turkern


Social choice

Social choice

Profile

R1

R1*

social choice mechanism

R2

R2*

Outcome

Rn

Rn*

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


Social choice1

Social Choice

and Computer Science

Computational thinking + optimization algorithms

CS

Social Choice

21th Century

Strategic thinking + methods/principles of aggregation

PLATO

4thC. B.C.

LULL

13thC.

BORDA

18thC.

CONDORCET

18thC.

ARROW

20thC.

TURING et al.

20thC.

PLATO et al.

4thC. B.C.---20thC.


Applications real world

Applications: real world

  • People/agents often have conflicting preferences, yet they have to make a joint decision


Applications academic world

Applications: academic world

  • Multi-agent systems [Ephrati and Rosenschein91]

  • Recommendation systems [Ghoshet al. 99]

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

  • Belief merging [Everaereet al. 07]

  • Human computation (crowdsourcing) [Mao et al. AAAI-13]

  • etc.


A burgeoning area

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-129 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 (UlleEndriss)

    • RPI (2013 fall Lirong Xia)

  • Book in progress: Handbook of Computational Social Choice


Flavor of this tutorial

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


How to design a good social choice mechanism

How to design a good social choice mechanism?

What is being “good”?


Two goals for social choice mechanisms

Two goals for social choice mechanisms

GOAL1: democracy

GOAL2: truth

1. Classical Social Choice

3. Statistical approaches

2. Computational aspects


Outline

Outline

1. Classical Social Choice

45 min

5 min

2.1 Computational aspects

Part 1

55 min

NP-

Hard

15 min

2.2 Computational aspects

Part 2

30 min

NP-

Hard

5 min

3. Statistical approaches

75 min

NP-

Hard


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

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 vectors1,...,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

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

2 1 0

2 1 0

2 1 0


Plurality with runoff

Plurality with runoff

  • 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]

a > b > c > d

a > d

c > d > a >b

d > a

d >a

  • d > a > b > c

  • d > a

  • b > c > d >a

d


Single transferable vote stv

Single transferable vote (STV)

  • Also called instant run-off voting or alternative vote

  • The election has m-1rounds, 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]

a > b > c > d

a > c > d

a > c

a > c

c > d > a

c > d > a >b

c > a

c > a

  • d > a > b > c

  • d > a > c

  • b > c > d >a

  • c > d >a

a


The kemeny rule

The Kemeny rule

  • Kendall tau distance

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

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

  • For single winner, choose the top-ranked alternative in Kemeny(D)

  • [Has a statistical interpretation]

K( b ≻c≻a,a≻b≻c ) =

2

1

1


And many others

…and many others

  • Approval, Baldwin, Black, Bucklin, Coombs, Copeland, Dodgson, maximin, Nanson, Range voting, Schulze, Slater, ranked pairs, etc…


Computational social choice

  • Q: How to evaluate rules in terms of achieving democracy?

  • A: Axiomatic approach


Axiomatic approach what has been done in the past 50 years

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


Which axiom is more important

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?


An easy fact

An easy fact

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

    • proof:

>

>

Alice

>

>

Bob

W.O.L.G.

Anonymity

Neutrality


Another easy fact fishburn apsr 74

Another easy fact [Fishburn APSR-74]

  • Thm. No positional scoring rule is Condorcet consistent:

    • suppose s1 >s2 >s3

is the Condorcet winner

> >

3 Voters

CONTRADICTION

> >

2 Voters

: 3s1 + 2s2+ 2s3

<

> >

1Voter

: 3s1 + 3s2+ 1s3

> >

1Voter


Not so easy facts

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&LevenglickSIAMAM-78]


Food for thought

Food for thought

  • 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]


Outline1

Outline

1. Classical Social Choice

45 min

5 min

2.1 Computational aspects

Part 1

55 min

15 min

2.2 Computational aspects

Part 2

30 min

5 min

3. Statistical approaches

75 min


Computational axioms

Computational axioms

  • Easy to compute:

    • the winner can be computed in polynomial time

  • Hard to manipulate:

    • computing a beneficial false vote is hard


Computational axioms1

Computational axioms

  • Easy to compute:

    • the winner can be computed in polynomial time

  • Hard to manipulate:

    • computing a beneficial false vote is hard


Which rule is easy to compute

Which rule is easy to compute?

  • 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. SchudySTOC-07]

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


Really easy to compute

Really easy to compute?

  • Easy to compute axiom: computing the winner takes polynomial time in the input size

    • input size: nmlogm

  • What if m is extremely large?


Combinatorial domains multi issue domains

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={} ×{ }


Multiple referenda

Multiple referenda

  • In California, voters voted on 11 binary issues ( / )

    • 211=2048 combinations in total

    • 5/11 are about budget and taxes

  • Prop.30 Increase sales and some income tax for education

  • Prop.38 Increase income tax on almost everyone for education


Overview

Overview

Combinatorial voting

  • Preference

  • representation

  • New voting rule

  • Evaluation


Preference representation cp nets boutilier et al jair 04

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

Variables:x,y,z.

Graph CPTs

This CP-net encodes the following partial order:

x

y

z


Sequential voting rules lang ijcai 07

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: ( , )


Research topics

Research topics

  • 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]


Other combinatorial domains

Other combinatorial domains

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

  • Judgment aggregation [List and Pettit EP-02]

merging operator

K1

K2

Kn


Computational axioms2

Computational axioms

  • Easy to compute:

    • the winner can be computed in polynomial time

  • Hard to manipulate:

    • computing a beneficial false vote is hard


Strategic behavior of the agents

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

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

>

>

> >

Alice

Plurality rule

> >

Bob

> >

Carol


Any strategy proof voting rule

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!


A f ew ways out

A few ways out

  • 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


Computational thinking

Computational thinking

  • 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 hard


Overview1

Overview

Manipulation is inevitable

(Gibbard-Satterthwaite Theorem)

Can we use computational complexity as a barrier?

Why prevent manipulation?

  • Yes

  • May lead to very

  • undesirable outcomes

Is it a strong barrier?

  • No

How often?

Other barriers?

  • Seems not very often

  • Limited information

  • Limited communication


Manipulation a computational complexity perspective

Manipulation: A computational complexity perspective

If it is computationallytoo 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?

NP-

Hard


Computing a manipulation

Computing a manipulation

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

  • Votes are weighted or unweighted

    • Bounded number of alternatives [Conitzer, Sandholm, &Lang JACM-07]

    • Unweightedmanipulation: 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!


Unweighted coalitional manipulation ucm problem

Unweightedcoalitional 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


The stunningly big table for ucm

The stunningly big table for UCM


What can we conclude

What can we conclude?

  • 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


Probably not a strong barrier

Probably NOT a strong barrier

1. Frequency of manipulability

2. Easiness of Approximation

3. Quantitative G-S


A first angle frequency of manipulability

A first angle: frequency of manipulability

  • 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]


A general result xia conitzer ec 08a

A general result[Xia&ConitzerEC-08a]

  • 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]

All-powerful

Θ(√n)

# manipulators

No power


A second angle approximation

A second angle: approximation

  • Unweighted coalitional optimization (UCO):compute the smallest number of manipulators that can make cwin

    • A greedy algorithm has additive error no more than 1 for Borda[Zuckerman, Procaccia, &Rosenschein AIJ-09]


An approximation algorithm for positional scoring rules xia conitzer procaccia ec 10

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

  • 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)


The scheduling problems q pmtn c max

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


Thinking about uco pos

Thinking about UCOpos

  • Let p,p1,…,pm-1 be the total points that c,c1,…,cm-1 obtain in the non-manipulators’ profile

V1

PNM

∪{V1=[c>c1>c2>c3]}

=

c

c

p

c1

c1

s1=s1-s2

(J1)

p1

p

p1 -p

p1 –p-(s1-s2)

s1-s2

c3

c2

(J2)

s2=s1-s3

s1-s3

p

p2

p2 -p

p2 –p-(s1-s4)

c2

c3

(J3)

p

p3

p3 -p

p3 –p-(s1-s3)

s3=s1-s4

s1-s4


The approximation algorithm

The approximation algorithm

Scheduling problem

Original UCO

No more than

OPT+m-2

[Gonzalez&Sahni

JACM 78]

Solution to the

UCO

Solution to the

scheduling problem

Rounding


Complexity of ucm for borda

Complexity of UCM for Borda

  • 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 bestpaper]


Computational social choice

A third angle: quantitative G-S

  • 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]


Next steps

Next steps

  • 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


Limited information

Limited information

  • 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]


Limited communication among manipulators

Limited communication among manipulators

  • 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]


Overview2

Overview

  • Manipulation is inevitable

  • (Gibbard-Satterthwaite Theorem)

Can we use computational complexity as a barrier?

Why prevent manipulation?

  • Yes

  • May lead to very

  • undesirable outcomes

Is it a strong barrier?

  • No

How often?

Other barriers?

  • Seems not very often

  • Limited information

  • Limited communication


Research questions

Research questions

  • 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]

Optimal welfare when agents are truthful

Worst welfare when agents are fully strategic


Simultaneous move voting games

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


Equilibrium selection problem

Equilibrium selection problem

>

>

Alice

> >

Plurality rule

>

>

Bob

> >

>

>

Carol

> >


Stackelberg voting games xia conitzer aaai 10

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]


General paradoxes ordinal poa

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-2voters

    • (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

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)


Other types of strategic behavior of the chairperson

Other types of strategic behavior (of the chairperson)

  • 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, &HemaspaandraJAIR-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


Food for thought1

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


Outline2

Outline

1. Classical Social Choice

45 min

5 min

2.1 Computational aspects

Part 1

55 min

15 min

2.2 Computational aspects

Part 2

30 min

5 min

3. Statistical approaches

75 min


Ranking pictures pgm aaai 121

Ranking pictures [PGM+ AAAI-12]

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Two goals for social choice mechanisms1

Two goals for social choice mechanisms

GOAL1: democracy

GOAL2: truth

1. Classical Social Choice

3. Statistical approaches

2. Computational aspects


Outline statistical approaches

Outline: statistical approaches

  • Condorcet’s MLE model

  • (history)

Why MLE?

Why Condorcet’s model?

  • A General framework

  • Random Utility Models

  • Model selection

76


The condorcet jury theorem condorcet 1785

The Condorcet Jury theorem [Condorcet 1785]

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 truth


Condorcet s mle approach

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)

    • EachPis a ranking

  • For any profileD=(P1,…,Pn),

    • The likelihood of ΘisL(Θ|D)=Pr(D|Θ)=∏P∈D Pr(P|Θ)

    • The MLE mechanism MLE(D)=argmaxΘL(Θ|D)

    • Break ties randomly

“Ground truth”Θ

P1

P2

Pn


Condorcet s model condorcet 1785

Condorcet’s model [Condorcet 1785]

  • Parameterized by aranking

  • Given a “ground truth” ranking W and p>1/2, generate each pairwise comparison in V independently as follows (suppose c ≻ din 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)

  • p

    c≻d in V

    c≻d in W

    d≻c in V

    1-p

    Pr( b ≻c≻a|a≻b≻c ) =

    p (1-p)

    p (1-p)2

    (1-p)

    Constant

    <1


    Recent studies on condorcet s model

    Recent studies on Condorcet’s model

    • Learning [Lu and Boutilier ICML-11]

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


    Outline statistical approaches1

    Outline: statistical approaches

    • Condorcet’s MLE model

    • (history)

    Why MLE?

    Why Condorcet’s model?

    • A General framework

    81


    Statistical decision framework azari parkes and xia draft 13

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

    Decision

    (winner, ranking, etc)

    Given Mr

    Step 2: decision making

    Information about the

    ground truth

    Mr

    ground truth Θ

    Step 1: statistical inference

    ……

    P1

    Pn

    P1

    P2

    Pn

    Data D


    Example kemeny

    Example: Kemeny

    Winner

    Mr= Condorcet’ model

    Step 1: MLE

    Step 2: top-alternative

    Step 2: top-1 alternative

    The most probable ranking

    Step 1: MLE

    P1

    P2

    Pn

    Data D


    Frequentist vs bayesian in general

    Frequentist vs. Bayesian in general

    • 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?

    Credit: Panos Ipeirotis

    & Roy Radner

    • 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!

    • 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!


    Kemeny frequentist approach

    Kemeny = Frequentist approach

    Winner

    Mr= Condorcet’ model

    Step 2: top-1 alternative

    The most probable ranking

    This is the Kemeny rule

    (for single winner)!

    Step 1: MLE

    P1

    P2

    Pn

    Data D


    Example bayesian

    Example: Bayesian

    Winner

    Step 2: mostly likely top-1

    Mr = Condorcet’ model

    Posterior over rankings

    This is a new rule!

    Step 1: Bayesian update

    P1

    P2

    Pn

    Data D


    Frequentist vs bayesian azari parkes and xia draft 13

    Frequentistvs. Bayesian[Azari, Parkes, and Xia draft 13]


    Outline statistical approaches2

    Outline: statistical approaches

    • Condorcet’s MLE model

    • (history)

    Why MLE?

    Why Condorcet’s model?

    • A General framework

    88


    Classical voting rules as mles conitzer sandholm uai 05

    Classical voting rules as MLEs [Conitzer&SandholmUAI-05]

    • 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]


    Classical voting rules as mles conitzer sandholm uai 051

    Classical voting rules as MLEs [Conitzer&SandholmUAI-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&LevenglickSIAMAM-78]


    Are we happy

    Are we happy?

    • Condorcet’s model

      • not very natural

      • computationally hard

    • Other classic voting rules

      • Most are not MLEs

      • Models are not very natural either


    New mechanisms via the statistical decision framework

    New mechanisms via the statistical decision framework

    Decision

    Model selection

    • How can we evaluate fitness?

  • Frequentist or Bayesian?

    • Focus on frequentist

  • Computation

    • How can we compute MLE efficiently?

  • decision making

    Information about the

    ground truth

    inference

    Data D


    Why not just a problem of machine learning or statistics

    Why not just a problem of machine learning or statistics?

    • 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)


    Outline statistical approaches3

    Outline: statistical approaches

    • Condorcet’s MLE model

    • (history)

    Why MLE?

    Why Condorcet’s model?

    • A General framework

    • Random Utility Models

    94


    Random utility model rum thurstone 27

    Random utility model(RUM)[Thurstone 27]

    • 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 utilityUi 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)

    θ2

    θ3

    θ1

    U3

    U1

    U2


    Generating a preference profile

    Generating a preference-profile

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

    Parameters

    θ3

    θ2

    θ1

    Agent n

    Agent 1

    Pn= c1≻c2≻c3

    P1= c2≻c1≻c3


    Rums with gumbel distributions

    RUMs with Gumbel distributions

    • μi’s are Gumbeldistributions

      • 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 well

    c1 is the top choice in { c1,…,cm }

    c2 is the top choice in { c2,…,cm }

    cm-1 is preferred to cm


    Rum with normal distributions

    RUM with normal distributions

    • μi’sare 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

    Um: from -∞ to ∞

    Um-1: from Umto ∞

    U1: from U2to ∞


    Unimodality of likelihood ap x nips 12

    Unimodality of likelihood[APX.NIPS-12]

    • 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 μiis 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 convex


  • Mc em algorithm for rums ap x nips 12

    MC-EM algorithm for RUMs [APXNIPS-12]

    • Utility distributions μl’sbelong 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 = argmaxΘELL(Θ| Data, Θt)

    • Until |Pr(D|Θt)-Pr(D|Θt+1)|< ε

    Approximately computed

    by Gibbs sampling


    Outline statistical approaches4

    Outline: statistical approaches

    • Condorcet’s MLE model

    • (history)

    Why MLE?

    Why Condorcet’s model?

    • A General framework

    • Random Utility Models

    • Model selection

    101


    M odel selection

    Model selection

    • 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


    Recent progress

    Recent progress

    • Generalized RUM [APX UAI-13]

      • Learn the relationship between agent featuresand 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)


    Computational social choice

    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

    Computational thinking + optimization algorithms

    CS

    Social Choice

    Thank you!

    Strategic thinking + methods/principles of aggregation


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