computational social choice
Download
Skip this Video
Download Presentation
Computational Social Choice

Loading in 2 Seconds...

play fullscreen
1 / 105

Computational Social Choice - PowerPoint PPT Presentation


  • 112 Views
  • Uploaded on

Computational Social Choice. Lirong Xia. IJCAI-13 Tutorial Aug 4, 2013. A shameless advertisement…. About RPI The first technological institutes in English-speaking countries Graduate school rankings in US 47 th Computer Science 28 th Computer Engineering 17 th Applied Math

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Computational Social Choice' - kenna


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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]

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

>

>

.

.

.

.

.

.

.

.

.

.

.

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-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 (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
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…
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
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]
slide61

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]

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

>

>

.

.

.

.

.

.

.

.

.

.

.

A

C

B

C

C

B

A

B

>

>

A

B

>

>

Turker1

Turker2

Turkern

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

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)
slide105

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

ad