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Computational Social Choice

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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

- 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

A profile

> >

A

B

C

Alice

social choice mechanism

> >

A

C

A

B

Bob

> >

C

B

A

Carol

Ranking pictures [PGM+ AAAI-12]

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Turker1

Turker2

Turkern

Social choice

Profile

R1

R1*

social choice mechanism

R2

R2*

Outcome

…

…

Rn

Rn*

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

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

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

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

- 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

- 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?

What is being “good”?

Two goals for social choice mechanisms

GOAL1: democracy

GOAL2: truth

1. Classical Social Choice

3. Statistical approaches

2. Computational aspects

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)

- 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

- 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

- 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)

- 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

- 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

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

- Q: How to evaluate rules in terms of achieving democracy?
- A: Axiomatic approach

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?

- Some of these axiomatic properties are not compatible with others
- Food for thought: how to evaluate partial satisfaction of axioms?

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]

- 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

- 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

- 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]

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

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

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

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?

- 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?

- 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)

- 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

- 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

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]

- 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

- 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

- Belief merging [Gabbay et al. JLC-09]
- Judgment aggregation [List and Pettit EP-02]

merging operator

K1

K2

…

Kn

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)

- 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 )

>

>

> >

Alice

Plurality rule

> >

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!

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

- 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

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

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

- 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!

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?

- 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

1. Frequency of manipulability

2. Easiness of Approximation

3. Quantitative G-S

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&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

- 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]

- 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|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 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

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

- 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]

- Borda manipulation corresponds to scheduling where the machines speeds are m-1, m-2, …, 0

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

- 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

- 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

- 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]

- The leader broadcast a vote W, and the potential followers decide whether to cast W or not

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

- 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]

- Not very applicable in the social choice setting

Optimal welfare when agents are truthful

Worst welfare when agents are fully strategic

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

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)

- 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

- 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)

- 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 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

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-12]

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Turker1

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Two goals for social choice mechanisms

GOAL1: democracy

GOAL2: truth

1. Classical Social Choice

3. Statistical approaches

2. Computational aspects

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.

- 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

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

- 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)

- 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

- Learning [Lu and Boutilier ICML-11]
- Approximation by common voting rules [Caragiannis, Procaccia& Shah EC-13]

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]

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

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

- 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

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

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

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

Outline: statistical approaches

- Condorcet’s MLE model
- (history)

Why MLE?

Why Condorcet’s model?

- A General framework

88

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&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?

- 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 Frequentist or Bayesian? Computation

Decision

Model selection

- How can we evaluate fitness?

- Focus on frequentist

- 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?

- 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 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]

- 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

- 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

- 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]

- Analytical solution to the likelihood function

- Computationally tractable
- 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

- 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[APX.NIPS-12] 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

- Location family: RUMs where each μi is parameterized by its mean θi
- Normal distributions with fixed variance
- P-L

- Local optimality = global optimality
- The set of global maxima solutions is convex

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))

- Computes the expected log likelihood (ELL)
- 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 approaches

- Condorcet’s MLE model
- (history)

Why MLE?

Why Condorcet’s model?

- A General framework

- Random Utility Models

- Model selection

101

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

- 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)

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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