Transcript 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.

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
• 47th Computer Science
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• 17th Applied Math
• I am generally interested in both theory and
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– Hiring highly motivated Ph.D. students
– If interested, please send me an email
([email protected])
1
2011 UK Referendum
• The second nationwide
referendum in UK
– 1st was in 1975
• Member of Parliament election:
Plurality rule  Alternative vote rule?
• 68% No vs. 32% Yes
2
Ordinal Preference Aggregation: Social Choice
A profile
Alice
Bob
Carol
A
>
B
>
C
A
>
B
>
C
B
>
C
>
social choice
mechanism
A
A
3
Ranking pictures [PGM+ AAAI-12]
...
.. .
.
A
A > B > C
Turker 1
.
.
..
.
>
.. . ..
.
. . ..
.
B
B > A
Turker 2
>
. ..
.
C
…
B > C
Turker n
4
Social choice
Profile
R1
R1*
R2
R2*
social choice mechanism
Outcome
…
…
Rn
Rn*
Ri, Ri*: full rankings over a set A of alternatives
5
Social
andChoice
Computer Science
Computational thinking + optimization algorithms
CS
Social
Choice
21th Century
Strategic thinking + methods/principles of aggregation
PLATO
LULL
PLATO et13
al.thC.
4thC. B.C.
4thC. B.C.---20thC.
BORDA
18thC.
CONDORCET
ARROW
TURING et al.
18thC.20thC.
20thC.
6
Applications: real world
• People/agents often have conflicting
preferences, yet they have to make a
joint decision
7
Applications: academic world
• Multi-agent systems [Ephrati and Rosenschein 91]
• Recommendation systems [Ghosh et al. 99]
• Meta-search engines [Dwork et al. 01]
• Belief merging [Everaere et al. 07]
• Human computation (crowdsourcing) [Mao et al.
AAAI-13]
• etc.
8
A burgeoning area
• Recently has been drawing a lot of attention
– IJCAI-11:
– AAAI-11:
– AAMAS-11:
15 papers, best paper
6 papers, best paper
10 full papers, best paper runner-up
– AAMAS-12
– EC-12:
9 full papers, best student paper
3 papers
• Workshop: COMSOC Workshop 06, 08, 10, 12, 14
• Courses:
– Technical University Munich (Felix Brandt)
– Harvard (Yiling Chen)
– U. of Amsterdam (Ulle Endriss)
– RPI (2013 fall Lirong Xia)
• Book in progress: Handbook of Computational Social Choice
9
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
10
How to design a good social
What
is
being
“good”?
choice mechanism?
11
Two goals for social choice mechanisms
GOAL1: democracy
GOAL2: truth
1. Classical Social Choice
3. Statistical approaches
2. Computational aspects
12
Outline
45 min
1. Classical Social Choice
5 min
55 min
2.1 Computational aspects
Part 1
NPHard
15 min
30 min
2.2 Computational aspects
Part 2
NPHard
5 min
75 min
3. Statistical approaches
NPHard
13
Common voting rules
(what has been done in the past two centuries)
• Mathematically, a social choice mechanism (voting rule)
is a mapping from {All profiles} to {outcomes}
– an outcome is usually a winner, a set of winners, or a ranking
– m : number of alternatives (candidates)
– n : number of agents (voters)
– D=(P1,…,Pn) a profile
• Positional scoring rules
• A score vector s1,...,sm
– For each vote V, the alternative ranked in the i-th position gets si points
– The alternative with the most total points is the winner
– Special cases
• Borda, with score vector (m-1, m-2, …,0)
• Plurality, with score vector (1,0,…,0) [Used in the US]
An example
• Three alternatives {c1, c2, c3}
• Score vector (2,1,0) (=Borda)
• 3 votes,
c1  c2  c3
2
1
0
c2  c1  c3
2
1
0
• c1 gets 2+1+1=4, c2 gets 1+2+0=3,
c3 gets 0+0+2=2
• The winner is c1
c3  c1  c2
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 dd >> aa > b > c c > d > a >b
10
7
6
d
b > c > dd >a
>a
3
16
Single transferable vote (STV)
• Also called instant run-off voting or
alternative vote
• The election has m-1 rounds, in each round,
– The alternative with the lowest plurality score
drops out, and is removed from all votes
– The last-remaining alternative is the winner
• [used in Australia and Ireland]
a > b > cc >> dd dd >> aa >> b > c c > d > aa >b
10
7
6
a
b > c > d >aa
3
17
The Kemeny rule
• Kendall tau distance
– K(V,W)= # {different pairwise comparisons}
K( b ≻ c ≻ a , a ≻ b ≻ c ) = 12
• Kemeny(D)=argminW K(D,W)=argminW ΣP∈DK(D,W)
• For single winner, choose the top-ranked
alternative in Kemeny(D)
• [Has a statistical interpretation]
18
…and many others
• Approval, Baldwin, Black, Bucklin,
Coombs, Copeland, Dodgson, maximin,
Nanson, Range voting, Schulze, Slater,
ranked pairs, etc…
19
• Q: How to evaluate rules in terms of
achieving democracy?
• A: Axiomatic approach
20
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
21
Which axiom is more important?
Condorcet
consistency
Consistency
Easy to compute
Positional
scoring rules
N
Y
Y
Kemeny
Y
N
N
Ranked pairs
Y
N
Y
• Some of these axiomatic properties are not
compatible with others
• Food for thought: how to evaluate partial
satisfaction of axioms?
22
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
23
Another easy fact
[Fishburn APSR-74]
• Thm. No positional scoring rule is
Condorcet consistent:
– suppose s1 > s2 > s3
>
>
is the Condorcet winner
2 Voters
>
>
: 3s1 + 2s2 + 2s3
1 Voter
>
>
: 3s1 + 3s2 + 1s3
1 Voter
>
>
<
3 Voters
24
Not-So-Easy facts
• Arrow’s impossibility theorem
– Google it!
• Gibbard-Satterthwaite theorem
– Next section
• Axiomatic characterization
– Template: A voting rule satisfies axioms A1, A2, A2  if it is rule X
– If you believe in A1 A2 A3 are the most desirable properties then X is
optimal
– (anonymity+neutrality+consistency+continuity)  positional scoring
rules [Young SIAMAM-75]
– (neutrality+consistency+Condorcet consistency)  Kemeny
[Young&Levenglick SIAMAM-78]
25
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]
26
Outline
45 min
1. Classical Social Choice
5 min
55 min
2.1 Computational aspects
Part 1
15 min
30 min
2.2 Computational aspects
Part 2
5 min
75 min
3. Statistical approaches
27
Computational axioms
• Easy to compute:
– the winner can be computed in polynomial time
• Hard to manipulate:
– computing a beneficial false vote is hard
28
Computational axioms
• Easy to compute:
– the winner can be computed in polynomial time
• Hard to manipulate:
– computing a beneficial false vote is hard
29
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. Schudy STOC-07]
– Fixed-parameter analysis [Betzler et al. TCS-09]
30
Really easy to compute?
• Easy to compute axiom: computing the
winner takes polynomial time in the input
size
– input size: nmlog m
• What if m is extremely large?
31
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={
} ×{
}
32
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
33
Overview
Combinatorial voting
Preference
representation
New voting rule
Evaluation
34
Preference representation: CP-nets
[Boutilier et al. JAIR-04]
Variables: x,y,z. Dx  {x, x}, Dy  { y, y}, Dz  {z, z}.
x
y
z
Graph
CPTs
This CP-net encodes the following partial order:
35
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
• Winner:
(
,
,
is the winner for wine
)
36
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]
37
Other combinatorial domains
• Belief merging [Gabbay et al. JLC-09]
K1
merging operator
K2
…
Kn
• Judgment aggregation [List and Pettit EP-02]
Action P
Action Q
Liable? (P∧Q)
Judge 1
Y
Y
Y
Judge 2
Y
N
N
Judge 3
N
Y
N
Majority
Y
Y
N
38
Computational axioms
• Easy to compute:
– the winner can be computed in polynomial time
• Hard to manipulate:
– computing a beneficial false vote is hard
39
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
>
>
Bob
>
>
Carol
>
>
)
Plurality 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 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
43
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
44
Overview
Manipulation is inevitable
(Gibbard-Satterthwaite Theorem)
Can we use computational complexity as a barrier?
Why prevent manipulation?
Yes
Is it a strong barrier?
No
May lead to very
undesirable outcomes
How often?
Other barriers?
Limited information
Limited communication
Seems not very often
45
Manipulation: A computational
complexity perspective
If it is computationally too hard for a
manipulator to compute a manipulation,
she is best off voting truthfully
– Similar as in cryptography
NPHard
For which common voting rules
manipulation is computationally hard?
46
Computing a manipulation
• Initiated by [Bartholdi, Tovey, &Trick
SCW-89b]
• Votes are weighted or unweighted
• Bounded number of alternatives [Conitzer, Sandholm, &Lang
JACM-07]
– Unweighted manipulation: easy for most common rules
– Weighted manipulation: depends on the number of
manipulators
• Unbounded number of alternatives (next few
slides)
• Assuming the manipulators have complete
information!
47
Unweighted coalitional manipulation
(UCM) problem
• Given
– The voting rule r
– The non-manipulators’ profile PNM
– The number of manipulators n’
– The alternative c preferred by the manipulators
• We are asked whether or not there exists a
profile PM (of the manipulators) such that c is
the winner of PNM∪PM under r
48
The stunningly big table for UCM
#manipulators
Copeland
STV
One manipulator
P [BTT SCW-89b]
NPC [BO SCW-91]
At least two
NPC [FHS AAMAS-08,10]
NPC [BO SCW-91]
Veto
P [ZPR AIJ-09]
P [ZPR AIJ-09]
Plurality with runoff
P [ZPR AIJ-09]
P [ZPR AIJ-09]
Cup
P [CSL JACM-07]
P [CSL JACM-07]
Borda
P [BTT SCW-89b]
NPC
Maximin
P [BTT SCW-89b]
NPC [XZP+ IJCAI-09]
NPC [XZP+ IJCAI-09]
NPC [XZP+ IJCAI-09]
P [XZP+ IJCAI-09]
P [XZP+ IJCAI-09]
Ranked pairs
Bucklin
[DKN+ AAAI-11]
[BNW IJCAI-11]
Nanson’s rule
NPC [NWX AAA-11]
NPC [NWX AAA-11]
Baldwin’s rule
NPC [NWX AAA-11]
NPC [NWX AAA-11]
49
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
50
Probably NOT a strong barrier
1. Frequency of
manipulability
2. Easiness of
Approximation
3. Quantitative G-S
51
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]
52
A general result
[Xia&Conitzer EC-08a]
• Theorem. For any generalized scoring rule
– Including many common voting rules
All-powerful
# manipulators
Θ(√n)
No power
• 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]
53
A second angle: approximation
• Unweighted coalitional optimization
(UCO): compute the smallest number of
manipulators that can make c win
– A greedy algorithm has additive error no more
than 1 for Borda [Zuckerman, Procaccia,
&Rosenschein AIJ-09]
54
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)
55
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
56
Thinking about UCOpos
• Let p,p1,…,pm-1 be the total points that c,c1,…,cm-1
obtain in the non-manipulators’ profile
=
c
V1
PNM ∪{V1=[c>c1>c2>c3]}
p
c
∨
c1 (J1) p
p1 –p-(s
p1 p11-s-p2)
s1=s
s1-s
1-s
22
c1
∨
c2 (J2) p
p2 –p-(s
p21p-s2 4-p
)
s2=s
s1-s
1-s
33
c3
∨
c3 (J3) p
p3 –p-(s
p3 p1-s
3 -p
3)
s3s=s
1-s14-s4
c2
57
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
58
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 best paper]
59
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]
60
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
61
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]
62
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]
63
Overview
Manipulation is inevitable
(Gibbard-Satterthwaite Theorem)
Can we use computational complexity as a barrier?
Why prevent manipulation?
Yes
Is it a strong barrier?
No
May lead to very
undesirable outcomes
How often?
Other barriers?
Limited information
Limited communication
Seems not very often
64
Research questions
• How to predict the outcome?
– Game theory
• How to evaluate the outcome?
• Price of anarchy [Koutsoupias&Papadimitriou STACS-99]
–
Optimal welfare when agents are truthful
Worst welfare when agents are fully strategic
– 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]
65
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
66
Equilibrium selection problem
Alice
>
>
>
>
Plurality rule
Bob
Carol
>
>
>
>
>
>
>
>
67
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]
68
General paradoxes (ordinal PoA)
• Theorem. For any voting rule r that satisfies
majority consistency and any n, there exists an nprofile P such that:
– (many voters are miserable) SGr(P) is ranked
somewhere in the bottom two positions in the true
preferences of n-2 voters
– (almost Condorcet loser) SGr(P) loses to all but one
alternative in pairwise elections
• Strategic behavior of the voters is extremely
harmful in the worst case
69
Simulation results
(a)
(b)
• 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
70
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 IJCAI09]
• Bribery [Faliszewski, Hemaspaandra, &Hemaspaandra JAIR-09]
• See [Faliszewski, Hemaspaandra, &Hemaspaandra CACM-10] for a
survey on their computational complexity
• See [Xia Axriv-12] for a framework for studying many of
these for generalized scoring rules
71
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
72
Outline
45 min
1. Classical Social Choice
5 min
55 min
2.1 Computational aspects
Part 1
15 min
30 min
2.2 Computational aspects
Part 2
5 min
75 min
3. Statistical approaches
73
Ranking pictures [PGM+ AAAI-12]
...
.. .
.
A
A > B > C
Turker 1
.
.
..
.
>
.. . ..
.
. . ..
.
B
B > A
Turker 2
>
. ..
.
C
…
B > C
Turker n
74
Two goals for social choice mechanisms
GOAL1: democracy
GOAL2: truth
1. Classical Social Choice
3. Statistical approaches
2. Computational aspects
75
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
77
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)
“Ground truth” Θ
P1
P2
…
Pn
– Each P is a ranking
• For any profile D=(P1,…,Pn),
– The likelihood of Θ is L(Θ|D)=Pr(D|Θ)=∏P∈D Pr(P|Θ)
– The MLE mechanism
MLE(D)=argmaxΘ L(Θ|D)
– Break ties randomly
78
Condorcet’s model
[Condorcet 1785]
• Parameterized by a ranking
• Given a “ground truth” ranking W and p>1/2, generate each
pairwise comparison in V independently as follows (suppose
p
c ≻ d in W)
c≻d in V
c≻d in W
1-p
d≻c in V
p (1-p)2
Pr( b ≻ c ≻ a | a ≻ b ≻ c ) = (1-p)
• MLE ranking is the Kemeny rule [Young JEP-95]
K (P,W )
– Pr(P|W) = pnm(m-1)/2-K(P,W) (1-p) K(P,W) = p nm(m-1)/2 æç 1- p ö÷
è poføp (>1/2)
– The winning rankings are insensitive Constant
to the choice
<1
79
Recent studies on Condorcet’s model
• Learning [Lu and Boutilier ICML-11]
• Approximation by common voting rules
[Caragiannis, Procaccia & Shah EC-13]
80
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
Mr
Step 2: decision making
Information about the
ground truth
ground
truth Θ
Step 1: statistical inference
P1
……
Pn
P1
P2
…
Pn
Data D
82
Example: Kemeny
Winner
Step 2: top-1 alternative
Mr = Condorcet’ model
Step 1: MLE
The most probable ranking
Step 2: top-alternative
Step 1: MLE
P1
P2
…
Pn
Data D
83
Frequentist vs. Bayesian in general
• You have a biased coin: head w/p p
– You observe 10 heads, 4 tails
Credit: Panos Ipeirotis
& Roy Radner
– Do you think the next two tosses will be two heads in a row?
• Frequentist
• Bayesian
– there is an unknown
but fixed ground truth
– the ground truth is
captured by a belief
distribution
– p = 10/14=0.714
– Compute Pr(p|Data)
assuming uniform prior
– Pr(2heads|p=0.714)
=(0.714)2=0.51>0.5
– Yes!
– Compute
Pr(2heads|Data)=0.485
<0.5
– No!
84
Kemeny = Frequentist approach
Winner
Step 2: top-1 alternative
Mr = Condorcet’ model
The most probable ranking
This is the Kemeny rule
(for single winner)!
Step 1: MLE
P1
P2
…
Pn
Data D
85
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
86
Frequentist vs. Bayesian
[Azari, Parkes, and Xia draft 13]
Anonymity,
neutrality,
monotonicity
Consistency
Frequentist
(Kemeny)
Y
Bayesian
Condorcet
Easy to
compute
Y
N
N
Y
N
87
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]
• 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] 89
Classical voting rules as MLEs
[Conitzer&Sandholm UAI-05]
• When the outcomes are winning rankings
– MLE rules must satisfy reinforcement (the
counterpart of consistency for rankings)
– All classical voting rules except positional
scoring rules and Kemeny are NOT MLEs
• This is not (completely) a coincidence!
– Kemeny is the only preference function (that
outputs rankings) that satisfies neutrality,
reinforcement, and Condorcet consistency
[Young&Levenglick SIAMAM-78]
90
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
91
New mechanisms via the statistical
decision framework
Decision
Model selection
– How can we evaluate fitness?
• Frequentist or Bayesian?
– Focus on frequentist
decision making
Information about the
ground truth
inference
Data D
• Computation
– How can we compute MLE efficiently?
92
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)
93
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 utility Ui for alternative ci is generated
independently according to μi(Ui)
• Agents rank alternatives according to their perceived utilities
– Pr(c2≻c1≻c3|θ1, θ2, θ3) = PrUi ∼ μi (U2>U1>U3)
θ3
U3
θ2
θ1
U1 U2
95
Generating a preference-profile
• Pr(Data |θ1, θ2, θ3) = ∏R∈Data Pr(R |θ1, θ2, θ3)
Parameters
θ3
Agent 1
P1= c2≻c1≻c3
θ2
…
θ1
Agent n
Pn= c1≻c2≻c3
96
RUMs with Gumbel distributions
• μi’s are Gumbel distributions
– A.k.a. the Plackett-Luce (P-L) model [BM 60, Yellott 77]
• Equivalently, there exist positive numbers λ1,…,λm
Pr(c1
c2
• Pros:
cm | l1
lm ) =
l1 +
l1
+ lm
´
l2 +
l2
+ lm
´
lm-1
´
lm-1 + lm
c21 is the
cm-1top
is preferred
choice in to
{ cc21,…,c
,…,c
m
m}
– 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
97
RUM with normal distributions
• μi’s are normal distributions
– Thurstone’s Case V [Thurstone 27]
• Pros:
– Intuitive
– Flexible
• Cons: believed to be computationally intractable
– No analytical solution for the likelihood function Pr(P |
Θ) is known
Pr(c1
cm | Q) =
Um: from -∞ to ∞
ò ò
¥
¥
-¥
Um
ò
¥
U2
mm (Um )mm-1 (Um-1 ) m1 (U1 )dU1 dUm-1 dUm
Um-1: from Um to ∞ …
U1: from U2 to ∞
98
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 μi is log-concave, then for any
preference-profile D, the likelihood function
Pr(D|Θ) is log-concave
– Local optimality = global optimality
– The set of global maxima solutions is convex
99
MC-EM algorithm for RUMs
[APX NIPS-12]
• Utility distributions μl’s belong to the exponential
family (EF)
– Includes normal, Gamma, exponential, Binomial, Gumbel,
etc.
• In each iteration t
• E-step, for any set of parameters Θ
– Computes the expected log likelihood (ELL)
ELL(Θ| Data, Θt) = f (Θ, g(Data, Θt))
• M-step
Approximately computed
by Gibbs sampling
– Choose Θt+1 = argmaxΘ ELL(Θ| Data, Θt)
• Until |Pr(D|Θt)-Pr(D|Θt+1)|< ε
100
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
Value(Normal)
- Value(PL)
LL
Pred. LL
AIC
BIC
44.8(15.8)
87.4(30.5)
-79.6(31.6)
-50.5(31.6)
Red: statistically significant with 95% confidence
102
Recent progress
• Generalized RUM [APX UAI-13]
– Learn the relationship between agent features
and alternative features
• Preference elicitation based on experimental
design [APX UAI-13]
– c.f. active learning
• Faster algorithms [ACPX, ACP in submission]
– Generalized Method of Moments (GMM)
103
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