under Metric Preferences Elliot Anshelevich Onkar Bhardwaj John Postl Rensselaer Polytechnic Institute RPI Troy NY Voting and Social Choice m candidatesalternatives A B C D ID: 429578
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Slide1
Approximating Optimal Social Choiceunder Metric Preferences
Elliot
Anshelevich
Onkar
Bhardwaj
John
Postl
Rensselaer Polytechnic Institute (RPI), Troy, NYSlide2
Voting and Social Choice
m
candidates/alternatives
A, B, C, D, …
n
voters/agents: have preferences over alternatives
Elections
Recommender systems
Search engines
Preference aggregationSlide3
Voting and Social Choice
m
candidates/alternatives
A, B, C, D, …
n
voters/agents: have preferences over alternatives Usually specify total order over alternativesVoting mechanism decides outcome given these preferences (e.g., which alternative is chosen; ranking of alternatives; etc)
1. A > B > C
2. A > B > C
3. A > B > C
4. B > A > C
5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > BSlide4
Voting Mechanisms
m
candidates/alternatives
A, B, C, D, …
n
voters/agents: have preferences over alternatives Usually specify total order over alternativesMajority/ Plurality does not work very well: C wins even though A
pairwise
preferred to C
. E.g., Bush-Gore-Nader
1. A > B > C
2. A > B > C
3. A > B > C
4. B > A > C
5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > B
B
A
CSlide5
Voting Mechanisms
m
candidates/alternatives
A, B, C, D, …
n
voters/agents: have preferences over alternatives Usually specify total order over alternativesMajority/ Plurality does not work very well: C wins even though A
pairwise
preferred to C
. E.g., Bush-Gore-Nader
1. A > B > C
2. A > B > C
3. A > B > C
4. B > A > C
5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > B
B
A
CSlide6
Voting Mechanisms
Condorcet Cycle
1. A > B > C
2. B > C > A
3. C > A > B
B
A
CSlide7
Voting Mechanisms
Condorcet Cycle
So, what is “best” outcome?
All voting mechanisms have weaknesses.
“Axiomatic” approach: define some properties, see which mechanisms satisfy them
1. A > B > C
2. B > C > A
3. C > A > B
B
A
CSlide8
Arrow’s Impossibility Theorem (1950)
No mechanism for more than 2 alternatives can satisfy the following “reasonable” properties
Formally, no mechanism obeys all 3 of following properties
Unanimity (if A preferred to B by all voters, than A should be ranked higher)
Independence of Irrelevant Alternatives (how A is ranked relative to B only depends on order of A and B in voter preferences)
Non-dictatorship (voting mechanism does not just do what one voter says)Common approaches“Axiomatic” approach: analyze lots of different mechanisms, show good properties about eachMake extra assumptions on preferences(Nobel
prize in economics)Slide9Slide10
Our Approach: Metric Preferences
Metric preferences
Also called spatial preferences
Additional structure on who prefers which alternativeSlide11
Example: Political Spectrum
Left
Right
B
A
CSlide12
Example: Political SpectrumSlide13
Example: Political SpectrumSlide14
Example: Political Spectrum
xkcdSlide15
Example: Political Spectrum
xkcd
Downsian
proximity model (1957):
Each dimension is a different issueSlide16
Our Model
Voters and candidates are points in an arbitrary metric space
Each voter prefers candidates closer to themselves
Best alternative:
min
Σ d(i,A)Ai
B
A
CSlide17
Our Model
Voters and candidates are points in an arbitrary metric space
Each voter prefers candidates closer to themselves
Best alternative:
min
Σ d(i,A)Ai
B
A
C
B > A > CSlide18
Our Model
Voters and candidates are points in an arbitrary metric space
Each voter prefers candidates closer to themselves
Best alternative:
min
Σ d(i,A)Ai
B
A
CSlide19
Our Model
Voters and candidates are points in an arbitrary metric space
Each voter prefers candidates closer to themselves
Best alternative:
Finding best alternative is easy
min Σ d(i,A)Ai
B
A
CSlide20
Our Model
Voters and candidates are points in an arbitrary metric space
Each voter prefers candidates closer to themselves
Best alternative:
Usually don’t know numerical values!
min Σ d(i,A)Ai
B
A
CSlide21
Our Model
Given: Ordinal preferences of all voters
These preferences come from an unknown arbitrary metric space
Goal: Return best alternative
1. A > B > C
2. A > B > C3. A > B > C4. B > A > C5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > B
.....
.Slide22
Our Model
Given: Ordinal preferences of all voters
These preferences come from an unknown arbitrary metric space
Goal: Return provably good approximation to the best alternative
1. A > B > C
2. A > B > C3. A > B > C4. B > A > C5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > B
....
B = OPT
A
C
Σ
d(
i,C
)
i
Σ
d(
i,B
)
i
smallSlide23
Model Summary
Given: Ordinal preferences p of all voters
These preferences come from an unknown arbitrary metric space
Want mechanism which has small
distortion:
1. A > B > C2. A > B > C3. A > B > C
4. B > A > C
5. B > A > C
6. C > A > B
7. C > A > B
8. C > A > B
9. C > A > B
....
Σ
d(
i,winner
)
i
i
max
d
ϵ
D(p)
A
min
Σ
d(
i,A
)
Approximate median using
only ordinal informationSlide24
Easy Example: 2 candidates
2 candidates
n-k
voters have A > B
k
voters have B > A Slide25
Easy Example: 2 candidates
2 candidates
n-k
voters have A > B
k
voters have B > A BA
k
n-k
B may be optimal even if k=1Slide26
Easy Example: 2 candidates
2 candidates
n-k
voters have A > B
k
voters have B > A BA
k
n-k
B may be optimal even if k=1
But, if use majority, then distortion ≤ 3Slide27
Easy Example: 2 candidates
2 candidates
n/2
voters have A > B
n/2
voters have B > A BA
n/2
n/2
B may be optimal even if k=1
But, if use majority, then distortion ≤ 3
Also shows that no deterministic mechanism can have
distortion < 3Slide28
Our Results
Sum
Median
Plurality
2m-1
UnboundedBorda2m-1Unboundedk-approval2n-1UnboundedVeto2n-1UnboundedCopeland55Uncovered Set55Lower Bound3
5
Σ d(
i,winner)
i
i
max
dϵD(p)
A
min
Σ
d(
i,A
)
Sum Distortion =
Median Distortion =
replace sum with medianSlide29
Copeland Mechanism
Majority Graph:
Edge (A,B) if A
pairwise
defeats B
Copeland Winner: Candidate who defeats most othersBAC
E
DSlide30
Copeland Mechanism
Majority Graph:
Edge (A,B) if A
pairwise
defeats B
Copeland Winner: Candidate who defeats most othersBAC
E
D
Tournament winner: has one or two-hop path to all other nodes
Always exists, Copeland chooses one such winnerSlide31
Our Results
Sum
Median
Plurality
2m-1
UnboundedBorda2m-1Unboundedk-approval2n-1UnboundedVeto2n-1UnboundedCopeland55Uncovered Set55Lower Bound3
5
Σ d(
i,winner)
i
i
max
dϵD(p)
A
min
Σ
d(
i,A
)
Sum Distortion =
Median Distortion =
replace sum with medianSlide32
Distortion at most 5
Tournament winner W
Optimal candidate X
X
W
Distortion ≤ 3XWB
Distortion ≤ 5Slide33
Our Results
Sum
Median
Plurality
2m-1
UnboundedBorda2m-1Unboundedk-approval2n-1UnboundedVeto2n-1UnboundedCopeland55Uncovered Set55Lower Bound3
5
Σ d(
i,winner)
i
i
max
dϵD(p)
A
min
Σ
d(
i,A
)
Sum Distortion =
Median Distortion =
replace sum with medianSlide34
Our Results
Sum
Median
Plurality
2m-1
UnboundedBorda2m-1Unboundedk-approval2n-1UnboundedVeto2n-1UnboundedCopeland55Uncovered Set55Lower Bound3
5
med d(i,winner
)
max
dϵD(p)
A
min med
d(
i,A
)
Median Distortion =
Median instead of average
voter happiness
i
iSlide35
Bounds on Percentile Distortion
Percentile distortion: happiness of top
α
-percentile with outcome
α=1: minimize maximum voter unhappiness α=1/2: minimize median voter unhappiness α=0: minimize minimum voter unhappinessSlide36
Bounds on Percentile Distortion
Percentile distortion: happiness of top
α
-percentile with outcome
α=1: minimize maximum voter unhappiness α=1/2: minimize median voter unhappiness α=0: minimize minimum voter unhappinessLower Bounds on Distortion
α
0
1
Unbounded
5
3
2/3Slide37
Bounds on Percentile Distortion
Percentile distortion: happiness of top
α
-percentile with outcome
α=1: minimize maximum voter unhappiness α=1/2: minimize median voter unhappiness α=0: minimize minimum voter unhappinessLower Bounds on Distortion
α
0
1
Unbounded
5
3
2/3
Upper Bounds on Distortion
α
0
1
Unbounded
(Copeland) 5
(Plurality) 3
(m-1)/mSlide38
Our Results
Sum
Median
Plurality
2m-1
UnboundedBorda2m-1Unboundedk-approval2n-1UnboundedVeto2n-1UnboundedCopeland55Uncovered Set55Lower Bound3
5
Σ d(
i,winner)
i
i
max
dϵD(p)
A
min
Σ
d(
i,A
)
Sum Distortion =
Median Distortion =
replace sum with medianSlide39
Conclusions and Future Work
Closing gap between 5 and 3
Randomized Mechanisms can do better: Get distortion ≤ 3, but lower bound becomes 2
Multiple winners, k-median, k-center
Manipulation by voters or by candidates
Special voter distributions (e.g., never have many voters far away from a candidate)Slide40
Conclusions and Future Work
Closing gap between 5 and 3
Randomized Mechanisms can do better: Get distortion ≤ 3, but lower bound becomes 2
Multiple winners, k-median, k-center
Manipulation by voters or by candidates
Special voter distributions (e.g., never have many voters far away from a candidate)What other problems can be approximated using only ordinal information?