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Approximating Optimal Social Choice
Approximating Optimal Social Choice

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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 Download Presentation

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voters distortion voter candidates distortion voters candidates voter preferences median metric min winner alternatives alternative model mechanisms unhappiness minimize percentile mechanism voting

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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)Slide9
Slide10

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?

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