Approximating Optimal Social Choice under Metric Preferences Elliot Anshelevich Onkar Bhardwaj John Postl Rensselaer Polytechnic Institute (RPI), Troy,

Approximating Optimal Social Choice

under Metric Preferences

Elliot Anshelevich

Onkar Bhardwaj

John Postl

Rensselaer Polytechnic Institute (RPI), Troy, NY

Voting and Social Choice

• m candidates/alternatives A, B, C, D, …• n voters/agents: have preferences over alternatives

• Elections• Recommender systems• Search engines• Preference aggregation

Voting and Social Choice


Usually specify total order over alternatives

• Voting mechanism decides outcome given these preferences

(e.g., which alternative is chosen; ranking of alternatives; etc)

1. A > B > C2. A > B > C3. A > B > C4. B > A > C5. B > A > C

6. C > A > B7. C > A > B8. C > A > B9. C > A > B

Voting Mechanisms



• Majority/ Plurality does not work very well: C wins even though A pairwise preferred to C.

E.g., Bush-Gore-Nader


6. C > A > B7. C > A > B8. C > A > B9. C > A > B

B

A

C

Voting Mechanisms



• Majority/ Plurality does not work very well: C wins even though A pairwise preferred to C.

E.g., Bush-Gore-Nader


6. C > A > B7. C > A > B8. C > A > B9. C > A > B

B

A

C

Voting Mechanisms

• Condorcet Cycle

1. A > B > C2. B > C > A3. C > A > B

B

A

C

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 > C2. B > C > A3. C > A > B

B

A

C

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 propertieso Unanimity (if A preferred to B by all voters, than A should be ranked higher)o Independence of Irrelevant Alternatives (how A is ranked relative to B only depends on order

of A and B in voter preferences)o Non-dictatorship (voting mechanism does not just do what one voter says)

• Common approacheso “Axiomatic” approach: analyze lots of different mechanisms, show good properties about

eacho Make extra assumptions on preferences

(Nobel prize in economics)

Our Approach: Metric Preferences

• Metric preferenceso Also called spatial preferences

• Additional structure on who prefers which alternative

Example: Political Spectrum

Left Right

BA C




xkcd


xkcd

Downsian proximity model (1957): Each dimension is a different issue

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)

A i

B

A C

Our Model


A i

B

A CB > A > C

Our Model


A i

B

A C

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)A i

B

A C

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)A i

B

A C

Our Model

• Given: Ordinal preferences of all voters• These preferences come from an unknown

arbitrary metric space• Goal: Return best alternative

1. A > B > C2. A > B > C3. A > B > C4. B > A > C5. B > A > C6. C > A > B7. C > A > B8. C > A > B9. C > A > B

.

.

.

.

.

.

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


.

.

.

.

B = OPT

A C

Σ d(i,C)i

Σ d(i,B)i

small

Model Summary

• Given: Ordinal preferences p of all voters• These preferences come from an unknown

arbitrary metric space

• Want mechanism which has small distortion:


.

.

.

. Σ d(i,winner)i

i

maxdϵD(p)

Amin Σ d(i,A) Approximate median using

only ordinal information

Easy Example: 2 candidates

• 2 candidateso n-k voters have A > B o k voters have B > A



BA

kn-k

B may be optimal even if k=1



BA

kn-k

B may be optimal even if k=1But, if use majority, then distortion ≤ 3


• 2 candidateso n/2 voters have A > B o n/2 voters have B > A

BA

n/2n/2

B may be optimal even if k=1But, if use majority, then distortion ≤ 3Also shows that no deterministic mechanism can have distortion < 3

Our Results

Sum Median

Plurality 2m-1 Unbounded

Borda 2m-1 Unbounded

k-approval 2n-1 Unbounded

Veto 2n-1 Unbounded

Copeland 5 5

Uncovered Set 5 5

Lower Bound 3 5

Σ d(i,winner)i

i

maxdϵD(p)

Amin Σ d(i,A)

Sum Distortion = Median Distortion = replace sum with median

Copeland Mechanism

Majority Graph:

Edge (A,B) if A pairwise defeats B

Copeland Winner: Candidate who defeats most others

B

A

C

E

D

Copeland Mechanism

Majority Graph:

Edge (A,B) if A pairwise defeats B

Copeland Winner: Candidate who defeats most others

B

A

C

E

D

Tournament winner: has one or two-hop path to all other nodesAlways exists, Copeland chooses one such winner

Our Results

Sum Median




Veto 2n-1 Unbounded

Copeland 5 5

Uncovered Set 5 5

Lower Bound 3 5

Σ d(i,winner)i

i

maxdϵD(p)

Amin Σ d(i,A)


Distortion at most 5

Tournament winner W

Optimal candidate X

XW Distortion ≤ 3

XW

B

Distortion ≤ 5

Our Results

Sum Median




Veto 2n-1 Unbounded

Copeland 5 5

Uncovered Set 5 5

Lower Bound 3 5

Σ d(i,winner)i

i

maxdϵD(p)

Amin Σ d(i,A)


Our Results

Sum Median




Veto 2n-1 Unbounded

Copeland 5 5

Uncovered Set 5 5

Lower Bound 3 5

med d(i,winner)maxdϵD(p)

Amin med d(i,A)Median Distortion =

Median instead of average voter happinessi

i

Bounds on Percentile DistortionPercentile 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 unhappiness



Lower Bounds on Distortion

α0 1

Unbounded

5

3

2/3



Lower Bounds on Distortion

α0 1

Unbounded

5

3

2/3

Upper Bounds on Distortion

α0 1

Unbounded

(Copeland) 5 (Plurality)

3

(m-1)/m

Our Results

Sum Median




Veto 2n-1 Unbounded

Copeland 5 5

Uncovered Set 5 5

Lower Bound 3 5

Σ d(i,winner)i

i

maxdϵD(p)

Amin Σ d(i,A)


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)

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?

Documents

Approximating Optimal Social Choice under Metric Preferences Elliot Anshelevich Onkar Bhardwaj John Postl Rensselaer Polytechnic Institute (RPI), Troy,