Approximating Optimal Social Choice under Metric Preferences Elliot Anshelevich Onkar Bhardwaj John Postl Rensselaer Polytechnic Institute (RPI), Troy, NY
Dec 18, 2015
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
• m candidates/alternatives A, B, C, D, …• n voters/agents: have preferences over alternatives
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
• m candidates/alternatives A, B, C, D, …• n voters/agents: have preferences over alternatives
Usually specify total order over alternatives
• Majority/ Plurality does not work very well: C wins even though A pairwise preferred to C.
E.g., Bush-Gore-Nader
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
B
A
C
Voting Mechanisms
• m candidates/alternatives A, B, C, D, …• n voters/agents: have preferences over alternatives
Usually specify total order over alternatives
• Majority/ Plurality does not work very well: C wins even though A pairwise preferred to C.
E.g., Bush-Gore-Nader
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
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
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
• 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 CB > A > C
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
• 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
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
.
.
.
.
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:
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
.
.
.
. 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
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=1But, if use majority, then distortion ≤ 3
Easy Example: 2 candidates
• 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
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
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
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
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
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
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
Upper Bounds on Distortion
α0 1
Unbounded
(Copeland) 5 (Plurality)
3
(m-1)/m
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
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?