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?