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 > 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

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 > 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 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 > 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 C

Voting Mechanisms Condorcet Cycle 1. A > B > C 2. B > C > A 3. 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 > C 2. B > C > A 3. 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 properties o 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 approaches o “Axiomatic” approach: analyze lots of different mechanisms, show good properties about each o Make extra assumptions on preferences (Nobel prize in economics)

Our Approach: Metric Preferences Metric preferences o Also called spatial preferences Additional structure on who prefers which alternative

Example: Political Spectrum Left Right BAC

Example: Political Spectrum

xkcd

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 AC

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 AC 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 AC

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 AC

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 AC

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 > 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

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 > 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 = OPT AC 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 > 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 Σ d(i,winner) i i max d ϵ D(p) A min Σ d(i,A) Approximate median using only ordinal information

Easy Example: 2 candidates 2 candidates o n-k voters have A > B o k voters have B > A

Easy Example: 2 candidates 2 candidates o n-k voters have A > B o k voters have B > A BA k n-k B may be optimal even if k=1

Easy Example: 2 candidates 2 candidates o n-k voters have A > B o k voters have B > A BA k n-k B may be optimal even if k=1 But, if use majority, then distortion ≤ 3

Easy Example: 2 candidates 2 candidates o n/2 voters have A > B o n/2 voters have B > A BA 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 < 3

Our Results SumMedian Plurality2m-1Unbounded Borda2m-1Unbounded k-approval2n-1Unbounded Veto2n-1Unbounded Copeland55 Uncovered Set55 Lower Bound35 Σ d(i,winner) i i max d ϵ D(p) A min 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 nodes Always exists, Copeland chooses one such winner

Our Results SumMedian Plurality2m-1Unbounded Borda2m-1Unbounded k-approval2n-1Unbounded Veto2n-1Unbounded Copeland55 Uncovered Set55 Lower Bound35 Σ d(i,winner) i i max d ϵ D(p) A min Σ d(i,A) Sum Distortion =Median Distortion = replace sum with median

Distortion at most 5 Tournament winner W Optimal candidate X XW Distortion ≤ 3 XW B Distortion ≤ 5

Our Results SumMedian Plurality2m-1Unbounded Borda2m-1Unbounded k-approval2n-1Unbounded Veto2n-1Unbounded Copeland55 Uncovered Set55 Lower Bound35 Σ d(i,winner) i i max d ϵ D(p) A min Σ d(i,A) Sum Distortion =Median Distortion = replace sum with median

Our Results SumMedian Plurality2m-1Unbounded Borda2m-1Unbounded k-approval2n-1Unbounded Veto2n-1Unbounded Copeland55 Uncovered Set55 Lower Bound35 med d(i,winner) max d ϵ D(p) A min med d(i,A) Median Distortion = Median instead of average voter happiness i i

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 unhappiness

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 unhappiness Lower Bounds on Distortion α 01 Unbounded 5 3 2/3

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 unhappiness Lower Bounds on Distortion α 01 Unbounded 5 3 2/3 Upper Bounds on Distortion α 01 Unbounded (Copeland) 5 (Plurality) 3 (m-1)/m

Our Results SumMedian Plurality2m-1Unbounded Borda2m-1Unbounded k-approval2n-1Unbounded Veto2n-1Unbounded Copeland55 Uncovered Set55 Lower Bound35 Σ d(i,winner) i i max d ϵ D(p) A min 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?