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Approximating Optimal Social Choice under Metric Preferences

This study explores the problem of approximating optimal social choice when voters have metric preferences over alternatives. It examines different voting mechanisms and analyzes their weaknesses, including the Arrow's Impossibility Theorem. The proposed model introduces metric preferences to determine the best alternative. The goal is to return a provably good approximation to the best alternative based on ordinal preferences of voters.

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Approximating Optimal Social Choice under Metric Preferences

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  1. Approximating Optimal Social Choiceunder Metric Preferences Elliot Anshelevich OnkarBhardwaj John Postl Rensselaer Polytechnic Institute (RPI), Troy, NY

  2. 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

  3. 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 6. C > A > B 2. A > B > C 7. C > A > B 3. A > B > C 8. C > A > B 4. B > A > C 9. C > A > B 5. B > A > C

  4. 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 Apairwise preferred to C. E.g., Bush-Gore-Nader 1. A > B > C 6. C > A > B A 2. A > B > C 7. C > A > B 3. A > B > C 8. C > A > B B 4. B > A > C 9. C > A > B 5. B > A > C C

  5. 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 Apairwise preferred to C. E.g., Bush-Gore-Nader 1. A > B > C 6. C > A > B A 2. A > B > C 7. C > A > B 3. A > B > C 8. C > A > B B 4. B > A > C 9. C > A > B 5. B > A > C C

  6. Voting Mechanisms • Condorcet Cycle A C B 1. A > B > C 2. B > C > A 3. C > A > B

  7. Voting Mechanisms • Condorcet Cycle • So, what is “best” outcome? • All voting mechanisms have weaknesses. • “Axiomatic” approach: define some properties, see which mechanisms satisfy them A C B 1. A > B > C 2. B > C > A 3. C > A > B

  8. Arrow’s Impossibility Theorem (1950) (Nobel prize in economics) • 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 each • Make extra assumptions on preferences

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

  10. Example: Political Spectrum Left Right A B C

  11. Example: Political Spectrum

  12. Example: Political Spectrum

  13. Example: Political Spectrum xkcd

  14. Example: Political Spectrum xkcd Downsian proximity model (1957): Each dimension is a different issue

  15. 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) i A B A C

  16. 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) i A B A C B > A > C

  17. 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) i A B A C

  18. 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) i A B A C

  19. 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) i A B A C

  20. Our Model 1. A > B > C • Given: Ordinal preferences of all voters • These preferences come from an unknown arbitrary metric space • Goal: Return best alternative 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 . . . . . .

  21. Our Model 1. A > B > C • Given: Ordinal preferences of all voters • These preferences come from an unknown arbitrary metric space • Goal: Return provably good approximation to the best alternative 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 Σ d(i,C) 9. C > A > B i . . . . small Σ d(i,B) i B = OPT A C

  22. Model Summary 1. A > B > C • Given: Ordinal preferences p of all voters • These preferences come from an unknown arbitrary metric space • Want mechanism which has small distortion: 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 max min Σ d(i,A) dϵD(p) Approximate median using only ordinal information i A

  23. Easy Example: 2 candidates • 2 candidates • n-k voters have A > B • k voters have B > A

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

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

  26. Easy Example: 2 candidates • 2 candidates • n/2 voters have A > B • n/2 voters have B > A n/2 n/2 A B 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

  27. Our Results Σ d(i,winner) i max Sum Distortion = Median Distortion = replace sum with median min Σ d(i,A) dϵD(p) i A

  28. Copeland Mechanism Majority Graph: Edge (A,B) if A pairwise defeats B Copeland Winner: Candidate who defeats most others A B E D C

  29. Copeland Mechanism Majority Graph: Edge (A,B) if A pairwise defeats B Copeland Winner: Candidate who defeats most others A B E D C Tournament winner: has one or two-hop path to all other nodes Always exists, Copeland chooses one such winner

  30. Our Results Σ d(i,winner) i max Sum Distortion = Median Distortion = replace sum with median min Σ d(i,A) dϵD(p) i A

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

  32. Our Results Σ d(i,winner) i max Sum Distortion = Median Distortion = replace sum with median min Σ d(i,A) dϵD(p) i A

  33. Our Results Median instead of average voter happiness med d(i,winner) max i Median Distortion = min med d(i,A) dϵD(p) i A

  34. 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

  35. 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 Unbounded 5 3 α 2/3 0 1 Lower Bounds on Distortion

  36. 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 Unbounded Unbounded 5 (Copeland) 5 3 (Plurality) 3 α 2/3 α (m-1)/m 0 1 0 1 Lower Bounds on Distortion Upper Bounds on Distortion

  37. Our Results Σ d(i,winner) i max Sum Distortion = Median Distortion = replace sum with median min Σ d(i,A) dϵD(p) i A

  38. 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)

  39. 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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