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Common Voting Rules as Maximum Likelihood Estimators

Common Voting Rules as Maximum Likelihood Estimators. Vincent Conitzer, Tuomas Sandholm Presented by Matthew Kay. Outline. Introduction Noise Models Terminology Voting Rules Results Positive Results Lemma 1 Negative Results Conclusion Summary of Results Conclusions and Contributions

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Common Voting Rules as Maximum Likelihood Estimators

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  1. Common Voting Rules as Maximum Likelihood Estimators Vincent Conitzer, Tuomas Sandholm Presented by Matthew Kay CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  2. Outline • Introduction • Noise Models • Terminology • Voting Rules • Results • Positive Results • Lemma 1 • Negative Results • Conclusion • Summary of Results • Conclusions and Contributions • Future Work CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  3. Introduction • Two views of voting: • Voters are idiosyncratic; the best we can do is try to maximize social welfare using a compromise • There is some prior “absolute” way that we can say one candidate is better than another, and votes represent the agents’ noisy perception of this • We consider the second case only CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  4. Introduction • Under these assumptions, voting becomes a way to infer the “absolute” or “objective” goodness of the candidates • One way to do this is a maximum likelihood estimate, or MLE CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  5. Noise Models • Paper assumes the votes are independent and identically distributed (i.i.d.) • Conditionally independent given the outcome • Each voter has the same conditional distribution CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  6. Noise Models • Without these restrictions, any rule is an MLE • Simply let the probability on all vote vectors that produce the correct outcome be positive, and all other probabilities be 0 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  7. Terminology • Set of agents (voters), N = {1, 2, …, n} • Set of candidates, C • Set of outcomes, O: • A winner: O is the set of single candidates, C • A ranking:O is the set of weak total orders of C • A set of strict total orders of C, L • A voting rule p : Ln→ O CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  8. Terminology • MLEWIV: maximum likelihood estimator for winner under i.i.d. votes • MLERIV: maximum likelihood estimator for ranking under i.i.d. votes • I will shorten these to MLEW and MLER CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  9. Outline • Introduction • Noise Models • Terminology • Voting Rules • Results • Positive Results • Lemma 1 • Negative Results • Conclusion • Summary of Results • Conclusions and Contributions • Future Work CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  10. Scoring Rules • Let α = (α1,…, αm) s.t. α1 ≥ α2 … ≥ αm • For each voter, a candidate receives αi points if the voter ranked them at position i • The candidate with the highest score wins • Examples: • Plurality: α = (1, 0, …, 0) • Veto: α = (1, …, 1, 0) • Borda: α = (m – 1, m – 2, …, 1, 0) CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  11. Single Transferable Vote (STV) • Series of m – 1 plurality votes • In each round, the lowest-ranked candidate is eliminated • The last remaining candidate wins CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  12. Bucklin • Approval: a voter “approves” their top l candidates • For candidate c, let B(c,l) be the number of voters with c in their top l candidates • Bucklin score: min{l : B(c,l) > n/2} • To calculate: increase l until a candidate is “approved” by > n/2 voters; this is their score • For ties, consider B(c,l) - n/2 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  13. Pairwise Rules • For these rules, we consider pairwise election graphs instead of the rankings themselves • Example: • 2 voters • Votes: • a > b > c • b > a > c • Note (Lemma 2): for any pairwise election graph with even weights, there is a set of rankings that produces that graph CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  14. maximin • Candidate’s rank = their worst score in a pairwise election • a: 6 • b: 8 • c: 10 • d: 12 • Outcome: a > b > c > d CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  15. Copeland • Candidate’s rank = wins – losses (outgoing edges - incoming edges) • a: 2 • b: 1 • c: 0 • d: -1 • e: -2 • Outcome: a > b > c > d > e CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  16. Ranked Pairs • Sort pairs of candidates by edge weights • Start with the highest-weighted pair and “lock in” that order • “lock in” the next-highest pair, etc • If an ordering cannot be “locked in”, skip it CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  17. Ranked Pairs • Sort pairs by weight: • (b,d), (a,b), (d,a), (b,c), (c,d) • “Lock in”: • (b,d) : b > d • (a,b) : a > b > d • (d,a) : skipped • (b,c) : a > b > c , a > b > d • (c,d) : a > b > c > d • Outcome: a > b > c > d CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  18. Outline • Introduction • Noise Models • Terminology • Voting Rules • Results • Positive Results • Lemma 1 • Negative Results • Conclusion • Summary of Results • Conclusions and Contributions • Future Work CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  19. Positive Results • Basic outline for proving a rule is an MLEW or MLER • Contrive a distribution (noise model) that in some way “mimics” the behaviour of the voting rule, so that finding the maximum likelihood estimate is done either by choosing the winning candidate (MLEW) or the winning ranking (MLER) CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  20. Positive Results • Able to show: • Any scoring rule is MLEW, MLER • STV is MLER (see paper for details) CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  21. Negative Results • Some voting rules are not MLER or MLEW • We can prove this using Lemma 1 (page 4): For a given type of outcome (e.g. winner or ranking), if there exist vectors of votes V1, V2 such that rule p produces the same outcome on V1 and V2, but a different outcome on V1+V2 (the votes in V1 and V2 taken together, then p is not a maximum likelihood estimator for that type of outcome under i.i.d. votes. CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  22. Proof of Lemma 1 • Consider a rule, p, that produces the same outcome, s, on V1 and V2, but a different outcome on V1+V2 MLE for V1 MLE for V2 MLE for V1 +V2 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  23. Proof of Lemma 1 • But s is not the outcome produced by p on V1+V2 • So p is not an MLE for this distribution MLE for V1 MLE for V2 MLE for V1 +V2 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  24. Lemma 1 Implications • In order to prove a rule is not an MLEW or MLER, we need to find two sets of votes that produce the same outcome, but when combined produce a different outcome. CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  25. Negative results:STV is not MLEW • V1: • 3 votes of c > a > b • 4 votes of a > b > c • 6 votes of b > a > c CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  26. Negative results:STV is not MLEW • V1: • 3 votes of c > a > b • 4 votes of a > b > c • 6 votes of b > a > c Round 1, c eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  27. Negative results:STV is not MLEW • V1: • 7 votes of a > b • 6 votes of b > a Round 1, c eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  28. Negative results:STV is not MLEW • V1: • 7 votes of a > b • 6 votes of b > a Round 2, b eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  29. Negative results:STV is not MLEW • V1: • a wins • V2: • 3 votes of b > a > c • 4 votes of a > c > b • 6 votes of c > a > b CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  30. Negative results:STV is not MLEW • V1: • a wins • V2: • 3 votes of b > a > c • 4 votes of a > c > b • 6 votes of c > a > b Round 1, b eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  31. Negative results:STV is not MLEW • V1: • a wins • V2: • 7 votes of a > c • 6 votes of c > a Round 1, b eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  32. Negative results:STV is not MLEW • V1: • a wins • V2: • 7 votes of a > c • 6 votes of c > a Round 2, c eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  33. Negative results:STV is not MLEW • V1: • a wins • V2: • a wins • a won V1and V2, so a must win V1+ V2 for STV to be an MLEW CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  34. Negative results:STV is not MLEW • V1 + V2: • 4 votes of a > c > b • 4 votes of a > b > c • 9 votes of c > a > b • 9 votes of b > a > c CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  35. Negative results:STV is not MLEW • V1 + V2: • 4 votes of a > c > b • 4 votes of a > b > c • 9 votes of c > a > b • 9 votes of b > a > c Round 1, a eliminated CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  36. Negative results:STV is not MLEW • V1 + V2: • 4 votes of a > c > b • 4 votes of a > b > c • 9 votes of c > a > b • 9 votes of b > a > c • STV is not an MLEW CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  37. Negative Results:Ranked Pairs • From introduction: a > b > c > d V1 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  38. Negative Results:Ranked Pairs • From introduction: a > b > c > d a > b > c > d V1 V2 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  39. Negative Results:Ranked Pairs b > c > d > a V1 +V2 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  40. Negative Results:Ranked Pairs • Result: the ranked pairs rule is not an MLEW or MLER • Proofs for other pairwise election results are similar (see paper): • Copeland is not MLEW or MLER • maximin is not MLEW or MLER CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  41. MLEW Not MLER Relationship Between MLEW and MLER • From STV, we note that MLER does not imply MLEW • In addition, MLEW does not imply MLER: consider a hybrid rule that chooses the winner according to an MLEW rule and the remaining candidates from a rule which is not MLER: a > b > c > d > … CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  42. Outline • Introduction • Noise Models • Terminology • Voting Rules • Results • Positive Results • Lemma 1 • Negative Results • Conclusion • Summary of Results • Conclusions and Contributions • Future Work CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  43. Summary of Results CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  44. Conclusion • Paper considers applications in which there is some prior, “objective” sense in which some candidates are better than others • Contributions: • Without any restrictions on the noise model, any voting rule is an MLE. • Noise models for scoring rules (showing it is an MLEW/MLER) and STV (showing it is an MLER) • Method (Lemma 1) for generating impossibility results, and shows that various voting methods are not MLEW/MLERs CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

  45. Issues and Future Work • The distributions used to prove the positive results were somewhat contrived • Need to evaluate how reasonable they are • If they are unreasonable, can they be refined? • Can we build new voting rules to match an observed noise model? • Are there rules which Lemma 1 cannot prove are not MLEW/MLER but which nevertheless are not MLEW/MLER (i.e. can Lemma 1 be used to show that a rule is an MLEW/MLER?) CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

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