11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 1 Common Voting Rules as Maximum Likelihood Estimators Vincent Conitzer, Tuomas Sandholm Presented by Matthew Kay

11/24/2008CS 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 3 Introduction Two views of voting: 1. Voters are idiosyncratic; the best we can do is try to maximize social welfare using a compromise 2. 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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 : L n → O

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 10 Scoring Rules Let α = (α 1,…, α m ) s.t. α 1 ≥ α 2 … ≥ α m For each voter, a candidate receives α i 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)

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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

11/24/2008CS 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)

11/24/2008CS 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)

11/24/2008CS 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 V 1, V 2 such that rule p produces the same outcome on V 1 and V 2, but a different outcome on V 1 +V 2 (the votes in V 1 and V 2 taken together, then p is not a maximum likelihood estimator for that type of outcome under i.i.d. votes.

11/24/2008CS 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 V 1 and V 2, but a different outcome on V 1 +V 2 MLE for V 1 MLE for V 2 MLE for V 1 +V 2

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 23 Proof of Lemma 1 But s is not the outcome produced by p on V 1 +V 2 So p is not an MLE for this distribution MLE for V 1 MLE for V 2 MLE for V 1 +V 2

11/24/2008CS 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.

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 25 Negative results: STV is not MLEW V 1 : 3 votes of c > a > b 4 votes of a > b > c 6 votes of b > a > c

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 26 Negative results: STV is not MLEW V 1 : 3 votes of c > a > b 4 votes of a > b > c 6 votes of b > a > c Round 1, c eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 27 Negative results: STV is not MLEW V 1 : 7 votes of a > b 6 votes of b > a Round 1, c eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 28 Negative results: STV is not MLEW V 1 : 7 votes of a > b 6 votes of b > a Round 2, b eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 29 Negative results: STV is not MLEW V 1 : a wins V 2 : 3 votes of b > a > c 4 votes of a > c > b 6 votes of c > a > b

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 30 Negative results: STV is not MLEW V 1 : a wins V 2 : 3 votes of b > a > c 4 votes of a > c > b 6 votes of c > a > b Round 1, b eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 31 Negative results: STV is not MLEW V 1 : a wins V 2 : 7 votes of a > c 6 votes of c > a Round 1, b eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 32 Negative results: STV is not MLEW V 1 : a wins V 2 : 7 votes of a > c 6 votes of c > a Round 2, c eliminated

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 33 Negative results: STV is not MLEW V 1 : a wins V 2 : a wins a won V 1 and V 2, so a must win V 1 + V 2 for STV to be an MLEW

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 34 Negative results: STV is not MLEW V 1 + V 2: 4 votes of a > c > b 4 votes of a > b > c 9 votes of c > a > b 9 votes of b > a > c

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 35 Negative results: STV is not MLEW V 1 + V 2: 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 36 Negative results: STV is not MLEW V 1 + V 2: 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 37 Negative Results: Ranked Pairs From introduction: a > b > c > d V 1

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 38 Negative Results: Ranked Pairs From introduction:a > b > c > d V 1 V 2

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 39 Negative Results: Ranked Pairs b > c > d > a V 1 + V 2

11/24/2008CS 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 41 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 > … MLEW Not MLER

11/24/2008CS 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

11/24/2008CS Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay 43 Summary of Results RuleMLEW?MLER? ScoringYes STVNoYes BucklinNo CopelandNo MaximinNo Ranked PairsNo

11/24/2008CS 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

11/24/2008CS 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?)