Common Voting Rules as Maximum Likelihood Estimators Vincent Conitzer (Joint work with Tuomas Sandholm) Early version of this work appeared in UAI-05

Voting (rank aggregation) rules Set of m candidates (alternatives) C n voters; each voter ranks the candidates (the voter’s vote) –E.g. b > a > c > d Voting rule f maps every (multi-)set of votes to either: –a winner in C, or –a complete ranking of C E.g. plurality: –every voter votes for a single candidate (equiv. we only consider the candidate’s top-ranked candidate) –candidate with most votes wins E.g. single transferable vote (STV): –candidate ranked first by fewest voters drops out and is removed from rankings –repeat –final ranking is inverse of order in which they dropped out

Two views of voting 1.Voters’ preferences are idiosyncratic; only purpose is to find a compromise winner/ranking 2.There is some absolute sense in which some candidates are better than others, independent of voters’ preferences; votes are merely noisy perceptions of candidates’ true quality a “correct” outcome a agents’ votes a “correct” outcome a vote 1 a vote 2 a vote n … conditional independence assumption Goal: given votes, find maximum likelihood estimate of correct outcome Different noise model  different maximum likelihood estimator/voting rule (outcome=winner or ranking) P(vote|outcome) P(all votes|outcome)

Marquis de Condorcet [1785] Condorcet was interested in the “correct ranking” model He assumed noise model where voter ranks any two candidates correctly with fixed probability p > 1/2, independently With some probability this gives a cycle… –E.g. if the correct ranking is a > b > c, then with probability p 2 (1- p) a voter will prefer a > b, b > c, c > a But, it does not matter for the MLE approach as long as we get a probability for each (acyclic) vote –Equivalently, we can renormalize the probabilities over the acyclic votes –Equivalently, we can say that if a cyclic vote is drawn, it must be redrawn Condorcet solved for the MLE rule for the cases of 2 and 3 candidates

The Kemeny rule [1959] Given a ranking r, a vote v, and two candidates a, b, let δ ab (r, v) = 1 if r and v disagree on the relative ranking of a and b, and 0 otherwise A Kemeny ranking r minimizes Σ ab Σ v δ ab (r, v) Young [1986]’s observation: the Kemeny rule is the solution to Condorcet’s problem! Drissi & Truchon [2002] extend to the case where p is allowed to vary with the distance between two candidates in correct ranking

What is next? Does this suggest using Kemeny rule? –Many other noise models possible –Some of these may correspond to other, better-known rules Goal of this work: Classify which common rules are a maximum likelihood estimator for some noise model –Positive and negative results –Positive results are constructive Motivation: –Rules corresponding to a noise model are more natural –Knowing a noise model can give us insight into the rule and its underlying assumptions –If we disagree with the noise model, we can modify it and obtain new version of the rule

Conditional independence restriction Without any independence restriction, it turns out that any rule has a noise model: P(vote set|outcome) > 0 if and only if f(vote set)=outcome a “correct” outcome a agents’ votes a “correct” outcome a vote 1 a vote 2 a vote n … conditional independence assumption So, will focus on conditionally independent votes If a rule has a noise model in this setup we call it an –MLEWIV rule if producing winner –MLERIV rule if producing ranking –(IV = Independent Votes)

Any scoring rule is MLEWIV and MLERIV Scoring rule gives a candidate a 1 points if it is ranked first, a 2 points if it is ranked second, etc. –plurality rule: a 1 = 1, a i = 0 otherwise –Borda rule: a i = m-i –veto rule: a m = 0, a i = 1 otherwise MLEWIV noise model: P(v|w) = 2 a l(v,w) where l(v,w) is the rank of w in v –want to choose w to maximize Π v 2 a l(v,w) = 2 Σ v a l(v,w) MLERIV noise model: P(v|r) = Π 1≤i≤m (m+1-i) a l(v,r i ) where r i is the candidate ranked ith in r

Single Transferable Vote (STV) is MLERIV STV rule: candidate ranked first by fewest voters drops out and is removed from rankings; repeat; final ranking is inverse of order in which they dropped out MLERIV noise model: –Let r i be the candidate ranked ith in r –Let δ v (r i ) = 1 if all the candidates ranked higher than r i in v are ranked lower in r (i.e. they are all contained in {r i+1, r i+2, …, r m }), otherwise 0 –P(v|r) = Π 1≤i≤m k i δ v (r i ) where k i+1 << k i < 1

Lemma to prove negative results For any noise model, if there is a single outcome that maximizes the likelihood of both vote set 1 and vote set 2, then it must also maximize the likelihood of vote set 3 Hence, a voting rule that produces the same outcome on both set 1 and set 2 but a different one on set 3 cannot be a maximum likelihood estimator correct outcome vote 1 vote k vote k+1 vote n vote set 1 vote set 2 vote set 3 … …

STV rule is not MLEWIV STV rule: candidate ranked first by fewest voters drops out and is removed from rankings; repeat. Final ranking is inverse of order in which they dropped out First vote set: –3 times c > a > b –4 times a > b > c –6 times b > a > c –c drops out first, then a wins Second vote set: –3 times b > a > c –4 times a > c > b –6 times c > a > b –b drops out first, then a wins But: taking all votes together, a drops out first! –(8 votes vs. 9 for the others)

Bucklin rule is not MLEWIV/MLERIV Bucklin rule: –For every candidate, consider the minimum k such that more than half of the voters rank that candidate among the top k –Candidates are ranked (inversely) by their minimum k –Ties are broken by the number of voters by which the “half” mark is passed First vote set: –2 times a > b > c > d > e –1 time b > a > c > d > e –gives final ranking a > b > c > d > e Second vote set: –2 times b > d > a > c > e –1 time c > e > a > b > d –1 time c > a > b > d > e –gives final ranking a > b > c > d > e But: taking all votes together gives final ranking b > a > c > d > e –(b goes over half at k=2, a does not)

Pairwise election graphs Pairwise election: take two candidates and see which one is ranked above the other in more votes Pairwise election graph has edge of weight k from a to b if a defeats b by k votes in the pairwise election E.g. votes a > b > c and b > a > c together produce pairwise election graph:

(Roughly) all pairwise election graphs can be realized Lemma: any graph with even weights is the pairwise election graph for some votes Proof: can increase the weight of edge from a to b by two by adding the following two votes: –a > b > c 1 > c 2 > … > c m-2 –c m-2 > c m-1 > … c 1 > a > b Hence, from here on, we will simply show the pairwise election graph rather than the votes that realize it

Copeland is not MLEWIV/MLERIV Copeland rule: candidate’s score = number of pairwise victories – number of pairwise defeats –i.e. outdegree – indegree of vertex in pairwise election graph a: 3-1 = 2 b: 2-1 = 1 c: 2-2 = 0 d: 1-2 = -1 e: 1-3 = -2 a: 3-1 = 2 b: 2-1 = 1 c: 2-2 = 0 d: 1-2 = -1 e: 1-3 = -2 b: 2-0 = 2 a: 2-1 = 1 c: 2-2 = 0 d: 1-2 = -1 e: 0-2 = -2 + =

Maximin is not MLEWIV/MLERIV maximin rule: candidate’s score = score in worst pairwise election –i.e. candidates are ordered inversely by weight of largest incoming edge a: 6 b: 8 c: 10 d: 12 + = a: 6 b: 8 c: 10 d: 12 c: 2 a: 4 d: 6 b: 8

Ranked pairs is not MLEWIV/MLERIV ranked pairs rule: pairwise elections are locked in according by margin of victory –i.e. larger edges are “fixed” first, an edge is discarded if it introduces a cycle b > d fixed a > b fixed d > a discarded b > c fixed c > d fixed result: a > b > c > d + = a > c fixed c > d fixed d > a discarded b > c fixed a > b fixed result: a > b > c > d d > a fixed c > d fixed a > c discarded b > d fixed a > b discarded b > c fixed result: b > c > d > a

Consistency & scoring rules A rule is consistent if, whenever it produces the same winner on two vote sets, it produces the same winner on the union of those sets Known result: A rule is consistent if and only if it determines the winner according to a scoring rule [Young 1975] Hence, the following are equivalent properties of a rule: –Consistency –Determining the winner according to a scoring rule –MLEWIV These questions are open (as far as I know): –What is the characterization of MLERIV rules? –What is the characterization of “ranking-consistent” voting rules? –What is the relationship between these?

Conclusions MLERIVnot MLERIV MLEWIVscoring rules (incl. plurality, Borda, veto) hybrids of MLEWIV and (not MLERIV) rules not MLEWIVSTV, KemenyBucklin, Copeland, maximin, ranked pairs, Slater Thank you for your attention! We asked the question: which common voting rules are maximum likelihood estimators (for some noise model)? If votes are not independent given outcome (winner/ranking), any rule is MLE If votes are independent given outcome, some rules are MLEWIV (MLE for winner), some are MLERIV (MLE for ranking), some are both: