Common Voting Rules as Maximum Likelihood Estimators Vincent

Common Voting Rules as Maximum Likelihood Estimators Vincent Conitzer and Tuomas Sandholm Carnegie Mellon University, Computer Science Department

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 vector 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/candidates are ranked by score

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 “correct”aoutcome agents’ a votes (outcome=winner or ranking) “correct”aoutcome vote a 1 vote a 2 … vote a n conditional independence assumption Goal: given votes, find maximum likelihood estimate of correct outcome Different noise model different maximum likelihood estimator/voting rule

History • [Condorcet 1785] assumed noise model where voter ranks any two candidates correctly with fixed probability p > 1/2, independently – Gives cyclical rankings with some probability, but does not affect MLE approach – Solved cases of 2 and 3 candidates • Two centuries pass… • [Young 1995] solved case of arbitrary number of candidates under the same model – Showed that it coincided with rule proposed by Kemeny [Kemeny 1959] • [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 paper: 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

Independence restriction “correct”aoutcome agents’ a votes • Without any independence restriction, it turns out that any rule has a noise model: • P(vote vector|outcome) > 0 if and only if f(vote vector)=outcome • So, will focus on conditionally “correct”aoutcome independent votes • If a rule has a noise model in this vote a 1 a 2 … vote a n setup we call it an conditional independence assumption – 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, ai = 0 otherwise – Borda rule: ai = m-i – veto rule: am = 0, ai = 1 otherwise • MLEWIV noise model: P(v|w) = 2 al(v, w) where l(v, w) is the rank of w in v – want to choose w to maximize Πv 2 al(v, w) = 2Σval(v, w) • MLERIV noise model: P(v|r) = Π 1≤i≤m(m+1 -i)al(v, r ) where ri is the candidate ranked ith in r i

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

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

STV rule is not MLEWIV • STV rule: Candidate ranked first by fewest voters drops out and is removed from rankings, final ranking is inverse of order in which they dropped out • First vote vector: – – 3 times c > a > b 4 times a > b > c 6 times b > a > c c drops out first, then a wins • Second vote vector: – – 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 vector: – 2 times a > b > c > d > e – 1 time b > a > c > d > e – gives final ranking a > b > c > d > e • Second vote vector: – – 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 > … > cm-2 – cm-2 > cm-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 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 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

Conclusions • 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: MLERIV not MLERIV MLEWIV scoring rules (incl. plurality, Borda, veto) hybrids of MLEWIV and (not MLERIV) rules not MLEWIV STV Bucklin, Copeland, maximin, ranked pairs Thank you for your attention!