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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 )

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

voting rank aggregation rules
Voting (rank aggregation) rules
  • Set of mcandidates (alternatives) C
  • n voters; each voter ranks the candidates (the voter’s vote)
    • E.g. b > a > c > d
  • Voting rulef maps every (multi-)set of votes to either:
    • a winner in C, or
    • a complete ranking ofC
  • 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
Two views of voting
  • Voters’ preferences are idiosyncratic; only purpose is to find a compromise winner/ranking
  • 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

(outcome=winner or ranking)

“correct” outcome

“correct” outcome

a

a

P(vote|outcome)

P(all votes|outcome)

agents’ votes

vote 1

vote 2

vote n

a

a

a

a

conditional independence assumption

Goal: given votes, find maximum likelihood estimate of correct outcome

Different noise model different maximum likelihood estimator/voting rule

marquis de condorcet 1785
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 p2(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
The Kemeny rule [1959]
  • Given a ranking r, a vote v, and two candidates a, b, let δab(r, v) = 1 if r and v disagree on the relative ranking of a and b, and 0 otherwise
  • A Kemeny rankingr minimizes ΣabΣvδab(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
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
Conditional independence restriction

“correct” outcome

a

  • 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

agents’ votes

a

  • 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)

“correct” outcome

a

vote 1

vote 2

vote n

a

a

a

conditional independence

assumption

any scoring rule is mlewiv and mleriv
Any scoring rule is MLEWIV and MLERIV
  • Scoring rule gives a candidate a1 points if it is ranked first, a2 points if it is ranked second, etc.
    • plurality rule: a1 = 1, ai = 0 otherwise
    • Borda rule: ai = m-i
    • veto rule: am = 0, ai = 1 otherwise
  • MLEWIV noise model: P(v|w) = 2al(v,w) where l(v,w) is the rank of w in v
    • want to choose w to maximize Πv 2al(v,w) = 2Σval(v,w)
  • MLERIV noise model: P(v|r) = Π1≤i≤m(m+1-i)al(v,ri) where ri is the candidate ranked ith in r
single transferable vote stv is mleriv
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 ribe the candidate ranked ith in r
    • Let δv(ri) = 1 if all the candidates ranked higher than riin v are ranked lower in r (i.e. they are all contained in {ri+1, ri+2, …, rm}), otherwise 0
    • P(v|r) = Π1≤i≤mkiδv(ri) where ki+1 << ki < 1
lemma to prove negative results
Lemma to prove negative results

correct outcome

  • 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

vote n

vote k+1

vote k

vote 1

vote set 2

vote set 1

vote set 3

stv rule is not mlewiv
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 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 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
(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 > c1 > c2 > … > cm-2
    • cm-2 > cm-1 > … c1 > 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 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

=

+

b: 2-0 = 2

a: 2-1 = 1

c: 2-2 = 0

d: 1-2 = -1

e: 0-2 = -2

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

maximin is not mlewiv mleriv
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

=

+

c: 2

a: 4

d: 6

b: 8

a: 6

b: 8

c: 10

d: 12

a: 6

b: 8

c: 10

d: 12

ranked pairs is not mlewiv mleriv
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

=

+

d > a fixed

c > d fixed

a > c discarded

b > d fixed

a > b discarded

b > c fixed

result: b > c > d > a

a > c fixed

c > d fixed

d > a discarded

b > c fixed

a > b fixed

result: a > b > c > d

b > d fixed

a > b fixed

d > a discarded

b > c fixed

c > d fixed

result: a > b > c > d

consistency scoring rules
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
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:

Thank you for your attention!

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