Voting (rank aggregation) rules

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# Voting (rank aggregation) rules - PowerPoint PPT Presentation

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
• 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
• 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)

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]
• 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]
• 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?
• 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

“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

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

“correct” outcome

a

vote 1

vote 2

vote n

a

a

a

conditional independence

assumption

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
• 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

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: 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 > 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 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 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 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

b > d fixed

b > c fixed

result: b > c > d > a

a > c fixed

c > d fixed

b > c fixed

a > b fixed

result: a > b > c > d

b > d fixed

a > b fixed

b > c fixed

c > d fixed

result: a > b > c > d

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
• 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: