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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)

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]

- 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

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

- 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

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

- 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:

Thank you for your attention!

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