Common Voting Rules as Maximum Likelihood Estimators

11/24/2008 CS 886 - Common Voting Rules as Maximum Likelihood Estimators - Matthew Kay

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Common Voting Rules as Maximum Likelihood

Estimators

Vincent Conitzer, Tuomas Sandholm

Presented by Matthew Kay


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Outline Introduction

Noise Models Terminology

Voting Rules Results

Positive Results Lemma 1 Negative Results

Conclusion Summary of Results Conclusions and Contributions Future Work


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Introduction

Two views of voting:1. Voters are idiosyncratic; the best we can do is try

to maximize social welfare using a compromise

2. There is some prior “absolute” way that we can say one candidate is better than another, and votes represent the agents’ noisy perception of this

We consider the second case only


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Introduction

Under these assumptions, voting becomes a way to infer the “absolute” or “objective” goodness of the candidates

One way to do this is a maximum likelihood estimate, or MLE


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Noise Models

Paper assumes the votes are independent and identically distributed (i.i.d.)

Conditionally independent given the outcome

Each voter has the same conditional distribution


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Noise Models

Without these restrictions, any rule is an MLE

Simply let the probability on all vote vectors that produce the correct outcome be positive, and all other probabilities be 0


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Terminology

Set of agents (voters), N = {1, 2, …, n} Set of candidates, C Set of outcomes, O:

A winner: O is the set of single candidates, C A ranking: O is the set of weak total orders of C

A set of strict total orders of C, L A voting rule p : Ln → O


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Terminology

MLEWIV: maximum likelihood estimator for winner under i.i.d. votes

MLERIV: maximum likelihood estimator for ranking under i.i.d. votes

I will shorten these to MLEW and MLER


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Scoring Rules

Let α = (α1,…, αm) s.t. α1 ≥ α2 … ≥ αm

For each voter, a candidate receives αi points if the voter ranked them at position i

The candidate with the highest score wins Examples:

Plurality: α = (1, 0, …, 0) Veto: α = (1, …, 1, 0) Borda: α = (m – 1, m – 2, …, 1, 0)


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Single Transferable Vote (STV)

Series of m – 1 plurality votes In each round, the lowest-ranked candidate is

eliminated The last remaining candidate wins


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Bucklin

Approval: a voter “approves” their top l candidates For candidate c, let B(c,l) be the number of voters

with c in their top l candidates Bucklin score: min{l : B(c,l) > n/2} To calculate: increase l until a candidate is

“approved” by > n/2 voters; this is their score For ties, consider B(c,l) - n/2


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Pairwise Rules

For these rules, we consider pairwise election graphs instead of the rankings themselves

Example: 2 voters Votes:

a > b > c b > a > c

Note (Lemma 2): for any pairwise election graph with even weights, there is a set of rankings that produces that graph


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maximin

Candidate’s rank = their worst score in a pairwise election a: 6 b: 8 c: 10 d: 12

Outcome: a > b > c > d


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Copeland

Candidate’s rank = wins – losses(outgoing edges - incoming edges) a: 2 b: 1 c: 0 d: -1 e: -2

Outcome: a > b > c > d > e


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Ranked Pairs

Sort pairs of candidates by edge weights Start with the highest-weighted pair and “lock

in” that order “lock in” the next-highest pair, etc If an ordering cannot be “locked in”, skip it


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Ranked Pairs

Sort pairs by weight: (b,d), (a,b), (d,a), (b,c), (c,d)

“Lock in”: (b,d) : b > d (a,b) : a > b > d (d,a) : skipped (b,c) : a > b > c , a > b > d (c,d) : a > b > c > d

Outcome: a > b > c > d


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Positive Results

Basic outline for proving a rule is an MLEW or MLER Contrive a distribution (noise model) that in some

way “mimics” the behaviour of the voting rule, so that finding the maximum likelihood estimate is done either by choosing the winning candidate (MLEW) or the winning ranking (MLER)


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Positive Results

Able to show: Any scoring rule is MLEW, MLER STV is MLER

(see paper for details)


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Negative Results

Some voting rules are not MLER or MLEW We can prove this using Lemma 1 (page 4):

For a given type of outcome (e.g. winner or ranking), if there exist vectors of votes V1, V2 such that rule p produces the same outcome on V1 and V2, but a different outcome on V1+V2 (the votes in V1 and V2 taken together, then p is not a maximum likelihood estimator for that type of outcome under i.i.d. votes.


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Proof of Lemma 1

Consider a rule, p, that produces the same outcome, s, on V1 and V2, but a different outcome on V1+V2

)'|(maxarg

)'|()'|(maxarg

)'|(maxarg

)'|(maxarg

21'

21'

2'

1'

sSVVPs

sSVPsSVPs

sSVPs

sSVPs

s

s

s

s

MLE for V1

MLE for V2

MLE for V1 +V2


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Proof of Lemma 1

But s is not the outcome produced by p on V1+V2

So p is not an MLE for this distribution

)'|(maxarg

)'|()'|(maxarg

)'|(maxarg

)'|(maxarg

21'

21'

2'

1'

sSVVPs

sSVPsSVPs

sSVPs

sSVPs

s

s

s

s

MLE for V1

MLE for V2

MLE for V1 +V2


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Lemma 1 Implications

In order to prove a rule is not an MLEW or MLER, we need to find two sets of votes that produce the same outcome, but when combined produce a different outcome.


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Negative results:STV is not MLEW

V1 : 3 votes of c > a > b 4 votes of a > b > c 6 votes of b > a > c


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V1 : 3 votes of c > a > b 4 votes of a > b > c 6 votes of b > a > c

Round 1, c eliminated


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V1 : 7 votes of a > b 6 votes of b > a



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V1 : 7 votes of a > b 6 votes of b > a

Round 2, b eliminated


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V1 : a wins

V2 : 3 votes of b > a > c 4 votes of a > c > b 6 votes of c > a > b


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V1 : a wins

V2 : 3 votes of b > a > c 4 votes of a > c > b 6 votes of c > a > b



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V1 : a wins

V2 : 7 votes of a > c 6 votes of c > a



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V1 : a wins

V2 : 7 votes of a > c 6 votes of c > a



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V1 : a wins

V2 : a wins

a won V1 and V2, so a must win V1 + V2 for STV to be an MLEW


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V1 + V2:

4 votes of a > c > b 4 votes of a > b > c 9 votes of c > a > b 9 votes of b > a > c


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V1 + V2:


Round 1, a eliminated


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V1 + V2:


STV is not an MLEW


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Negative Results:Ranked Pairs

From introduction:

a > b > c > d

V1


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From introduction:

a > b > c > d a > b > c > d

V1 V2


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b > c > d > a

V1 + V2


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Result: the ranked pairs rule is not an MLEW or MLER

Proofs for other pairwise election results are similar (see paper): Copeland is not MLEW or MLER maximin is not MLEW or MLER


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Relationship Between MLEW and MLER

From STV, we note that MLER does not imply MLEW

In addition, MLEW does not imply MLER: consider a hybrid rule that chooses the winner according to an MLEW rule and the remaining candidates from a rule which is not MLER:

a > b > c > d > …MLEW

Not MLER


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Summary of Results

Rule MLEW? MLER?

Scoring Yes Yes

STV No Yes

Bucklin No No

Copeland No No

Maximin No No

Ranked Pairs No No


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Conclusion

Paper considers applications in which there is some prior, “objective” sense in which some candidates are better than others

Contributions: Without any restrictions on the noise model, any voting rule

is an MLE. Noise models for scoring rules (showing it is an

MLEW/MLER) and STV (showing it is an MLER) Method (Lemma 1) for generating impossibility results, and

shows that various voting methods are not MLEW/MLERs


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Issues and Future Work

The distributions used to prove the positive results were somewhat contrived Need to evaluate how reasonable they are If they are unreasonable, can they be refined?

Can we build new voting rules to match an observed noise model?

Are there rules which Lemma 1 cannot prove are not MLEW/MLER but which nevertheless are not MLEW/MLER (i.e. can Lemma 1 be used to show that a rule is an MLEW/MLER?)

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Common Voting Rules as Maximum Likelihood Estimators