Vote Elicitation with Probabilistic Preference Models: Empirical Estimation and Cost

Tradeoffs

Tyler Lu and Craig BoutilierUniversity of Toronto

IntroductionNew communication platforms can transform

the way people make group decisions.

How can computational social choice realize this shift?

ChoicesPeople

Computational Social Choice

Consensus

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Introduction• Computational social choice

– Aggregate full preferences (rankings)– Mostly study rank-based schemes (Borda, maximin, etc…)

• Rank-based voting schemes rarely used in practiceProblem: Cognitive and communication burdenOur approach (recent work): Elicit just the right preferences to make good enough group decisionsThis work: Multi-round elicitation and probabilistic preference models to further reduce burdensAlice Bob Cindy

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Outline

• Preliminaries

• Multi-round Probabilistic Vote Elicitation

• Methodology and Analysis for One-round

• Experimental Results

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Preliminaries• Voters N = {1..n}; alternatives/items A = {a1…am}

• Vote vi is a ranking of A

• Complete profile v = (v1, …, vn)

Alice Bob1

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voting rule r

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Cindy

Score-based Rules

• Many rules have score-based interpretation– Surrogate for “total group satisfaction”– E.g. Borda, Bucklin, maximin, Copeland, etc…

• Associates a score for each item given full rankings s(a, v)

• Winner has highest score6

s( , v) = 7

s( , v) = 6

s( , v) = 5

Alice Bob1

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Cindy Borda scores

Partial Preferences

• Partial vote pi is a partial order of A– Represented as a (consistent) set of pairwise comparisons– Higher order: top-k, bottom-k, …– Easy for humans to specify

• Partial profile p

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Alice

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> >

How to make decision with partial preferences?

Decision with Partial Preferences

• Possible and necessary co-winners [Konczak, Lang’05]

• Recently: minimax regret (MMR) [Lu, Boutilier’11]

– Provides worst-case guarantee on score loss w.r.t. true winner

– Small MMR means good enough decision– Zero MMR means decision is optimal

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Minimax Regret

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Adversarial

Bestresponse

Vote Elicitation

• MMR: good choices with “right” partial votes– How to minimize amount of partial preference

queries to make good decision?

• MMR-based incremental elicitation [Lu, Boutilier’11]

– Problem: must wait for response before next query

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Incremental Elicitation Woes

• Each query is a (voter, pairwise comparison) pair– Exploits MMR, depends on all previous responses

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ElicitorYES NO

…> ? > ?…

Bob annoyed at having to come back to answer query“interruption cost”

Our Solution:Multi-Round Batching

• Send queries to many voters in each round

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Elicitor Round: 1Give your top 2

1. 1. 1.

Round: 2Give your next top 1

3. 3. 3.

MMR ≤ εRecommendation:

Interruption cost reduced

2. 2. 2.

Multi-Round Probabilistic Vote Elicitation

• Query class: “rank top-5”, “is A > B?”, etc…– Single request of preferences from voter– Have different cognitive costs

• In each round π selects a subset of voters, and corresponding queries– Can be conditioned on previous round responses

• Function ω, selects winner and stops elicitation• How to design elicitation protocol with provably good

performance?– Worst-case not useful (for common rules)– Use probabilistic preference models to guide design

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Multi-Round Probabilistic Vote Elicitation

• Distribution P over vote profiles– Induced distribution over runs of protocol (π, ω)

• Can define distribution over performance metrics

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Quality of winner: Max regret, expected regret

Amount of information elicited: equivalent #pairwise comparisons, or bits.

Number of rounds of elicitation

Tradeoffs!

Depends on what costs are

important.

One-Round Protocol

• Query type: top-k– “Rank your top-k most preferred”

• Simple top-k heuristics [Kalech et al’11]

– Necessary and possible co-winners– No theoretical guarantees on winner quality– Don’t provide guidance on good k– No tradeoff between winner quality and k

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Probably Approximately Correct (PAC) One-Round Protocol

• Any rank-based voting rule• Any distribution P over profiles

• What is a good k?– p[k] are partial votes after eliciting top-k

k*: smallest k, with prob. ≥ 1 - δ, MMR(p[k]) ≤ ε

• As long as we can sample from P, we can find “approximately” good k…– Samples can come from historical datasets, surveying, or generated

from learned distribution

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Probably Approximately Correct One-Round Protocol

General Methodology• Input: sample of vote profiles: v1, …, vt

• MMR accuracy ε > 0• MMR confidence δ > 0• Sampling accuracy ξ > 0• Sampling confidence η > 0

Find best the smallest k with

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Probably Approximately Correct One-Round Protocol

Theorem: if sample size

then for any P, with probability 1 - η, we have

(a) ≤ k* (b) P[ MMR(p[ ]) ≤ ε ] ≥ 1 - δ - 2ξ

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Practical Considerations

• Sample size from theorem typically unnecessarily large

• Empirical methodology can be used heuristically

• Can generate histograms of MMR for profile samples from runs of elicitation– Can “eyeball” a good k– Can “eyeball” tradeoffs with MMR

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Experimental Results

• First experiments with Mallows distribution– Rankings generated i.i.d.– Unimodal, with dispersion parameter– t = 100 profiles (for guarantees, use bounds for t)

• Borda voting• Simulate runs of elicitation– Measure max regret and true regret– Normalize regret by number of voters

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Experimental Results

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x-axis is MMR per voter

Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

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Sushi 10 alternatives50 profiles, each with 100 rankings

Experimental Results

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Dublin North12 alternatives73 profiles, each with 50 rankings

Concluding Summary

• Model of multi-round elicitation protocol– Highlights tradeoffs between quality of winner,

amount of information elicited, and #rounds– Probabilistic preference profiles to guide design and

performance instead of worst-case• One-round, top-k elicitation– Simple, efficient empirical methodology for choosing k– PAC guarantees and sample complexity– With MMR solution concept, enables probabilistic and

anytime guarantees previous works cannot achieve

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Future Work

• Multi-round elicitation, top-k or pairwise comparisons

• Fully explore above tradeoffs (associative different costs)

• Assess expected regret and max regret

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The End

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