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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
2
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
>1
2
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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 Bob
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2
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voting rule r
5
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 Bob
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2
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CindyBorda 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 13
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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