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Vote Elicitation with Probabilistic Preference Models: Empirical Estimation and Cost Tradeoffs Tyler Lu and Craig Boutilier University of Toronto

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Introduction New communication platforms can transform the way people make group decisions. How can computational social choice realize this shift? Choices People Computational Social Choice Consensus 2

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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 practice Problem: Cognitive and communication burden Our approach (recent work): Elicit just the right preferences to make good enough group decisions This work: Multi-round elicitation and probabilistic preference models to further reduce burdens Alice BobCindy > 1 2 1 3

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Outline Preliminaries Multi-round Probabilistic Vote Elicitation Methodology and Analysis for One-round Experimental Results 4

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Preliminaries Voters N = {1..n}; alternatives/items A = {a 1 …a m } Vote v i is a ranking of A Complete profile v = (v 1, …, v n ) Alice Bob 1 2 3 voting rule r 5 Cindy

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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 score 6 s(, v) = 7 s(, v) = 6 s(, v) = 5 Alice Bob 1 2 3 Cindy Borda scores

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Partial Preferences Partial vote p i 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 7 Alice > >> How to make decision with partial preferences?

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Decision with Partial Preferences Possible and necessary co-winners [Konczak, Lang05] Recently: minimax regret (MMR) [Lu, Boutilier11] – Provides worst-case guarantee on score loss w.r.t. true winner – Small MMR means good enough decision – Zero MMR means decision is optimal 8

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Minimax Regret 9 Adversarial Best response

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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, Boutilier11] – Problem: must wait for response before next query 10

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Incremental Elicitation Woes Each query is a (voter, pairwise comparison) pair – Exploits MMR, depends on all previous responses 11 Elicitor YES NO … > ? > ? … Bob annoyed at having to come back to answer query interruption cost

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Our Solution: Multi-Round Batching Send queries to many voters in each round 12 Elicitor Round: 1 Give your top 2 1. Round: 2 Give your next top 1 3. MMR ε Recommendation: Interruption cost reduced 2.

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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

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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 14 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.

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One-Round Protocol Query type: top-k – Rank your top-k most preferred Simple top-k heuristics [Kalech et al11] – Necessary and possible co-winners – No theoretical guarantees on winner quality – Dont provide guidance on good k – No tradeoff between winner quality and k 15

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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 16

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Probably Approximately Correct One-Round Protocol General Methodology Input: sample of vote profiles: v 1, …, v t MMR accuracy ε > 0 MMR confidence δ > 0 Sampling accuracy ξ > 0 Sampling confidence η > 0 Find best the smallest k with 17

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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 18

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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 19

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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 20

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Experimental Results 21 x-axis is MMR per voter

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Experimental Results 22

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Experimental Results 23

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Experimental Results 24

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Experimental Results 25

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Experimental Results 26 Sushi 10 alternatives 50 profiles, each with 100 rankings

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Experimental Results 27 Dublin North 12 alternatives 73 profiles, each with 50 rankings

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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 28

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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 29

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The End 30

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