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Efficient Voting via the Top-k Elicitation

Scheme: A Probabilistic Approach

Joel Oren, University of Toronto

Joint work with Yuval Filmus, Institute of Advanced Study

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Motivation

• Common theme: communication-efficient group decision-making.• How to pick the single “best” item/candidate, without

extracting too much information from the customers/committee members.

• Hiring committees: members can’t rank all of the candidates.• Too many candidates.• Committee members may only be familiar with just

subsets of candidates in their own fields.

The Basic Setting

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• Candidates (alternatives/products): • Agents with preferences over the possible outcomes.

𝑐1≻𝑐 2≻⋯≻ 𝑐5≻𝑐3≻⋯≻ 𝑐7≻𝑐19≻…≻

1. A canonical task: select the “best” outcome.

2. What is the information need for doing so.

1. Social choice: methods for aggregating preferences.

2. Efficient preference elicitation.

Basic Definitions

• – candidates, – set of all orderings over .• Voter set with preferences .• Voting rule: .• Score-based voting rules: assign each candidate a score, pick the one

with maximal score (most voting rules).

• Positional scoring rules (PSR): score vector , ’th ranked candidate receives a score of . • Question: do we really need to know all of to pick

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𝑐1≻𝑐 2≻𝑐3⋯≻ 𝑐5≻𝑐3≻𝑐2⋯≻ 𝑐7≻𝑐19≻…≻𝑚 ,𝑚−1 ,𝑚−2… 𝑚 ,𝑚−1 ,𝑚−2… 𝑚 ,𝑚−1 ,𝑚−2…Borda1 ,12,13,… 1 ,

12,13,… 1 ,

12,13,…

Harmonic

𝑟 (𝝅 )=𝑎𝑟𝑔𝑚𝑎𝑥𝑐 ∑𝑖∈𝑁

𝑣 𝜋 𝑖 (𝑐 )

Geometric 𝜌 , 𝜌2 ,𝜌 3 ,… 𝜌 , 𝜌2 ,𝜌 3 ,… 𝜌 , 𝜌2 ,𝜌 3 ,…

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Top- Voting – The General Elicitation Scheme

•Given a prescribed voting rule , ask each voter to report only his the length- prefix of her preferences.•Voter reports only .

• The top- elicitation scheme: an algorithm that 1. Elicits 2. Computes a function that picks a winner `based

on and the voting rule .

• Question: how large does need to be so that ?

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Previous Work• Theoretically: many common voting rules require a lot of

information from the voters, in the worst case [Conitzer & Sandholm’02,05, Xia & Conitzer’08].• Top- voting: Complexity of finding possible/necessary winners

[Baumeister et al.’12].• Empirically: in practice, top- voting performs well [Kalech et

al.’11].

• Bridging the gap: assume a distributional model of preferences.• Lu & Boutilier’11: studied the performance of partial

preference elicitation for regret-minimization a objective, under distributional assumptions.• Recently: A lower-bound on , for selecting the Borda winner

in an impartial culture [O, Filmus, Boutilier’13].

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A Distributional Approach to Top- Voting • Input: • Scoring rule: . Focus: positional scoring rules.• Top- algorithm :given top- votes, selects a winning candidate. • Each preference : drawn i.i.d. from distribution over .• Output: .

Q: for which , , w.h.p.

𝑐1≻𝑐 2≻⋯≻ 𝑐5≻𝑐3≻⋯≻ 𝑐7≻𝑐19≻…≻

𝑫

𝑨𝒌𝒔𝒄 𝐴𝑘

𝑠𝑐 (𝝅𝒌)

𝑎𝑟𝑔𝑚𝑎𝑥 𝑠𝑐 (𝑐 ,𝝅)

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Three Distributional Models (and high-level Results)

• Three distributional models (increasing order of hardness):1. Biased Distributions: distributions that “favor” one particular

candidate over all else. (positionally-biased, pairwise-biased). • Overall results: As , is sufficient for many scoring rules. See paper

for details.2. Impartial Culture: is the uniform distribution over .• Main results:

a) PSRs: an almost tight threshold theorem.b) Additional scoring rule: Copeland.

3. Adversarial: is set by an adversary, but is fully known to us.• Main results:

a) Harmonic PSR: A worst-case distribution requiring .b) Geometric PSR: for any – sufficient; this is tight.

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Top- for PSRs: under an Impartial Culture

• Goal: Given a score vector , s.t. , determine a LB and an UB in the limit.• Possible algorithms the upper bound:1. OPT alg.: compute the winning prob. of candidates – not clear

how to rigorously analyze such an algorithm.2. Naïve: , , if , , otherwise.∀3. FairCutOff: like Naïve, but if , .

𝜋 𝑖−1 (1 )≻𝜋 𝑖

−2 (2 )≻⋯≻𝜋 𝑖−1 (𝑘 )≻𝜋 𝑖

−1 (𝑘+1 )≻⋯≻𝜋 𝑖−1 (𝑚 )

𝑣1 𝑣2 𝑣𝑘⋯ 0 0⋯𝝅 𝒊 𝐸 𝑗∈ {𝑘+1 ,… ,𝑚 }[𝑣 𝑗 ]𝝅 𝒊 ,𝒌

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Empirical Results for the three algorithms• , sample populations.

Borda Harmonic

FairCutoff overlaps with Opt

FairCutoff overlaps with Opt

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A Threshold Theorem for PSRs• The top- scores according to FairCutoff: for ,

• We consider a measure depending on the amount of “noise” in each of the two parts of the score vector, in terms of :• ,

• Set – the -partition variability ratio.

• Theorem: in the limit • LB: if , no top- can determine the right winner w.p. .• UB: if , then FairCutoff determines the right winner w.p. .

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Applications

Theorem: in the limit • LB: if , no top- can determine the right winner w.p. .• UB: if , then FairCutoff determines the right winner w.p. .

• Borda: assuming that , . Gives a LB of (improvement over the previous bound of ).•Harmonic: UB – is sufficient, LB – .•Geometric: () UB – -- sufficient, LB – .

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Proof Sketch – Order Statistics of Correlated RVs•Upper bound: FairCutoff guesses the right winner

w.h.p.

• For each candidate, partition voter scores based on top-/bottom-( partition:

𝜋 𝑖−1 (1 )≻𝜋 𝑖

−1 (2 )⋯≻𝜋 𝑖−1 (𝑘)

𝑣1 𝑣2 𝑣𝑘⋯𝝅 𝒊

𝐸 𝑗∈ {𝑘+1 ,… ,𝑚 }[𝑣 𝑗 ]

+

+

+

+

+

+

+

+

+

⋯

𝑐𝑚 ,𝑠𝑐 (𝑐𝑚 ) :

𝑠𝑐(1) (𝑐1 )+𝑠𝑐 (2 ) (𝑐1 )𝑠𝑐(1) (𝑐2 )+𝑠𝑐 (2 ) (𝑐2 ) 𝑠𝑐(1) (𝑐𝑚 )+𝑠𝑐 (2) (𝑐𝑚 )

Proof Sketch (continued)• Each candidate’s score is the sum: .• Score according to top- .• First and second order statistics: .

• First step: estimate the distribution of .• Complication: the scores are correlated – diagonalize covariance

matrix via a linear transformation, without affecting the distribution of the difference.• Approximate via CLT + order statistics analysis of Gaussian RVs.• Difference grows as variance of grows ().

• Second step: switch from to loses the “added noise” – has a bounded effect (use – above gap won’t be closed, w.h.p.

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Top- elicitation under Copleland’s rule• The scoring rule: for every two distinct candidates , set( if beats

[loses to] in a pairwise election. If there’s a tie, set . • A candidate’s score is: . • Theorem: for , no top- algorithm predicts the correct winner w.p. ().

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The Adversarial Model• The preference distribution is set by an adversary – the full

specifications are given to the decision maker.• How bad can the bounds be for that case?• Borda: already got a lower bound of .• Harmonic: we have a polylog UB for the impartial culture. • Theorem: For a fixed , there is a distribution , such that

any top- algorithm requires , under the harmonic scoring rule.

• Geometric: a UB under an impartial culture.• Theorem: if the decay factor is fixed, then is sufficient

under the geometric scoring rule, under any distribution, and this is tight.

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Conclusions & Future Directions• A rigorous, probabilistic analysis of the top- elicitation scheme.• Under impartial culture: a principled method of analysis with applications:

• Result for Copeland: extension to weighted majority graph based rules?• Can we extend our results to other rules?• More involved distributions: mixture models.• Top-k for proportional representation.

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Scoring rule LB UB

Borda [prev. ]

HarmonicGeometricCopeland ) (future:

Contact:oren@cs.toronto.eduHomepage: www.cs.toronto.edu/~oren

Thank you!

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