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Robust Winners and Winner Determination Policies under Candidate UncertaintyJOEL OREN, UNIVERSITY OF TORONTO

JOINT WORK WITH CRAIG BOUTILIER, JÉRÔME LANG AND HÉCTOR PALACIOS.

Motivation – Winner Determination under Candidate Uncertainty

•A committee, with preferences over alternatives: • Prospective projects.• Goals.

•Costly determination of availabilities:•Market research for determining the

feasibility of a project: engineering estimates, surveys, focus groups, etc.

• “Best” alternative depends on available ones.

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Efficient Querying Policies for Winner Determination•Voters submit votes in advance.

•Query candidates sequentially, until enough is known in order to a determine the winner.

•Example: a wins.

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The Formal Model•A set C of candidates.

•A vector, , of rankings (a preference profile).

•Set is partitioned:1. – a priori known availability.2. – the “unknown” set.

•Each candidate is available with probability .

•Voting rule: is the election winner.ac

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Y(available)

U(unknown)

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Querying & Decision Making•At iteration submit query q(x), .

•Information set .

•Initial available set .

•Upon querying candidate :• If available: add to .• If unavailable: remove from .

• – restriction of pref. profile to the candidate set .

•Stop when is -sufficient – no additional querying will change – the “robust” winner.

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Computing a Robust Winner•Robust winner: Given , is a robust winner if .

•A related question in voting: [Destructive control by candidate addition] Candidate set , disjoint spoiler set , pref. profile over , candidate , voting rule .• Question: is there a subset , s.t. ?

•Proposition: Candidate is a robust winner there is no destructive control against , where the spoiler set is .

Y Yx 𝑟 (𝑣 (𝑌 ) )=𝑥𝑟 (𝑣 (𝑌∪𝐷 ) )=𝑦

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Computing a Robust Winner•Proposition: Candidate is a robust winner there is no destructive control against , where the spoiler set is .

•Implication: Pluarlity, Bucklin, ranked pairs – coNP-complete; Copeland, Maximin -polytime tractable.

•Additional results: Checking if is a robust winner for top cycle, uncovered set, and Borda can be done in polynomial time.• Top-cycle & Uncovered set: prove useful criteria for the corresponding majority

graph.

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The Query Policy•Goal: design a policy for finding correct winner.

•Can be represented by a decision tree.

•Example for the vote profile (plurality):• abcde, abcde, adbec, • bcaed, bcead,• cdeab, cbade, cdbea

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Winner Determination Policies as Trees

•r-Sufficient tree:• Information set at each leaf is -sufficient.• Each leaf is correctly labelled with the winner.

• -- cost of querying candidate/node .

• – expected cost of policy, over dist. of .

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\{𝑎 ,𝑏}∈ 𝐴

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•Cost of a tree: .

•For each node – a training set: Possible true underlying sets A, that agree with .• Example 1: • Example 2: .

•Can solve using a dynamic-programming approach.

•Running time: -- computationally heavy.

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Recursively Finding Optimal Decision Trees

𝑥 𝑦

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Myopically Constructing Decision Trees•Well-known approach of maximizing information gain at every node until reached pure training sets – leaves (C4.5).

•Mypoic step: query the candidate for the highest “information gain” (decrease in entropy of the training set).

•Running time:

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Empirical Results• 100 votes, availability probability .

•Dispersion parameter . ( uniform distribution).

•Tested for Plurality, Borda, Copeland.

•Preference distributions drawn i.i.d. from Mallows -distribution: probabilities decrease exponentially with distance from a “reference” ranking.

= 0.3 0.5 0.9MethodPlurality, DP 4.1 3.4 2.7Plurality, Myopic 4.1 3.5 2.8Borda, DP 3.7 2.7 1.7Borda, Myopic 3.7 2.7 1.7

Average cost (# of queries)

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Empirical Results•Cost decrease as increases – [ less uncertainty about the available candidates set].

•Myopic performed very close to the OPT DP alg.

•Not shown:

•Cost increases with the dispersion parameter – “noisier”/more diverse preferences (not shown).

• -Approximation: stop the recursion when training set is – pure.• For plurality, , , .• For , .

0.3 0.5 0.9MethodPlurality, DP 4.1 3.4 2.7Plurality, Myopic 4.1 3.5 2.8Borda, DP 3.7 2.7 1.7Borda, Myopic 3.7 2.7 1.7

Average cost (# of queries)

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Additional Results•Query complexity: expected number of queries under a worst-case preference profile.• Result: For Plurality, Borda, and Copeland, worst-case exp. query

complexity is .

•Simplified policies: Assume for all . Then there is a simple iterative query policy that is asymptotically optimal as .

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Conclusions & Future Directions•A framework for querying candidates under a probabilistic availability model.

•Connections to control of elections.

•Two algorithms for generating decision trees: DP, Myopic.

•Future directions:1. Ways of pruning the decision trees (depend on the voting rules).2. Sample-based methods for reducing training set size.3. Deeper theoretical study of the query complexity.

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