Incentives for Participation and Abstention in Probabilistic Social Choice

Johannes Hofbauer

AAMAS 2015, Istanbul, Turkey May 8, 2015

Felix Brandt Johannes HofbauerFlorian Brandl

Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Two Voting Paradoxes

• Downs paradox: Individual vote might have no influence on the outcome

‣ all reasonable non-probabilistic choice functions affected

‣ does probabilistic social choice offer a way out?

• No-show paradox: Not voting might yield a preferred outcome

‣ first observed by Fishburn and Brams (1983)

‣ immunity to no-show paradox called participation by Moulin (1988)

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Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Preliminaries

• Social decision scheme (SDS) ff : 𝓡𝓕(𝓝) → ∆(A)

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• An SDS is majoritarian if its outcome only depends on the majority relation

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Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Stochastic Dominance

• How can we extend preferences over alternatives to preferences over lotteries?

‣ p is SD-preferred over q if the expected utility for p is at least as large as that for q for every vNM utility function

‣ p ≳ q iff ∑x≿y p(x) ≥ ∑x≿y q(x) for all y∈A

• p is SD-dominated by q if all agents SD-prefer q over p

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Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Degrees of Efficiency

• Unanimity: unique common top choice has to be selected uniquely

• Ex post efficiency: Pareto-dominated alternatives receive probability zero

• SD-efficiency: no SD-dominated lotteries selected

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‣ ½🍏+½🍓 SD-dominates ¼🍏+¼🍌+¼🍇+¼🍓

Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Degrees of Participation

• Participation: voting is never worse than abstaining

‣ f(R-i) ≻i f(R) for no R ∈ 𝓡𝓕(𝓝)

• Strong participation: voting is always at least as good as abstaining

‣ f(R) ≿i f(R-i) for all R ∈ 𝓡𝓕(𝓝)

• Very strong participation: voting is always better than abstaining

‣ f(R) ≻i f(R-i) for all R ∈ 𝓡𝓕(𝓝) (whenever this is possible)

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Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

A Tradeoff Between Efficiency and Participation?

• Random serial dictatorship (RSD) satisfies very strong SD-participation and ex post efficiency.

• The uniform lottery over all Borda winners (BOR) satisfies strong SD-participation and SD-efficiency.

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ex post efficiency

SD-efficiency

very strong SD-part.

strong SD-part.

RSD

BOR

&

&

Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Majoritarian SDSs

• Three impossibility theorems:

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very strong SD-part.

strong SD-part.

SD-part.

no efficiency

unanimity

ex post efficiency

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* holds even when preferences are required to be linear

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Incentives for Participation and Abstention in Probabilistic Social Choice Johannes Hofbauer

Summary and Conclusions

• Very strong participation can be satisfied (ex ante) by probabilistic functions

• Further results concerning

‣ pairwise functions

‣ group-participation

• Interesting open problems

‣ Is there an SDS satisfying SD-efficiency and very strong SD-participation?

‣ Is there a Condorcet extension satisfying strong SD-participation?

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