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Interference in Judgment Aggregation Dorothea Baumeister, Gbor Erdlyi, Olivia Erdlyi, and Jrg Rothe ILLC Workshop on Collective Decision Making, April 2013 Judgment Aggregation Penalty Area Foul Penalty Yes Yes Yes Yes Yes Yes

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Interference in Judgment Aggregation

ILLC Workshop on Collective Decision Making, April 2013 Dorothea Baumeister, Gábor Erdélyi, Olivia Erdélyi, and Jörg Rothe

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Dorothea Baumeister 2

Judgment Aggregation

Penalty Area Foul Penalty Yes Yes Yes Yes No No No Yes No Yes Yes No Yes Yes Yes Yes Yes Yes Yes No No No No No

Doctrinal Paradox / Discursive Dilemma

Majority

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SLIDE 3

Dorothea Baumeister 3

  • Formal Framework
  • Manipulation
  • Types of preferences
  • Strategyproofness
  • Bribery
  • Control by …
  • Adding Judges
  • Deleting Judges
  • Replacing Judges

Outline

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Dorothea Baumeister 4

Formal Framework

Judges Agenda Premises Conclusions Individual Judgment Sets Yes / No Collective Judgment Set Yes if quota is reached Penalty Area Foul Penalty Referee 1 Yes Yes Yes Referee 2 Yes No No Referee 3 No Yes No Quota ½ Yes Yes Yes Variants:

  • Uniform quota
  • Constant quota

Requirements:

  • Agenda is closed under propositional variables
  • Premises consists of all literals

Complete and consistent outcome Quota fraction for each premise We focus on:

  • PBP: Uniform premise-based quota rules for quota ½
  • Uniform constant premise-based quota rules
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Dorothea Baumeister 5

Forms of Interference

Manipulation: Provide untruthful information to obtain a better result. Bribery: Briber judges to obtain a better result. Control: Change the structure to obtain a better result. Widely studied in voting from a computational point of view!

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Dorothea Baumeister 6

Manipulation

Incentive: Provide untruthful information to obtain a better result.

  • Information = individual judgment set
  • Result = collective outcome
  • Better = ?

Different assumptions on the preferences:

  • Unrestricted
  • Top-respecting
  • Closeness-respecting
  • Hamming-distance induced
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Dorothea Baumeister 7

Preferences over collective JS

Preferences with respect to JS

1 0 0 1 1

  • Unrestriced (U): every preference is possible
  • Top-respecting (TR): >

1 0 0 1 1 ? ? ? ? ?

  • Closeness-respecting (CR): >

1 ? ? ? 1 1 1 1 0 1

  • Hamming-distance induced (HD):

>

0 0 0 0 1 1 1 1 0 1

The only complete relation is HD (by allowing equalities) A judgment aggregation procedure is strategyproof if a judge prefers the acutual outcome to all outcomes resulting from untruthful individual judgment sets of him.

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Dorothea Baumeister 8

Strategyproofness

Fix some induced preference >: A judgment aggregation procedure is necessarily/ possibly strategyproof if a judge necessarily/possible prefers the acutual outcome to all outcomes resulting from untruthful individual judgment sets of him. A judge necessarily prefers 𝑌 to 𝑍 if 𝑌 ≻ 𝑍 in every complete extension of >. A judge possibly prefers 𝑌 to 𝑍 if 𝑌 ≻ 𝑍 in some complete extension of >.

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Dorothea Baumeister 9

Manipulation

Question: Is it possible to obtain a „better outcome“ by reporting an inscincere judgment set?

A F A ∧ F Yes Yes Yes Yes No No No Yes No Yes Yes Yes

Manipulative judge

A F A ∧ F Yes Yes Yes Yes No No No No No Yes No No

HD, TR, CR-preferences regarding A ∧ F, Exact

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Dorothea Baumeister

Results for Manipulation

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Preferences Necessary Manipulation Possible Manipulation Unrestricted ? ? Top-respecting NP-complete ? Closeness-respecting NP-complete ? Hamming Distance NP-complete Exact NP-complete strategyproof Complete desired judgment set strategyproof in P in P Also holds for general quotas

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Dorothea Baumeister 11

Bribery (HD + Exact)

Question: Is it possible to obtain a „better outcome“ by bribing at most k judges?

  • Desired judgment set
  • Budget k

Exact Variant: Is it possible to reach the desired judgment set by bribing at most k judges?

Microbribery: Change up to k premise entries

A F A ∧ F Yes Yes Yes Yes No No No Yes No Yes Yes Yes A F A ∧ F Yes Yes Yes Yes No No No No No Yes No No

Bribe 1 judge

No

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Dorothea Baumeister

Results for Bribery

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Bribery Exact Bribery MicroBribery Exact MicroBribery # judges NP-comp. NP-comp. NP-comp. # of bribes NP-comp. W[2]-hard X X # of microbribes X X NP-comp. NP-comp. General problem NP-comp. NP-comp. NP-comp. NP-comp. Desired Judgment set:

  • complete
  • contains all premises
  • contains only premises

in P Reduction from Dominating Set Generalization of Optimal Lobbying

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Dorothea Baumeister 13

Control by Adding Judges

Question: Is it possible to obtain a „better outcome“ by adding at most k judges?

  • Desired judgment set
  • Set of potential new judges
  • Positive integer k

Exact Variant: Is it possible to reach the desired judgment set by adding at most k judges?

A F A ∧ F Yes Yes Yes Yes No No No Yes No Yes Yes Yes A F A ∧ F Yes Yes Yes Yes No No No Yes No No No No No No No No No No

Add 2 judges

No No No No

Non-constant number of judges: Difference between uniform and uniform constant premise- based quota rule

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Dorothea Baumeister 14

Control by Deleting Judges

Question: Is it possible to obtain a „better outcome“ by deleting at most k judges?

  • Desired judgment set
  • Positive integer k

Exact Variant: Is it possible to reach the desired judgment set by deleting at most k judges?

A F A ∧ F Yes Yes Yes Yes No No No Yes No Yes Yes Yes A F A ∧ F No Yes No No Yes No

Delete 2 judges

No

Non-constant number of judges: Difference between uniform and uniform constant premise based quota rule

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Dorothea Baumeister 15

Control by Replacing Judges

Question: Is it possible to obtain a „better outcome“ by replacing at most k judges?

  • Desired judgment set
  • Set of potential new judges
  • Positive integer k

Exact Variant: Is it possible to reach the desired judgment set by replacing at most k judges?

Constant number of judges: No difference between uniform and uniform constant premise-based quota rule

A F A ∧ F Yes Yes Yes Yes No No No Yes No Yes Yes Yes A F A ∧ F Yes Yes Yes Yes No No No No No Yes No No

Replace 1 judge

No No No No

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Dorothea Baumeister

Approach

Control is usually an undesired behavior

Immune Control is never possible Susceptible Not Immune Vulnerable Susceptible and polynomial-time solvable Resistant Susceptible but NP-hard

Computational hardness can be seen as a barrier against control

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Dorothea Baumeister 17

Results for Control

Uniform Constant Quota Uniform Quota = ½ Uniform Quota Adding Judges (HD) Resistant Resistant Adding Judges (Exact) Resistant Resistant Deleting Judges (HD) Resistant Resistant Deleting Judges (Exact) Resistant Resistant Replacing Judges (HD) Resistant Resistant Resistant Replacing Judges (Exact) Resistant Resistant Resistant Agenda contains only premises Reduction from Dominating Set Reduction from Exact Cover by 3-Sets

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Dorothea Baumeister 18

Concluding Remarks

  • Different Aggregation Procedures
  • New Control Problems
  • Typical-case analysis
  • Different types of induced preferences for Bribery

and Control

Thank you for your attention!