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Bundling Attacks in Judgment Aggregation

Bundling Attacks in Judgment Aggregation. Reshef Meir Joint work with Noga Alon , Dvir Falik and Moshe Tennenholtz. Example. A committee needs to decide on purchasing computing equipment for the school. There are three optional features:. Example.

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Bundling Attacks in Judgment Aggregation

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  1. Bundling Attacks in Judgment Aggregation Reshef Meir Joint work with NogaAlon, DvirFalik and Moshe Tennenholtz

  2. Example • A committee needs to decide on purchasing computing equipment for the school. There are three optional features:

  3. Example • A committee needs to decide on purchasing computing equipment for the school. There are three optional features:

  4. Example • By a majority decision, no feature is approved.

  5. Example • By a majority decision, no feature is approved. • Suppose the vendor offers all features in a single bundle:

  6. Example • By a majority decision, no feature is approved. • Suppose the vendor offers all features in a single bundle:

  7. Example • By a majority decision, no feature is approved. • Suppose the vendor offers all features in a single bundle: • Bundling is common in commercial and political settings

  8. Model • We consider a binary matrix A • m issues (columns) • n judges (rows) • The chair has some goal vector • W.l.o.g.: to approve all issues • Can partition the issues to bundles • Each judge approves or rejects each bundle P : C1 C2 C3

  9. Model • We consider a binary matrix A • m issues (columns) • n judges (rows) • The chair has some goal vector • W.l.o.g.: to approve all issues • Can partition the issues to bundles • Each judge approves or rejects each bundle P : C1 C2 C3

  10. Bundling attacks • We saw that sometimes the chair can revert the entire outcome • Even with a single bundle • Known as the “Ostrogorski paradox” • It seems that the chair has a lot of power • Even more power if we allow for arbitrary partitions

  11. NP-hard NP-hard The power of the chair • Does a bundling attack exist often? • According to what distribution? • Is it (computationally) easy to find a bundling attack? • Problem 1: find a perfect partition (approve all issues) • Reduction from IS-TRIPARTITE-GRAPH • Problem 2: find a good bundle (approve at least k issues) • Reduction from OPTIMAL-LOBBYING [Christian et al. ‘07] • Also follows from [Alon et al. ‘13]

  12. Frequency of bundling attacks • Consider a random preference matrix A • aij =1w.p. p, and otherwise 0 • Many issues and voters: m,n∞ • How often does a (perfect) partition exist? • How often can the chair approve at least k issues?

  13. Frequency of bundling attacks • Consider a random preference matrix A • aij =1w.p. p, and otherwise 0 • Many issues and voters: m,n∞ • How often does a (perfect) partition exist? • How often can the chair approve at least k issues? X V ?

  14. Frequency of bundling attacks • Consider a random preference matrix A • aij=1w.p. p=0.5 • Many issues: m∞, any number of voters n >1 Theorem: W.h.p, there is a perfect bundling attack Moreover, it can be found efficiently (thus the problem is easy in the average case) X V V

  15. Proof outline • Find many “copies” of the Ostrogorskiparadox • Put each copy in a bundle • Put all other columns in a single bundle C* • The density of C* is slightly more than 0.5 C1 C2 • All small bundles are approved • C* is approved w.h.p.

  16. Future directions • Adding constraints on allowed partitions • Only small bundles, etc. • Adding restrictions on allowed matrices • Interdependencies among issues [Conitzer, Lang, Xia ‘09] • Logical constraints [Endriss,Grandi, Porello ’10] • Gerrymandering Issues Voters

  17. Future directions • Adding constraints on allowed partitions • Only small bundles, etc. • Adding restrictions on allowed matrices • Interdependencies among issues [Conitzer, Lang, Xia ‘09] • Logical constraints [Endriss,Grandi, Porello ’10] • Gerrymandering Voters Issues “District”

  18. Thank you! Questions?

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