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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. - PowerPoint PPT Presentation
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Bundling Attacks in Judgment Aggregation
Reshef MeirJoint work with Noga Alon, Dvir Falik and Moshe Tennenholtz
• A committee needs to decide on purchasing computing equipment for the school. There are three optional features:
Example
WiFi GPU UPS
Member 1Member 2Member 3Member 4Member 5
Example
• A committee needs to decide on purchasing computing equipment for the school. There are three optional features:
WiFi GPU UPS
Member 1 X X XMember 2 X X XMember 3 V V XMember 4 V X VMember 5 X V V
Example
WiFi GPU UPS
Member 1 X X XMember 2 X X XMember 3 V V XMember 4 V X VMember 5 X V VDecision: X X X
• By a majority decision, no feature is approved.
Example
WiFi GPU UPS Vote:
Member 1 X X XMember 2 X X XMember 3 V V XMember 4 V X VMember 5 X V VDecision: X X X
• By a majority decision, no feature is approved.• Suppose the vendor offers all features in a single
bundle:
Example• By a majority decision, no feature is approved.• Suppose the vendor offers all features in a single
bundle:WiFi GPU UPS Vote:
Member 1 X X X XMember 2 X X X XMember 3 V V X VMember 4 V X V VMember 5 X V V VDecision: X X X
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
WiFi GPU UPS Vote:
Member 1 X X X XMember 2 X X X XMember 3 V V X VMember 4 V X V VMember 5 X V V VDecision: X X X V
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
A i1 i2 i3 .. .. im
j1 0 0 0 0 1 0j2 0 0 0 1 1 1j3 1 1 0 1 0 1j4 1 0 1 0 0 1j5 0 1 1 1 1 1
0 0 0
0 0 1
1 0 1
1 1 0
1 1 1
P : C1 C2 C3
•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
Model
A i1 i2 i3 .. .. im
j1 0 0 0 0 1 0j2 0 0 0 1 1 1j3 1 1 0 1 0 1j4 1 0 1 0 0 1j5 0 1 1 1 1 1
1 1 0 1 1 1
0 0 0
0 0 1
1 0 1
1 1 0
1 1 1
P : C1 C2 C3
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
A i1 i2 i3
j1 0 0 0
j2 0 0 0
j3 1 1 0
j4 1 0 1
j5 0 1 1
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]
NP-hard
NP-hard
Frequency of bundling attacks
• Consider a random preference matrix A–aij =1 w.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?
p<0.5 p=0.5 p>0.5
Frequency of bundling attacks
p<0.5 p=0.5 p>0.5Nothing works w.h.p.
A single bundle works w.h.pX V?
• Consider a random preference matrix A–aij =1 w.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?
Frequency of bundling attacks
• Consider a random preference matrix A–aij =1 w.p. p=0.5
–Many issues: m∞, any number of voters n >1
Theorem: W.h.p, there is a perfect bundling attackMoreover, it can be found efficiently(thus the problem is easy in the average case)
p<0.5 p=0.5 p>0.5Nothing works w.h.p.
A single bundle works w.h.pX VV
Proof outline• Find many “copies” of the
Ostrogorski paradox
• 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
A i1 i2 i3 .. .. .. .. .. .. .. .. .. im
j1 0 0 0 0 1 0 0 1 0 1 1 0 0
j2 0 0 0 1 1 0 1 0 1 1 1 0 1
j3 1 1 0 1 0 0 1 1 1 0 0 0 1
j4 1 0 1 0 0 1 0 0 0 0 0 1 1
j5 0 1 1 1 1 1 1 0 0 0 1 1 1
• All small bundles are approved• C* is approved w.h.p.
C1 C2
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 A i1 i2 i3 .. .. .. .. .. .. .. .. .. im
j1 0 0 0 0 1 0 0 1 0 1 1 0 0
j2 0 0 0 1 1 0 1 0 1 1 1 0 1
j3 1 1 0 1 0 0 1 1 1 0 0 0 1
j4 1 0 1 0 0 1 0 0 0 0 0 1 1
j5 0 1 1 1 1 1 1 0 0 0 1 1 1
Issues
Voters
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 A i1 i2 i3 .. .. .. .. .. .. .. .. .. im
j1 0 0 0 0 1 0 0 1 0 1 1 0 0
j2 0 0 0 1 1 0 1 0 1 1 1 0 1
j3 1 1 0 1 0 0 1 1 1 0 0 0 1
j4 1 0 1 0 0 1 0 0 0 0 0 1 1
j5 0 1 1 1 1 1 1 0 0 0 1 1 1
Voters
Issues
“District”
Thank you!Questions?
