Group Manipulation in Judgment Aggregation Sirin Botan, Arianna - - PowerPoint PPT Presentation
Group Manipulation in Judgment Aggregation Sirin Botan, Arianna Novaro and Ulle Endriss ILLC - Universiteit van Amsterdam November 20, 2015 Motivating Example Judgment Aggregation: Combine agents opinions about some issues into a
ILLC - Universiteit van Amsterdam November 20, 2015
Judgment Aggregation: Combine agents’ opinions about some issues into a collective decision on them.
p q p ∧ ∧ ∧ q Agent 1 ✓ ✓ ✓ Agent 2 ✓ × × Agent 3 × ✓ × PB-Rule ✓ ✓ ✓
We will talk about: ⇒ Different type of Rules ⇒ More general type of Preferences
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N = {1, . . . , n} is the set of agents. Φ is the agenda (finite non-empty set of propositional formulas and their negations). Ji ⊆ Φ is the individual judgment set for agent i. J = (J1, . . . , Jn) is the profile on agenda Φ. J (Φ) is the set of all complete & consistent subsets of Φ. An aggregation rule for an agenda Φ and a set of n agents is a function from profiles to (collective) judgment sets: F : J (Φ)n → 2Φ.
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A uniform quota rule is defined by q ∈ {0, 1, . . . , n + 1}: Fq(J) = {ϕ ∈ Φ | #{i ∈ N | ϕ ∈ Ji} ≥ q}.
r s t ¬r ¬s ¬t J1 × ✓ ✓ ✓ × × J2 ✓ × ✓ × ✓ × J3 ✓ ✓ × × × ✓ J4 × × × ✓ ✓ ✓ J5 × × × ✓ ✓ ✓ F3(J) × × × ✓ ✓ ✓
In this example, F3 is the Majority Rule.
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The Hamming Distance is defined as H(J, J′) := |J \ J′| + |J′ \ J|. The Hamming Preferences of agent i are such that J ⪰i J′ ⇔ H(J, Ji) ≤ H(J′, Ji). We will assume Hamming Preferences for our theorems.
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Agent i manipulates whenever she does not report her truthful judgment set Ji. Agent i has an incentive to manipulate if for some J′
i ∈ J (Φ):
F(J−i, J′
i) ≻i F(J).
A rule F is single-agent strategy-proof, if for no truthful profile J there is an agent with an incentive to manipulate.
Dietrich & List. Strategy-Proof Judgment Aggregation. Economics & Philosophy, 2007.
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A coalition C of agents is a subset of N. J′ is a C-variant of J if Ji = J′
i for all agents i not in C.
F is group strategy-proof against coalitions of size ≤ k, if for all truthful profiles J, for all coalitions C of size ≤ k, and for all C-variants J′ of J we have F(J) ⪰i F(J′) for all agents i ∈ C.
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against coalitions of manipulators of at most 2 agents.
1 agent Follows from previous theorem. ✓ 2 agents Formulas on which the agents agree: already both rejecting or both accepting them. ⇒ Changes useless or counterproductive. Formulas on which the agents disagree: if agent 1 changes her opinion on some ϕs, she goes against her interest to possibly help agent 2 (by changing the outcome). ⇒ Agent 1 needs ”in return” strictly more formulas from agent 2 (Hamming Distance preferences). ⇒ The reasoning is symmetric for both agents. ✓
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Theorem. If the (atomic) agenda Φ includes at least 3 (non-negated) formulas, then every Uniform Quota Rule Fq such that 3 ≤ q ≤ n (or 1 ≤ q ≤ n − 2) is not group strategy-proof against coalitions of size ≤ 3.
a general method for constructing a profile manipulable by three agents. By checking the Hamming Distances we see that they have an incentive to manipulate together.
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Consider the truthful profile J . . .
ϕ1 ϕ2 ϕ3 … ¬ϕ1 ¬ϕ2 ¬ϕ3 … J1 × ✓ ✓ … ✓ × × … J2 ✓ × ✓ … × ✓ × … J3 ✓ ✓ × … × × ✓ … J4 ✓ ✓ ✓ … × × × … . . . . . . . . . . . . … . . . . . . . . . … Jq ✓ ✓ ✓ … × × × … Jq+1 × × × … ✓ ✓ ✓ … . . . . . . . . . . . . … . . . . . . . . . … Jn × × × … ✓ ✓ ✓ … Fq(J) × × × … ? ? ? …
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. . . and the manipulated profile J′.
ϕ1 ϕ2 ϕ3 … ¬ϕ1 ¬ϕ2 ¬ϕ3 … J′
1
✓ ✓ ✓ … × × × … J′
2
✓ ✓ ✓ … × × × … J′
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✓ ✓ ✓ … × × × … J4 ✓ ✓ ✓ … × × × … . . . . . . . . . . . . … . . . . . . . . . … Jq ✓ ✓ ✓ … × × × … Jq+1 × × × … ✓ ✓ ✓ … . . . . . . . . . . . . … . . . . . . . . . … Jn × × × … ✓ ✓ ✓ … Fq(J′) ✓ ✓ ✓ … ? ? ? …
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⇒ What happens if agents in our construction are allowed to opt out of the jointly agreed plan? ⇒ What happens if agents are risk-averse (to the possibility
ϕ1 ϕ2 ϕ3 ¬ϕ1 ¬ϕ2 ¬ϕ3 J1 × ✓ ✓ ✓ × × J2 ✓ ✓ ✓ × × × J3 ✓ ✓ ✓ × × × J4 × × × ✓ ✓ ✓ J5 × × × ✓ ✓ ✓ F3(J) × ✓ ✓ ✓ × ×
then Uniform Quota Rules are group strategy-proof.
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We introduced the notion of group manipulation in JA. For Uniform Quota Rules we get the following results: ✓ Strategy-proof against single agent (D. & L., 2007). ✓ Strategy-proof against two manipulators. × Manipulable by three (or more) agents. ✓ Strategy-proof against unstable groups. Similar results for more general rules (Independent and Monotonic).
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