Decision Framing in Judgment Aggregation Fabrizio Cariani, Marc Pauly, Josh Snyder Philosophy Departments: University of California Berkeley, and Stanford University APA Pacific Meeting, April 7 th 2007
The Capital (aka Discursive Dilemma) A three people committee is faced with this question: in the reunited Germany should the German parliament and the seat of government move to Berlin or stay in Bonn ? Dilemma Parliament Moves? Government Moves? Both? 1 yes no no 2 no yes no 3 yes yes yes Majority Majority Voting does not guarantee logically consistent outcomes. Question : Which judgment aggregation procedures yield logically consistent group judgments?
The Capital (aka Discursive Dilemma) A three people committee is faced with this question: in the reunited Germany should the German parliament and the seat of government move to Berlin or stay in Bonn ? Dilemma Parliament Moves? Government Moves? Both? 1 yes no no 2 no yes no 3 yes yes yes Majority Majority Voting does not guarantee logically consistent outcomes. Question : Which judgment aggregation procedures yield logically consistent group judgments?
The Capital: Procedure 1 Conclusion-Based Procedure Parliament? Government? Both? 1 yes no no 2 no yes no 3 yes yes yes Majority no
The Capital: Procedure 2 Premise-Based Procedure Parliament? Government? Both? 1 yes no no 2 no yes no 3 yes yes yes Majority yes yes Outcome yes
An alternative decision frame Designate as premises: ‘P’ (says that the Parliament should move to Berlin), and‘SC’ (says that Parliament and Government should be in the same city). Premise-Based Procedure 2 P? SC? Both Move ( P ∧ SC )? 1 yes no no 2 no no no 3 yes yes yes Majority yes no Outcome no
Our Central Question Which judgment aggregation procedures are invariant under switches of decision frame? Modeling assumption: switches of decision frame as changes of modeling language (languages are interpreted!). The underlying assumption is that within each language the atomic sentences express the proposition that are designated as premises in the associated frame. Language-Frames for the German Capital Language 1 P G P ≡ G P ∧ G [Language 2] [ P ] [ P ≡ SC ] [ SC ] [ P ∧ SC ] 1 yes no no no 2 no yes no no 3 yes yes yes yes Majority
The Road Ahead Different formal languages are related by means of translations. In this context, the question of which procedures are invariant under switches of decision frame becomes the question of which procedures are invariant with respect to translation. After defining this concept, we present two impossibility-like result for translation-invariant aggregation rules. (i.e. that there is no aggregation rule satisfying translation invariance together with certain other additional conditions). Finally, we investigate ways of finding some logical space for translation invariance by weakening one of the additional conditions.
The Road Ahead Different formal languages are related by means of translations. In this context, the question of which procedures are invariant under switches of decision frame becomes the question of which procedures are invariant with respect to translation. After defining this concept, we present two impossibility-like result for translation-invariant aggregation rules. (i.e. that there is no aggregation rule satisfying translation invariance together with certain other additional conditions). Finally, we investigate ways of finding some logical space for translation invariance by weakening one of the additional conditions.
The Road Ahead Different formal languages are related by means of translations. In this context, the question of which procedures are invariant under switches of decision frame becomes the question of which procedures are invariant with respect to translation. After defining this concept, we present two impossibility-like result for translation-invariant aggregation rules. (i.e. that there is no aggregation rule satisfying translation invariance together with certain other additional conditions). Finally, we investigate ways of finding some logical space for translation invariance by weakening one of the additional conditions.
The Road Ahead Different formal languages are related by means of translations. In this context, the question of which procedures are invariant under switches of decision frame becomes the question of which procedures are invariant with respect to translation. After defining this concept, we present two impossibility-like result for translation-invariant aggregation rules. (i.e. that there is no aggregation rule satisfying translation invariance together with certain other additional conditions). Finally, we investigate ways of finding some logical space for translation invariance by weakening one of the additional conditions.
Basic Notation n individual agents (index set: { 1 , ..., n } also referred to as N ). language L of propositional logic over finitely many atomic sentences ( L 0 ), with classical logic and semantics. given L , V L is the set of classical semantic valuations for L . judgment set: a subset of L . Definition A judgment aggregation procedure A is a (possibly partial) map taking n individual judgment sets (the inputs ) to a collective judgment set A ( X 1 , . . . , X n ) = Y , where X 1 , . . . , X n , Y ⊆ L
Decisive Judgment Aggregation Procedures If A is a procedure, � X a profile, and X a member of � X , then we require X to be: consistent (there is at least one valuation satisfying X ) complete (there is at most one valuation satisfying X ) deductively closed ( X is closed under logical consequence) We make a similar assumption about judgment sets in the output of A . We call aggregation functions defined on all and only such inputs and yielding only such outputs decisive . We mostly represent them by the functions: A : ( V L ) n �→ V L that they induce.
Properties of Judgment Aggregation Procedures Definition (Anonymity) An aggregation procedure A is anonymous iff for any permutation f of the set N of agents and any profile v 1 , ..., v n , A ( v 1 , ..., v n ) = A ( v f (1) , ..., v f ( n ) ) Definition (Dictatorship) An aggregation procedure A is a dictatorship iff there is some i ∈ N such that for all v 1 , ..., v n we have A ( v 1 , ..., v n ) = v i
Expressive equivalence. Remarks on our translation-invariance requirement. We do not require translation-invariance across all possible 1 languages. We restrict ourselves to languages that are expressively equivalent : these are languages that can express exactly the same possible world propositions. What does this restriction imply? If we required translation 2 invariance across more languages (say all possible languages) the notion of translation invariance would be more restrictive (and so fewer procedures would count as invariant). We could require invariance across fewer languages. Then 3 translation-invariance would be more permissive. [Maybe in future research].
Semantic Translation We think that it makes sense to talk about at least two concepts of translations between formal languages. (i) sensitive to logical structure ( syntactic translations) (ii) insensitive to logical structure ( semantic translations). In the semantic sense, two sentences of a language are translations of each other just in case they express the same possible world proposition. Definition (Semantic Translation) A semantic translation is any map τ : V L 1 �→ V L 2 that acts as a bijection on the set of valuations for L 1 and L 2 .
Translation Invariance The relations between syntactic and semantic translations are intricate and very interesting (and discussed in our paper!). However, we are focused on decisive judgment aggregation, and there we don’t need to look at logical structure. Definition An aggregation function A is translation invariant iff for all translations τ and profiles � v , τ ( A ( � v )) = A ( τ ( v 1 ) , ..., τ ( v n )) . Question Which aggregation functions are translation invariant ?
Translation Invariance The relations between syntactic and semantic translations are intricate and very interesting (and discussed in our paper!). However, we are focused on decisive judgment aggregation, and there we don’t need to look at logical structure. Definition An aggregation function A is translation invariant iff for all translations τ and profiles � v , τ ( A ( � v )) = A ( τ ( v 1 ) , ..., τ ( v n )) . Question Which aggregation functions are translation invariant ?
Main Result In our paper, we characterize the class of decisive translation invariant judgment aggregation function. The main consequence of that result is the following: Theorem If | L 0 | ≥ log 2 ( n + 2) , every decisive judgment aggregation function must either: (i) violate anonymity or (ii) violate translation invariance. We should be careful in interpreting this result.
Significance 1 We do not mean to use this result to make this argument: First Argument (P1) Translation invariance is an unconditional desideratum. (P2) Anonymity is an unconditional desideratum. (C) Therefore decisive judgment aggregation functions are normatively defective (unconditionally). Note, that even (P2) may be questionable.
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