Decision Framing in Judgment Aggregation Fabrizio Cariani, Marc Pauly, Josh Snyder Philosophy Departments: University of California Berkeley, and Stanford University APA Pacific Meeting, April 2007
The Great and the Good Instability Bovens & Rabinowicz pointed out that the premise-based procedure is not stable under re-identification of the premises. Applying PBP to ‘( A ≡ B ) ∧ ( C ≡ D )’ may yield different results if we identify our premises as the atoms, or as the two biconditionals. The general problem of sensitivity to the different ways of framing an issue has been studied under the rubric of agenda manipulability .
The Great and the Good Instability Bovens & Rabinowicz pointed out that the premise-based procedure is not stable under re-identification of the premises. Applying PBP to ‘( A ≡ B ) ∧ ( C ≡ D )’ may yield different results if we identify our premises as the atoms, or as the two biconditionals. The general problem of sensitivity to the different ways of framing an issue has been studied under the rubric of agenda manipulability .
The Great and the Good Instability Bovens & Rabinowicz pointed out that the premise-based procedure is not stable under re-identification of the premises. Applying PBP to ‘( A ≡ B ) ∧ ( C ≡ D )’ may yield different results if we identify our premises as the atoms, or as the two biconditionals. The general problem of sensitivity to the different ways of framing an issue has been studied under the rubric of agenda manipulability .
Agenda Manipulability as Language Variance Standard presentation: a single aggregation problem can be framed in two different ways by applying restricted proposition-wise majority to different agendas. In the example, frame 1 has agenda { A, B, C, D, negations } , while frame 2 has agenda { A ≡ B , C ≡ D , negations } Given a formal language, we assume that there is only one agenda for it–the one with atomic statements and negations. We represent switches of frames as switches of the underlying language. In the example we might have: ‘( A ≡ B ) ∧ ( C ≡ D )’ and ‘ α ∧ β ’.
Agenda Manipulability as Language Variance Standard presentation: a single aggregation problem can be framed in two different ways by applying restricted proposition-wise majority to different agendas. In the example, frame 1 has agenda { A, B, C, D, negations } , while frame 2 has agenda { A ≡ B , C ≡ D , negations } Given a formal language, we assume that there is only one agenda for it–the one with atomic statements and negations. We represent switches of frames as switches of the underlying language. In the example we might have: ‘( A ≡ B ) ∧ ( C ≡ D )’ and ‘ α ∧ β ’.
Agenda Manipulability as Language Variance Standard presentation: a single aggregation problem can be framed in two different ways by applying restricted proposition-wise majority to different agendas. In the example, frame 1 has agenda { A, B, C, D, negations } , while frame 2 has agenda { A ≡ B , C ≡ D , negations } Given a formal language, we assume that there is only one agenda for it–the one with atomic statements and negations. We represent switches of frames as switches of the underlying language. In the example we might have: ‘( A ≡ B ) ∧ ( C ≡ D )’ and ‘ α ∧ β ’.
We relate different frames by connecting their formal languages by means of translations . A procedure is manipulable by the ‘agenda setter’, in the sense we are interested in, if, given two different languages, applying the procedure in one language or in the other yields results that are not translations of each other. We will call such procedures translation variant . We only look at translation invariance across expressively equivalent languages . The A − B − C − D and the α − β languages are not expressively equivalent, hence not instances of the kind of agenda manipulability that is interesting to us. Besides expressive equivalence we require an equal number of atomic statements in both language.
We relate different frames by connecting their formal languages by means of translations . A procedure is manipulable by the ‘agenda setter’, in the sense we are interested in, if, given two different languages, applying the procedure in one language or in the other yields results that are not translations of each other. We will call such procedures translation variant . We only look at translation invariance across expressively equivalent languages . The A − B − C − D and the α − β languages are not expressively equivalent, hence not instances of the kind of agenda manipulability that is interesting to us. Besides expressive equivalence we require an equal number of atomic statements in both language.
We relate different frames by connecting their formal languages by means of translations . A procedure is manipulable by the ‘agenda setter’, in the sense we are interested in, if, given two different languages, applying the procedure in one language or in the other yields results that are not translations of each other. We will call such procedures translation variant . We only look at translation invariance across expressively equivalent languages . The A − B − C − D and the α − β languages are not expressively equivalent, hence not instances of the kind of agenda manipulability that is interesting to us. Besides expressive equivalence we require an equal number of atomic statements in both language.
Our Aims In this talk we will: Develop a rigorous theory of translations and use it to 1 formally define translation invariance. We will present a characterization theorem for translation 2 invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results. We will finally investigate ways of finding some logical space 3 for translation invariance by weakening the full rationality requirements.
Our Aims In this talk we will: Develop a rigorous theory of translations and use it to 1 formally define translation invariance. We will present a characterization theorem for translation 2 invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results. We will finally investigate ways of finding some logical space 3 for translation invariance by weakening the full rationality requirements.
Our Aims In this talk we will: Develop a rigorous theory of translations and use it to 1 formally define translation invariance. We will present a characterization theorem for translation 2 invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results. We will finally investigate ways of finding some logical space 3 for translation invariance by weakening the full rationality requirements.
Basic Notation n individual voters (label by means of the index set: N ) propositional atoms: L 0 language L of propositional logic over finitely many atoms L 0 , with classical logic and semantics. judgment set: a subset of L judgment aggregation procedure A maps n individual judgment sets to a collective judgment set A ( X 1 , . . . , X n ) = Y , where X 1 , . . . , X n , Y ⊆ L
Decisive Judgment Aggregation Procedures For most of this talk, we assume that both the input judgment sets X i as well as the output judgment set Y are consistent: there is at least one valuation satisfying X i and Y complete: there is at most one valuation satisfying X i and Y deductively closed: X i and Y are closed under logical consequence We call aggregation functions defined on all and only such inputs and yielding only such outputs decisive . We will mostly represent them by the functions: A : ( V L ) n �→ V L that they induce.
Decisive Judgment Aggregation Procedures For most of this talk, we assume that both the input judgment sets X i as well as the output judgment set Y are consistent: there is at least one valuation satisfying X i and Y complete: there is at most one valuation satisfying X i and Y deductively closed: X i and Y are closed under logical consequence We call aggregation functions defined on all and only such inputs and yielding only such outputs decisive . We will mostly represent them by the functions: A : ( V L ) n �→ V L that they induce.
Syntactic Translation Definition A syntactic translation is any map τ : L 1 �→ L 2 that: (1) preserves the logical operations [i.e. τ ( α ∧ β ) = τ ( α ) ∧ τ ( β ) , τ ( ∼ α ) = ∼ ( τ ( α ))]. (2) preserves deducibility [i.e. α ⊢ β ⇔ τ ( α ) ⊢ τ ( β )] Translations are fully specified by their behavior on the atomic statements.
An Example Suppose L 1 is generated by the atoms { A, B } , L 2 is generated by { C, D } . A �→ C Let f be a function mapping: B �→ C ≡ D It is obvious that f can be uniquely extended to a τ satisfying (1) [component-wise definition]. It is less obvious, but true, that the unique extension of f that satisfies (1) also satisfies (2).
An Example Suppose L 1 is generated by the atoms { A, B } , L 2 is generated by { C, D } . A �→ C Let f be a function mapping: B �→ C ≡ D It is obvious that f can be uniquely extended to a τ satisfying (1) [component-wise definition]. It is less obvious, but true, that the unique extension of f that satisfies (1) also satisfies (2).
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