Decision Framing in Judgment Aggregation Fabrizio Cariani, Marc - - PowerPoint PPT Presentation

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:

1

Develop a rigorous theory of translations and use it to formally define translation invariance.

2

We will present a characterization theorem for translation invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results.

3

We will finally investigate ways of finding some logical space for translation invariance by weakening the full rationality requirements.

Our Aims

In this talk we will:

1

Develop a rigorous theory of translations and use it to formally define translation invariance.

2

We will present a characterization theorem for translation invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results.

3

We will finally investigate ways of finding some logical space for translation invariance by weakening the full rationality requirements.

Our Aims

In this talk we will:

1

Develop a rigorous theory of translations and use it to formally define translation invariance.

2

We will present a characterization theorem for translation invariant aggregation rules. Given assumptions of full rationality, this theorem will imply some impossibility results.

3

We will finally investigate ways of finding some logical space for translation invariance by weakening the full rationality requirements.

Basic Notation

n individual voters (label by means of the index set: N) propositional atoms: L0 language L of propositional logic over finitely many atoms L0, 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(X1, . . . , Xn) = Y , where X1, . . . , Xn, Y ⊆ L

Decisive Judgment Aggregation Procedures

For most of this talk, we assume that both the input judgment sets Xi as well as the output judgment set Y are consistent: there is at least one valuation satisfying Xi and Y complete: there is at most one valuation satisfying Xi and Y deductively closed: Xi 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 : (VL)n → VL that they induce.

Decisive Judgment Aggregation Procedures

For most of this talk, we assume that both the input judgment sets Xi as well as the output judgment set Y are consistent: there is at least one valuation satisfying Xi and Y complete: there is at most one valuation satisfying Xi and Y deductively closed: Xi 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 : (VL)n → VL that they induce.

Syntactic Translation

Definition A syntactic translation is any map τ : L1 → L2 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 L1 is generated by the atoms {A, B}, L2 is generated by {C, D}. Let f be a function mapping: A → C 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 L1 is generated by the atoms {A, B}, L2 is generated by {C, D}. Let f be a function mapping: A → C 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).

Semantic Translation

Definition (Semantic Translation) A semantic translation is any map ˆ τ : VL1 → VL2 that acts as a bijection on the set of valuations for L1 and L2. Say that τ and ˆ τ correspond if ˆ τ(VL1(α)) = VL2[τ(α)]. Lemma (Representation) (a) Given any syntactic translation there is a unique corresponding semantic translation. (b) Given any semantic translation there is a unique corresponding syntactic translation (up to logical equivalence).

Semantic Translation

Definition (Semantic Translation) A semantic translation is any map ˆ τ : VL1 → VL2 that acts as a bijection on the set of valuations for L1 and L2. Say that τ and ˆ τ correspond if ˆ τ(VL1(α)) = VL2[τ(α)]. Lemma (Representation) (a) Given any syntactic translation there is a unique corresponding semantic translation. (b) Given any semantic translation there is a unique corresponding syntactic translation (up to logical equivalence).

The Example Again

Consider again the syntactic translation generated by A → C B → C ≡ D This induces the following corresponding semantic translation: A, B → C, D 1, 1 → 1, 1 1, 0 → 1, 0 0, 1 → 0, 0 0, 0 → 0, 1

Translation Invariance

Given the Representation Lemma, we will feel authorized to talk of translations exclusively semantically Definition An aggregation function A is translation invariant iff for all translations τ and profiles v, τ(A( v)) = A(τ(v1), ..., τ(vn)). Question Which aggregation functions are translation invariant ?

Translation Invariance

Given the Representation Lemma, we will feel authorized to talk of translations exclusively semantically Definition An aggregation function A is translation invariant iff for all translations τ and profiles v, τ(A( v)) = A(τ(v1), ..., τ(vn)). Question Which aggregation functions are translation invariant ?

Sectarian Functions

Definition Given a profile v = v1, ..., vn, let P

v be the partition of N where i

and j are in the same partition element iff vi = vj. For example, if v1 = 1, 0, v2 = 1, 1, v3 = 1, 1, P

v = {{1}, {2, 3}}

Definition Let P be the set of all partitions of N. An aggregation function A is sectarian iff there is a function O : P → N such that for all

• v = v1, . . . , vn, we have A(v1, . . . , vn) = vO(P

v).

So say that A( v) = v1, then consider the profile w1 = 1, 1, w2 = 0, 0, w3 = 0, 0, if A is sectarian, then A( w) = w1. Sectarian functions have ‘local’ dictators

Sectarian Functions

Definition Given a profile v = v1, ..., vn, let P

v be the partition of N where i

and j are in the same partition element iff vi = vj. For example, if v1 = 1, 0, v2 = 1, 1, v3 = 1, 1, P

v = {{1}, {2, 3}}

Definition Let P be the set of all partitions of N. An aggregation function A is sectarian iff there is a function O : P → N such that for all

• v = v1, . . . , vn, we have A(v1, . . . , vn) = vO(P

v).

So say that A( v) = v1, then consider the profile w1 = 1, 1, w2 = 0, 0, w3 = 0, 0, if A is sectarian, then A( w) = w1. Sectarian functions have ‘local’ dictators

Sectarian Functions

Definition Given a profile v = v1, ..., vn, let P

v be the partition of N where i

and j are in the same partition element iff vi = vj. For example, if v1 = 1, 0, v2 = 1, 1, v3 = 1, 1, P

v = {{1}, {2, 3}}

Definition Let P be the set of all partitions of N. An aggregation function A is sectarian iff there is a function O : P → N such that for all

• v = v1, . . . , vn, we have A(v1, . . . , vn) = vO(P

v).

So say that A( v) = v1, then consider the profile w1 = 1, 1, w2 = 0, 0, w3 = 0, 0, if A is sectarian, then A( w) = w1. Sectarian functions have ‘local’ dictators

Characterization Result

Theorem If |L0| ≥ log2(n + 2), a judgment aggregation function is translation invariant iff it is sectarian. Corollary If |L0| ≥ log2(n + 2), there are no anonymous translation invariant judgment aggregation functions. Recall that we are talking about ‘decisive’ (i.e.‘satisfying full rationality’) aggregation functions.

Translation Invariance and Voter Manipulability

We use Dietrich and List’s definition of manipulability by voters: Definition An aggregation function A is a manipulable iff there is a voter i ∈ N, a proposition p, and profile v1, . . . , vn such that we have A(v1, . . . , vi, . . . , vn)(p) = vi(p) but A(v1, . . . , v ∗

i , . . . , vn)(p) = vi(p) for some alternative valuation v ∗ i .

We can then prove: Theorem Given |L0| log2(n + 2), translation invariant judgment aggregation functions are either dictatorships or are manipulable.

Life without Completeness

It is natural to ask whether we can find some logical space for translation invariant aggregation procedures:

(1) by weakening or relativizing the concept of translation invariance (2) by dropping some of the properties we assumed regarding aggregation functions (These were full rationality and anonymity)

We’ll explore (2). Natural implementation: weaken

• utcome-completeness to deductive closure.

Life without Completeness

It is natural to ask whether we can find some logical space for translation invariant aggregation procedures:

(1) by weakening or relativizing the concept of translation invariance (2) by dropping some of the properties we assumed regarding aggregation functions (These were full rationality and anonymity)

We’ll explore (2). Natural implementation: weaken

• utcome-completeness to deductive closure.

Life without Completeness

It is natural to ask whether we can find some logical space for translation invariant aggregation procedures:

(1) by weakening or relativizing the concept of translation invariance (2) by dropping some of the properties we assumed regarding aggregation functions (These were full rationality and anonymity)

We’ll explore (2). Natural implementation: weaken

• utcome-completeness to deductive closure.

Generalized Aggregation Functions

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!) Semantically, we end up with generalized aggregation functions: A : (2VL)n → 2VL We lift the notion of translation invariance to this level by saying that that A is translation invariant iff for all translations τ and all X1, ..., Xn ⊆ VL we have τ[A(X1, ..., Xn)] = A(τ[X1], ..., τ[Xn])

Generalized Aggregation Functions

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!) Semantically, we end up with generalized aggregation functions: A : (2VL)n → 2VL We lift the notion of translation invariance to this level by saying that that A is translation invariant iff for all translations τ and all X1, ..., Xn ⊆ VL we have τ[A(X1, ..., Xn)] = A(τ[X1], ..., τ[Xn])

Generalized Aggregation Functions

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!) Semantically, we end up with generalized aggregation functions: A : (2VL)n → 2VL We lift the notion of translation invariance to this level by saying that that A is translation invariant iff for all translations τ and all X1, ..., Xn ⊆ VL we have τ[A(X1, ..., Xn)] = A(τ[X1], ..., τ[Xn])

... some of which are Translation Invariant

Quite trivially, Union Rules are translation invariant. More interestingly, the function AAV (which we call Approval Voting) is translation invariant: AAV = {v ∈ VL|∀v ′ ∈ VL : |{i|v ∈ Xi}| |{i|v ′ ∈ Xi}|} To repeat: our interest is to prove the consistency of translation invariance at the generalized level. Procedures such as AAV might be objectionable on other grounds but they:

(i) supply the desired possibility result (ii) raise the general suspicion that the natural place for translation invariance is in a context without full rationality.

... some of which are Translation Invariant

Quite trivially, Union Rules are translation invariant. More interestingly, the function AAV (which we call Approval Voting) is translation invariant: AAV = {v ∈ VL|∀v ′ ∈ VL : |{i|v ∈ Xi}| |{i|v ′ ∈ Xi}|} To repeat: our interest is to prove the consistency of translation invariance at the generalized level. Procedures such as AAV might be objectionable on other grounds but they:

(i) supply the desired possibility result (ii) raise the general suspicion that the natural place for translation invariance is in a context without full rationality.

... some of which are Translation Invariant

Quite trivially, Union Rules are translation invariant. More interestingly, the function AAV (which we call Approval Voting) is translation invariant: AAV = {v ∈ VL|∀v ′ ∈ VL : |{i|v ∈ Xi}| |{i|v ′ ∈ Xi}|} To repeat: our interest is to prove the consistency of translation invariance at the generalized level. Procedures such as AAV might be objectionable on other grounds but they:

(i) supply the desired possibility result (ii) raise the general suspicion that the natural place for translation invariance is in a context without full rationality.

Checkout

1

We have offered a systematic framework to study translations formally: this might (we hope!) have applications even

• utside judgment aggregation

2

Translation invariance as a kind of agenda manipulability can easily be studied within our formal framework for translations. We think that informally translation invariance is a normatively desirable condition. We hope this intuition carries

• ver to our formalized concept.

3

In a framework with full rationality assumptions, translation invariance is an essentially impossible ideal.

4

Without full rationality, translation invariance, while still a strong constraint, is achievable.

5

Absent an alternative framework, this is a reason to pay close attention to frameworks without full rationality.

Checkout

1

We have offered a systematic framework to study translations formally: this might (we hope!) have applications even

• utside judgment aggregation

2

Translation invariance as a kind of agenda manipulability can easily be studied within our formal framework for translations. We think that informally translation invariance is a normatively desirable condition. We hope this intuition carries

• ver to our formalized concept.

3

In a framework with full rationality assumptions, translation invariance is an essentially impossible ideal.

4

Without full rationality, translation invariance, while still a strong constraint, is achievable.

5

Absent an alternative framework, this is a reason to pay close attention to frameworks without full rationality.

Checkout

1

We have offered a systematic framework to study translations formally: this might (we hope!) have applications even

• utside judgment aggregation

2

Translation invariance as a kind of agenda manipulability can easily be studied within our formal framework for translations. We think that informally translation invariance is a normatively desirable condition. We hope this intuition carries

• ver to our formalized concept.

3

In a framework with full rationality assumptions, translation invariance is an essentially impossible ideal.

4

Without full rationality, translation invariance, while still a strong constraint, is achievable.

5

Absent an alternative framework, this is a reason to pay close attention to frameworks without full rationality.

The End

The Bound

Most of our results require |L0| log2(n + 2) the bound is tight. the bound is really low (logarithmic in the number of voters) ultimately we believe we should be concerned with aggregation methods. Ways of aggregating judments that apply regardless of the number of voters. We can make talk of methods formal by defining them as functions M : N × N → A taking the number of voters and the number of atoms into an appropriate aggregation procedure. The standard properties of JA-procedures can be extended to methods, and a version of

• ur impossibility results proven for them.

A Note on the Proof on the result on Manipulability

We rely on a weakening of a result by Dietrich and List (2004): Theorem (Dietrich and List (2004)) If A is an aggregation functions that fails independence, then A is manipulable. Our piece of the proof is the following result: Theorem There are no independent, non-dictatorial, sectarian aggregation functions.

On the Example

Note that τ is not be onto (e.g. ‘D’ is not in its range), but this does not affect the fact that the languages are expressively equivalent. Technical Explanation τ induces an isomorphism on the boolean algebras generated by the atomic statements.