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 7th 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

• f 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 (L0), with classical logic and semantics. given L, VL 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(X1, . . . , Xn) = Y , where X1, . . . , Xn, 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

• f 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 : (VL)n → VL 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 v1, ..., vn, A(v1, ..., vn) = A(vf (1), ..., vf (n)) Definition (Dictatorship) An aggregation procedure A is a dictatorship iff there is some i ∈ N such that for all v1, ..., vn we have A(v1, ..., vn) = vi

Expressive equivalence.

Remarks on our translation-invariance requirement.

1

We do not require translation-invariance across all possible

• languages. We restrict ourselves to languages that are

expressively equivalent: these are languages that can express exactly the same possible world propositions.

2

What does this restriction imply? If we required translation 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).

3

We could require invariance across fewer languages. Then 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 : VL1 → VL2 that acts as a bijection on the set of valuations for L1 and L2.

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(v1), ..., τ(vn)). 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(v1), ..., τ(vn)). 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 |L0| ≥ log2(n + 2), every decisive judgment aggregation function must either: (i) violate anonymity

• r

(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.

Significance 2

Rather, all we need is that translation invariance and anonymity are sometimes jointly desirable. I.e. there are cases C such that, Second Argument (P1) Translation invariance (as we defined it) is a desideratum in C. (P2) Anonymity is a desideratum in C. (C) Therefore in cases of kind C, we must avoid decisive aggregation procedures. We think the German capital case is an example. We also are not committed to there being a unique satisfactory way of avoiding decisive judgment aggregation.

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), decisive translation invariant judgment aggregation functions are either dictatorships or are manipulable.

Weakening 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 decisiveness and anonymity)

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

• utcome-completeness to deductive closure.

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!)

Weakening 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 decisiveness and anonymity)

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

• utcome-completeness to deductive closure.

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!)

Weakening 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 decisiveness and anonymity)

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

• utcome-completeness to deductive closure.

We can also drop the completeness requirement at the level of the inputs. (Note: this does not facilitate our task!)

Generalized Aggregation Functions

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

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}|} Procedures such as AAV are objectionable on other grounds but they supply an interesting possibility result. A result by C. List and R. Goodin (2006), properly re-interpreted, guarantees that, among translation-invariant generalized aggregation functions , approval voting is unique in satisfying a few interesting, and relatively weak, properties.

... 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}|} Procedures such as AAV are objectionable on other grounds but they supply an interesting possibility result. A result by C. List and R. Goodin (2006), properly re-interpreted, guarantees that, among translation-invariant generalized aggregation functions , approval voting is unique in satisfying a few interesting, and relatively weak, properties.

... 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}|} Procedures such as AAV are objectionable on other grounds but they supply an interesting possibility result. A result by C. List and R. Goodin (2006), properly re-interpreted, guarantees that, among translation-invariant generalized aggregation functions , approval voting is unique in satisfying a few interesting, and relatively weak, properties.

... 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}|} Procedures such as AAV are objectionable on other grounds but they supply an interesting possibility result. A result by C. List and R. Goodin (2006), properly re-interpreted, guarantees that, among translation-invariant generalized aggregation functions , approval voting is unique in satisfying a few interesting, and relatively weak, properties.

Checkout

1

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

2

A requirement of translation invariance can be used, in JA, to model a requirement of robustness under switches of decision frame.

3

Our general attitude towards translation invariance in JA is that in some form of other it is often (though not always!)

• desirable. The particular notion we define is, at least

sometimes, desirable.

4

If we stick to decisive judgment aggregation, translation invariance is an essentially impossible ideal (exception: when there are reasons to forego anonymity).

5

Moving from decisive to generalized aggregation functions,

• ne can have both translation invariance and anonymity.

Checkout

1

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

2

A requirement of translation invariance can be used, in JA, to model a requirement of robustness under switches of decision frame.

3

Our general attitude towards translation invariance in JA is that in some form of other it is often (though not always!)

• desirable. The particular notion we define is, at least

sometimes, desirable.

4

If we stick to decisive judgment aggregation, translation invariance is an essentially impossible ideal (exception: when there are reasons to forego anonymity).

5

Moving from decisive to generalized aggregation functions,

• ne can have both translation invariance and anonymity.

Checkout

1

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

2

A requirement of translation invariance can be used, in JA, to model a requirement of robustness under switches of decision frame.

3

Our general attitude towards translation invariance in JA is that in some form of other it is often (though not always!)

• desirable. The particular notion we define is, at least

sometimes, desirable.

4

If we stick to decisive judgment aggregation, translation invariance is an essentially impossible ideal (exception: when there are reasons to forego anonymity).

5

Moving from decisive to generalized aggregation functions,

• ne can have both translation invariance and anonymity.

Checkout

1

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

2

A requirement of translation invariance can be used, in JA, to model a requirement of robustness under switches of decision frame.

3

Our general attitude towards translation invariance in JA is that in some form of other it is often (though not always!)

• desirable. The particular notion we define is, at least

sometimes, desirable.

4

If we stick to decisive judgment aggregation, translation invariance is an essentially impossible ideal (exception: when there are reasons to forego anonymity).

5

Moving from decisive to generalized aggregation functions,

• ne can have both translation invariance and anonymity.

Checkout

1

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

2

A requirement of translation invariance can be used, in JA, to model a requirement of robustness under switches of decision frame.

3

Our general attitude towards translation invariance in JA is that in some form of other it is often (though not always!)

• desirable. The particular notion we define is, at least

sometimes, desirable.

4

If we stick to decisive judgment aggregation, translation invariance is an essentially impossible ideal (exception: when there are reasons to forego anonymity).

5

Moving from decisive to generalized aggregation functions,

• ne can have both translation invariance and anonymity.

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.