Analyzing Arrow's Theorem Through Dependence and ......Analyzing Arrow’s Theorem Through...

Analyzing Arrow’s Theorem Through Dependenceand Independence Logic

Fan YangUniversity of Helsinki

UH-CAS Workshop on Mathematical LogicHelsinki, Oct. 30 - Nov. 2, 2018

———————————————–Partly based on joint work with Eric Pacuit (Maryland)

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Outline

1 Arrow’s Impossiblity Theorem

2 Formalizing Arrow’s Theorem in dependence and independencelogic

3 Arrow’s Theorem as a dependency strengthening theorem

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Social Choice Theory

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Preference Aggregation

A set of alternatives: A = {a,b, c,d , . . . }A finite set of voters: {v1, . . . , vn}A ranking R ⊆ A× A is a transitive and complete relation on A.A linear ranking is a ranking that is a linear relation.Denote by L(A) the set of all linear rankings of A.

groupyear voter 1 voter 2 · · · voter ndecision

2000 abc cab · · · acb ?2001 bac cba · · · cba ?2002 cba bca · · · acb ?2003 cab cba · · · acb ?

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Condorcet Paradox (18th century)

Problem with the Majority Rule:

groupvoter 1 voter 2 voter 3decision

a c bb a c ?c b a

Does the group prefer a over b? Yes.Does the group prefer b over c? Yes.Does the group prefer a over c? No.

a > b > c > a

5/27

Condorcet Paradox (with arbitrary rankings)

Problem with the Majority Rule:

groupvoter 1 voter 2 voter 3 voter 4 voter 5decision

a a c b bb∼c b a c c ?

c b a a

Does the group prefer a strictly over b? Yes.Does the group prefer b strictly over c? Yes.Does the group prefer a strictly over c? No.

a > b > c > a

May’s Theorem (1952): When |A| ≤ 2, the majority rule is the only“fair" aggregation rule.

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Desiderata

1 The voters’ votes should completely determine the group decision.

2 The voters’ votes are not constrained in any way.

3 The group decision should depend in the right way on the voters’ votes.

Dictatorship

Theorem (Arrow 1963)If |A| ≥ 3, then any preference aggregation rule F : D → L(A)(D ⊆ L(A)n) satisfying Universal Domain, Independence of IrrelevantAlternatives and Unanimity is a Dictatorship.

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Arrow’s Impossibility Theorem


Nobel Prize in Economics, 1972Kenneth Arrow

Nobel Prize in Economics, 1998

Amartya Kumar Sen

“Liberal paradox”

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groupvoter 1 voter 2 · · · voter ndecision

abc cab · · · acb abcbac cba · · · cba cbacba bca · · · abc bcacab cba · · · acb cab

v1 v2 · · · vn uabc cab · · · acb abccab bca · · · cba cbacba cab · · · abc bcacab bca . . . acb cab

a profile R

= F (R)dom(F ) = L(A)n

v1 v2 v3 uabc bac bca cbacab bca bac bcaabc bac bca bac

v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab

v1 · · · vi · · · vn uabc · · · acb · · · acb acbbca · · · abc · · · bac abccba · · · cab · · · bca cab

Goal of the talk:

Formalize & analyze Arrow’s Theorem in

Dependence and Independence Logic

Previous formalizations :in propositional logic: [Tang, Lin 2009]in first-order logic: [Grandi, Endriss 2013]in modal logic: [Ågotnes, van der Hoek, Wooldridge 2011], [Cinà, Endriss 2015]in higher order logic: [Nipkow 2009], [Wiedijk 2007]

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a profile R

= F (R)

dom(F ) = L(A)n




Goal of the talk:




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a profile R

= F (R)dom(F ) = L(A)n




Goal of the talk:




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Goal of the talk:




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Formalizing Arrow’s Theorem in dependence andindependence logic

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Logics for expressing dependencies

Henkin Quantifiers (1961):

∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ

Independence-Friendly Logic (Hintikka and Sandu, 1989):

∀u∃v∀x∃y/{u}φ

Dependence Logic (Väänänen 2007):

∀u∃v∀x∃y( =(x ; y) ∧ φ)

∃fIndependence Logic (Grädel, Väänänen 2013):

∀u∃v∀x∃y(u ⊥ y ∧ φ)×

Inclusion Logic (Galliani 2012):∀u∃x(u ⊆ x ∧ φ)

Thm. (Enderton, Walkoe, Hintikka)FO+Henkin quantifiers ≡ IF-logic≡ Σ11Thm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME

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Henkin Quantifiers (1961): ∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)


∀u∃v∀x∃y(u ⊥ y ∧ φ)×



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∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)


∀u∃v∀x∃y(u ⊥ y ∧ φ)×



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∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)


∀u∃v∀x∃y(u ⊥ y ∧ φ)×


Thm. (Enderton, Walkoe, Hintikka)FO+Henkin quantifiers ≡ IF-logic≡ Σ11

Thm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME

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∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)

∃f

Independence Logic (Grädel, Väänänen 2013):

∀u∃v∀x∃y(u ⊥ y ∧ φ)×


Thm. (Enderton, Walkoe, Hintikka)FO+Henkin quantifiers ≡ IF-logic≡ Σ11


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∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)


∀u∃v∀x∃y(u ⊥ y ∧ φ)×


Thm. (Enderton, Walkoe, Hintikka, Väänänen, Grädel)FO+Henkin quantifiers ≡ IF-logic ≡ (In)dependence logic≡ Σ11 ≡finite NP


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∀u ∃v ∀x ∃y φ

(∀u ∃v∀x ∃y

)φ




∀u∃v∀x∃y( =(x ; y) ∧ φ)


∀u∃v∀x∃y(u ⊥ y ∧ φ)×


Thm. (Enderton, Walkoe, Hintikka, Väänänen, Grädel)FO+Henkin quantifiers ≡ IF-logic ≡ (In)dependence logic≡ Σ11 ≡finite NPThm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME

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Functional dependency expressed in team semantics

v1 v2 v3 u

s1 bac cab abc abcs2 bac cab abc abcs3 cab bca acb cbas4 cab bca acb cba

Team semantics (Hodges 1997)

a set T of assignmentssa team:

M |=s2 =(~v ; u)?

iff for any s, s′ ∈ T ,

s(~t) = s′(~t) =⇒ s(~t ′) = s′(~t ′)

The functionality of aggregation rule:

θF := =(v1, . . . , vn; u)

Define ΓDM = “dom(M) = L(A).”

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v1 v2 v3 u




M |=T =(~v ; u) iff for any s, s′ ∈ T ,s(~v) = s′(~v) =⇒ s(u) = s′(u).

s(~t) = s′(~t) =⇒ s(~t ′) = s′(~t ′)

The functionality of aggregation rule:θF := =(v1, . . . , vn; u)


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v1 v2 v3 u




M |=T =(~t ; ~t ′) iff for any s, s′ ∈ T ,

s(~t) = s′(~t) =⇒ s(~t ′) = s′(~t ′)

The functionality of aggregation rule:

θF := =(v1, . . . , vn; u)


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IIA: The group decision on the relative preference between two alternatives a,b depends only on how the individual voters rank these two alternatives. It isindependent of their rankings with respect to other alternatives.

v1 v2 v3 us1 abc bac bca cbas2 cab bca bac bcas3 abc bac bca bac

− θIIA :=∧{ =(Pab(v1), . . . ,Pab(vn); Pab(u)

)| a,b ∈ A}

The signature LA consists of unary function symbols Pab for each(a,b) ∈ A× A.

for all >∈ L(A), PMab(>) = 1 iff a > b; PMab(>) = 0 iff a 6> b.

Define ΓOrd :={∀x((Pab(x)=1 ∧ Pbc(x) = 1)→ Pac(x)=1

)| a, b, c ∈ A} ∪ . . .

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Independence of irrelevant alternatives (IIA)



− θIIA :=∧{

=(Pab(v1), . . . ,Pab(vn); Pab(u)

)

| a,b ∈ A}




)| a, b, c ∈ A} ∪ . . .

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Independence of irrelevant alternatives (IIA)



− θIIA :=∧{ =(Pab(v1), . . . ,Pab(vn); Pab(u)

)| a,b ∈ A}




)| a, b, c ∈ A} ∪ . . .

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Independence

Universal domain: dom(F ) = L(A)n

v1 v2 v3 us1 abc bac cab bcas2 acb acb bac cbas3 acb bac cab acbs4 abc acb bac abc...

ss′

s′′

• M |=T iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′() = s() and s′′() = s′().

Independence: θI :=∧{vi ⊥ 〈vj〉j 6=i | 1 ≤ i ≤ n}

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Independence



ss′

s′′

• M |=T v1 ⊥ v2v3 iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(v1) = s(v1) and s′′(v2v3) = s′(v2v3).


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Independence



ss′

s′′

• M |=T ~t ⊥ ~t ′ iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(~t) = s(~t) and s′′(~t ′) = s′(~t ′).


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Inclusion

Universal domain: dom(F ) = L(A)n.

x v1 v2 v3 u

s1 abc acb cab bac bcas2 acb bac bac abc cbas3 bac abc cab acb acbs4 bca cba acb abc abcs5 cab bca acb abc abcs6 cba bac acb bac bac

...

M |=T x ⊆ vi iff for all s ∈ T , there exists s′ ∈ T such thats(x) = s′(vi).

All ranking: θAR :=

∧{

∀x(x ⊆ vi)

: 1 ≤ i ≤ n}

(Recall: dom(M) = L(A).)

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Inclusion


x v1 v2 v3 u


...

M |=T x ⊆ vi iff for all s ∈ T , there exists s′ ∈ T such thats(x) = s′(vi).

All ranking: θAR :=∧{∀x(x ⊆ vi) : 1 ≤ i ≤ n}


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Inclusion


x v1 v2 v3 u


...

M |=T ~t ⊆ ~t ′ iff for all s ∈ T , there exists s′ ∈ T such thats(~t) = s′(~t ′).

All ranking: θAR :=∧{∀x(x ⊆ vi) : 1 ≤ i ≤ n}


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Independence Logic & first-order formulas

Independence Logic = first-order logic + =(̄t ; t̄ ′) + t̄ ⊥ t̄ ′ + t̄ ⊆ t̄ ′

Def. For every formula α of first-order logic,M |=T α ⇐⇒ ∀s ∈ T : M |=s α.


Unanimity:

θUN :=∧

a,b∈A

(

(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1

)

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Independence Logic & first-order formulas

Independence Logic = first-order logic + =(̄t ; t̄ ′) + t̄ ⊥ t̄ ′ + t̄ ⊆ t̄ ′

Def. For every formula α of first-order logic,M |=T α ⇐⇒ ∀s ∈ T : M |=s α.


Unanimity:

θUN :=∧

a,b∈A

((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1

)

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Decisive sets and dictatorship


v1 v2 v3 ucba cab acb abcbca abc bac abccab acb abc cab

v1 · · · vi · · · vn uabc · · · acb · · · acb acbbca · · · bac · · · bac baccba · · · cba · · · bca cba

Unanimity:

θUN :=∧

a,b∈A

((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1

)The set {vi1 , . . . , vik} is decisive:

δ(vi1 , . . . , vik ) :=∧

a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)θUN = δ(v1, . . . , vn)

Dictatorship: θD := δ(vi)

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Unanimity:

θUN :=∧

a,b∈A

((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1


δ(vi1 , . . . , vik ) :=∧

a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)θUN = δ(v1, . . . , vn)

Dictatorship: θD :=∨n

i=1 δ(vi)

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Unanimity:

θUN :=∧

a,b∈A

((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1


δ(vi1 , . . . , vik ) :=∧

a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)θUN = δ(v1, . . . , vn)

Dictatorship: θD :=

> ni=1δ(vi)

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Unanimity:

θUN :=∧

a,b∈A

((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1

)→ Pab(u) = 1


δ(vi1 , . . . , vik ) :=∧

a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)θUN = δ(v1, . . . , vn)


i=1 δ(vi)

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Formalization of Arrow’s Theorem in independence logic


Let ΓArrow = ΓOrd ∪ {θF , θI , θAR, θIIA, θUN}.

ΓArrow |= θDor ΓArrow ,∼ θD |= ⊥

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non-dictatorship unanimity

universal domainirrelevant alternativesindependence of



M. C. Escher, 1960

Local consistency vs. global inconsistency

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Local consistency vs. global inconsistency

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Local consistency vs. global inconsistency19/27

A formal proof of Arrow’s Theorem

Let ΓArrow = {ΓOrd , θF , θAR, θI , θIIA, θU}.

ΓArrow |= θDor ΓArrow ,∼ θD |= ⊥

=⇒ Γ ` θD (by Completeness Theorem, and by (Pacuit, Y. 2017))

Theorem ((Kontinen, Väänänen, 2012), (Hannula 2015), (Y. 2016))For any set Γ ∪ {θ} of formulas of independence logic with θ(essentially) first-order,

Γ ` θ ⇐⇒ Γ |= θ.

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Proof idea (based on [Arrow 51], [Blau 72], etc.)

Lemma. If {vi1 , . . . , vik} ∩ {vj1 , . . . , vjm} = ∅, then

ΓArrow , δ(vi1 , . . . , vik , vj1 , . . . , vjm ) ` δ(vi1 , . . . , vik ) ∨ δ(vj1 , . . . , vjm )

v1 . . . ~vi . . . ~vj . . . vn u

δ(vi1 , . . . , vik ) =(vi1 , . . . , vik ; u)

=∧a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)

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Proof idea (based on [Arrow 51], [Blau 72], etc.)



Therefore,

ΓArrow , δ(v1, . . . , vn) ` δ(v1) ∨ δ(v2, . . . , vn)` δ(v1) ∨ δ(v2) ∨ δ(v3, . . . , vn)...` δ(v1) ∨ δ(v2) ∨ · · · ∨ δ(vn).

δ(vi1 , . . . , vik ) =(vi1 , . . . , vik ; u)

=∧a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)

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Proof idea



Therefore,

ΓArrow , δ(v1, . . . , vn) ` δ(v1) ∨ δ(v2, . . . , vn)` δ(v1) ∨ δ(v2) ∨ δ(v3, . . . , vn)...` δ(v1) ∨ δ(v2) ∨ · · · ∨ δ(vn).

δ(vi1 , . . . , vik ) =(vi1 , . . . , vik ; u)

=∧a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)21/27

Proof idea


ΓArrow , =(vi1 , . . . , vik , vj1 , . . . , vjm ; u) ` =(vi1 , . . . , vik ; u)∨=(vj1 , . . . , vjm ; u)

Therefore,

ΓArrow , =(v1, . . . , vn; u) `=(v1; u)∨ =(v2, . . . , vn; u)`=(v1; u)∨ =(v2; u)∨ =(v3, . . . , vn; u)...` =(v1; u) ∨=(v2; u) ∨ · · · ∨=(vn; u).

δ(vi1 , . . . , vik ) =(vi1 , . . . , vik ; u)

=∧a,b∈A

((Pab(vi1 ) = 1 ∧ · · · ∧ Pab(vik ) = 1

)→ Pab(u) = 1

)21/27

Arrow’s Theorem as a dependency strengtheningtheorem (work in progress)

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A “dependency strengthening" theorem

=(v1, . . . , vn; u) i.e., u = F (v1, . . . , vn)

=(v1; u) ∨ · · · ∨ =(vn; u) i.e., u = G(vi )

Γ

i.e., vi is a dictatorδ(v1) ∨ · · · ∨ δ(vn) i.e., u = vi(u = v1) ∨ · · · ∨ (u = vn)

v1 . . . . . . vn ucabcbabcacba

Work in progress: Find in general Γ and α(x) s.t.

Γ, =(α(v1), . . . , α(vn);α(u)) `n∨

i=1

(α(vi )↔ α(u)).

Arrow’s Thm and its generalizations (e.g., Kalai-Muller-SatterthwaiteThm) are special cases.

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=(v1, . . . , vn; u) i.e., u = F (v1, . . . , vn)

=(v1; u) ∨ · · · ∨ =(vn; u) i.e., u = G(vi )

Γ

i.e., vi is a dictatorδ(v1) ∨ · · · ∨ δ(vn) i.e., u = vi(u = v1) ∨ · · · ∨ (u = vn)

v1 . . . vi . . . vn ucab cabcba cbabca bcacba cba


Γ, =(α(v1), . . . , α(vn);α(u)) `n∨

i=1

(α(vi )↔ α(u)).


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=(v1, . . . , vn; u) i.e., u = F (v1, . . . , vn)

=(v1; u) ∨ · · · ∨ =(vn; u) i.e., u = G(vi )

Γ

i.e., vi is a dictatorδ(v1) ∨ · · · ∨ δ(vn)

i.e., u = vi(u = v1) ∨ · · · ∨ (u = vn)



Γ, =(α(v1), . . . , α(vn);α(u)) `n∨

i=1

(α(vi )↔ α(u)).


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=(v1, . . . , vn; u) i.e., u = F (v1, . . . , vn)

=(v1; u) ∨ · · · ∨ =(vn; u) i.e., u = G(vi )

Γ

i.e., vi is a dictatorδ(v1) ∨ · · · ∨ δ(vn)

i.e., u = vi(u = v1) ∨ · · · ∨ (u = vn)



Γ, =(α(v1), . . . , α(vn);α(u)) `n∨

i=1

(α(vi )↔ α(u)).

Arrow’s Thm and its generalizations (e.g., Kalai-Muller-SatterthwaiteThm) are special cases. 23/27

Analyzing Arrow’s Theorem

v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb

v1Pab(v1) Pbc(v1) Pac(v1)

1 0 1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb

x1y1z1 x2y2z2 x3y3z3 xyz101 011 100 010111 101 010 111011 011 100 101


Judgement aggregation: (Dietrich, List 2007), etc.

xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)= xiyizi

Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)

IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)

Dictatorship: θD :=n∨

i=1

xyz = xiyizi

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x1y1z1 x2y2z2 x3y3z3 xyz101 011 100 010111 101 010 111011 011 100 101



xyz = F (x1y1z1, . . . , xnynzn)

= f (x̄)g(ȳ)h(z̄)= xiyizi




i=1

xyz = xiyizi

24/27





x1y1z1 x2y2z2 x3y3z3 xyz101 011 100 010111 101 010 111011 011 100 101



xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)

= xiyizi




i=1

xyz = xiyizi

24/27





x1y1z1 x2y2z2 x3y3z3 xyz101 011 100 010111 101 010 111011 011 100 101







i=1

xyz = xiyizi

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i=1 xyz = xiyizi

Unanimity: θUN := (x̄ = 1̄→ x = 1) ∧ (x̄ = 0̄→ x = 0)∧(ȳ = 1̄→ y = 1) ∧ (ȳ = 0̄→ y = 0)

∧(z̄ = 1̄→ z = 1) ∧ (z̄ = 0̄→ z = 0)

Fixed values

Collective Rationality: θCR :=(x = 1 ∧ y = 1→ z = 1) ∧ (x = 0 ∧ y = 0→ z = 0)

Universal Domain: θUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = [1}

∧∧{xiyizi ⊥ 〈xjyjzj | j 6= i〉 | 1 ≤ i ≤ n}

Excluded values and interactions

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Dependency strengthening theorems

v1 v2 v3 uabc bac cab bcaacb acb bac abcacb bac cab acb





i=1 xyz = id(xi)id(yi)id(zi)

Unanimity: θUN := (x̄ = 1̄→ x = 1) ∧ (x̄ = 0̄→ x = 0)∧(ȳ = 1̄→ y = 1) ∧ (ȳ = 0̄→ y = 0)

∧(z̄ = 1̄→ z = 1) ∧ (z̄ = 0̄→ z = 0)

Fixed values





Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ

gD.

26/27








Unanimity: θUN := (x̄ = 1̄→ x = id(1)) ∧ (x̄ = 0̄→ x = id(0))∧(ȳ = 1̄→ y = 1) ∧ (ȳ = 0̄→ y = 0)

∧(z̄ = 1̄→ z = 1) ∧ (z̄ = 0̄→ z = 0)

Fixed values






gD.

26/27






Dictatorship: θidD :=∨n


Unanimity: θidUN := (x̄ = 1̄→ x = id(1)) ∧ (x̄ = 0̄→ x = id(0))∧(ȳ = 1̄→ y = id(1)) ∧ (ȳ = 0̄→ y = id(0))

∧(z̄ = 1̄→ z = id(1)) ∧ (z̄ = 0̄→ z = id(0))

Fixed values

Collective Rationality: θidCR :=(x = 1 ∧ y = 1→ z = id(1)) ∧ (x = 0 ∧ y = 0→ z = id(0))

Universal Domain:∧∧{xiyizi ⊥ 〈xjyjzj | j 6= i〉 | 1 ≤ i ≤ n}

θidUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = id([1)}



gD.

26/27






Dictatorship: θexD :=∨n

i=1 xyz = ex(xi)ex(yi)ex(zi)ex(1) = 0ex(0) = 1


∧(z̄ = 1̄→ z = id(1)) ∧ (z̄ = 0̄→ z = id(0))

Fixed values



θidUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = id([1)}



gD.

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Dictatorship: θexD :=∨n

i=1 xyz = ex(xi)ex(yi)ex(zi)ex(1) = 0ex(0) = 1


∧(z̄ = 1̄→ z = id(1)) ∧ (z̄ = 0̄→ z = id(0))

Fixed values



θidUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = id([1)}



gD.

C.f. [Tang, Lin 2009]

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Dictatorship: θid,id,exD :=∨n

i=1 xyz = id(xi)id(yi)ex(zi)ex(1) = 0ex(0) = 1

Unanimity: θid,id,exUN :=(x̄ = 1̄→ x = id(1)) ∧ (x̄ = 0̄→ x = id(0))∧(ȳ = 1̄→ y = id(1)) ∧ (ȳ = 0̄→ y = id(0))

∧(z̄ = 1̄→ z = ex(1)) ∧ (z̄ = 0̄→ z = ex(0))

Fixed values



θidUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = id([1)}



gD.

C.f. [Tang, Lin 2009]

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Dictatorship: θid,id,exD :=∨n

i=1 xyz = id(xi)id(yi)ex(zi)ex(1) = 0ex(0) = 1

Unanimity: θid,id,exUN :=(x̄ = 1̄→ x = id(1)) ∧ (x̄ = 0̄→ x = id(0))∧(ȳ = 1̄→ y = id(1)) ∧ (ȳ = 0̄→ y = id(0))

∧(z̄ = 1̄→ z = ex(1)) ∧ (z̄ = 0̄→ z = ex(0))

Fixed values



θidUD :=∧{[1[2[3 ⊆ xiyizi | [1, [2, [3 ∈ {0, 1}, [1 = [2 → [3 = id([1)}



gD.

Thm. θF , θIIA, θidUD, θexCR , θid,id,exUN ` θ

id,id,exD ; θF , θIIA, θ

exUD, θ

idCR , θ

ex,ex,idUN ` θ

ex,ex,idD .

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Thank you!

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Arrow's Impossiblity TheoremFormalizing Arrow's Theorem in dependence and independence logicArrow's Theorem as a dependency strengthening theorem

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