Analyzing Arrow's Theorem Through Dependence and ......Analyzing Arrow’s Theorem Through...
Transcript of Analyzing Arrow's Theorem Through Dependence and ......Analyzing Arrow’s Theorem Through...
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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
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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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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
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.
Nobel Prize in Economics, 1972Kenneth Arrow
Nobel Prize in Economics, 1998
Amartya Kumar Sen
“Liberal paradox”
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Arrow’s Impossibility Theorem
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.
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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Arrow’s Impossibility Theorem
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.
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]
9/27
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Arrow’s Impossibility Theorem
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.
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]
9/27
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Arrow’s Impossibility Theorem
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.
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]
9/27
-
Arrow’s Impossibility Theorem
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.
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]
9/27
-
Arrow’s Impossibility Theorem
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.
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]
9/27
-
Arrow’s Impossibility Theorem
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.
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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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
11/27
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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
11/27
-
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
11/27
-
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
11/27
-
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
11/27
-
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≡ Σ11
Thm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME
11/27
-
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)
∃f
Independence 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≡ Σ11
Thm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME
11/27
-
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, Väänänen, Grädel)FO+Henkin quantifiers ≡ IF-logic ≡ (In)dependence logic≡ Σ11 ≡finite NP
Thm. (Galliani and Hella 2013) Inclusion logic ≡ GFP+ ≡finite,< PTIME
11/27
-
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, 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).”
12/27
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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 |=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)
Define ΓDM = “dom(M) = L(A).”
12/27
-
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 |=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)
Define ΓDM = “dom(M) = L(A).”
12/27
-
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 |=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)
Define ΓDM = “dom(M) = L(A).”
12/27
-
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 |=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)
Define ΓDM = “dom(M) = L(A).”
12/27
-
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 |=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)
Define ΓDM = “dom(M) = L(A).”
12/27
-
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} ∪ . . .
13/27
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Independence of irrelevant alternatives (IIA)
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} ∪ . . .
13/27
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Independence of irrelevant alternatives (IIA)
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} ∪ . . .
13/27
-
Independence of irrelevant alternatives (IIA)
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} ∪ . . .
13/27
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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}
14/27
-
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 v1 ⊥ v2v3 iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(v1) = s(v1) and s′′(v2v3) = s′(v2v3).
Independence: θI :=∧{vi ⊥ 〈vj〉j 6=i | 1 ≤ i ≤ n}
14/27
-
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 v1 ⊥ v2v3 iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(v1) = s(v1) and s′′(v2v3) = s′(v2v3).
Independence: θI :=∧{vi ⊥ 〈vj〉j 6=i | 1 ≤ i ≤ n}
14/27
-
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 v1 ⊥ v2v3 iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(v1) = s(v1) and s′′(v2v3) = s′(v2v3).
Independence: θI :=∧{vi ⊥ 〈vj〉j 6=i | 1 ≤ i ≤ n}
14/27
-
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 ~t ⊥ ~t ′ iff for all s, s′ ∈ T , there exists s′′ ∈ T s.t.s′′(~t) = s(~t) and s′′(~t ′) = s′(~t ′).
Independence: θI :=∧{vi ⊥ 〈vj〉j 6=i | 1 ≤ i ≤ n}
14/27
-
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).)
15/27
-
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).)
15/27
-
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).)
15/27
-
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).)
15/27
-
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).)
15/27
-
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).)
15/27
-
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).)
15/27
-
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 ~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}
(Recall: dom(M) = L(A).)
15/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
(
(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
(
(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
(
(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
(
(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
(
(Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
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.
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
Unanimity:
θUN :=∧
a,b∈A
((Pab(v1) = 1 ∧ · · · ∧ Pab(vn) = 1
)→ Pab(u) = 1
)
16/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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)
17/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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)
17/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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)
17/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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 :=∨n
i=1 δ(vi)
17/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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 :=
> ni=1δ(vi)
17/27
-
Decisive sets and dictatorship
v1 v2 v3 uabc cab acb abcbca abc bac abccab acb abc cab
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 :=∨n
i=1 δ(vi)
17/27
-
Formalization of Arrow’s Theorem in independence logic
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.
Let ΓArrow = ΓOrd ∪ {θF , θI , θAR, θIIA, θUN}.
ΓArrow |= θDor ΓArrow ,∼ θD |= ⊥
18/27
-
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
M. C. Escher, 1960
Local consistency vs. global inconsistency
19/27
-
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
Local consistency vs. global inconsistency
19/27
-
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
Local consistency vs. global inconsistency
19/27
-
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
non-dictatorship unanimity
universal domainirrelevant alternativesindependence of
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.
20/27
-
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
)
21/27
-
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
)
21/27
-
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 )
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
Lemma. If {vi1 , . . . , vik} ∩ {vj1 , . . . , vjm} = ∅, then
ΓArrow , δ(vi1 , . . . , vik , vj1 , . . . , vjm ) ` δ(vi1 , . . . , vik ) ∨ δ(vj1 , . . . , vjm )
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
Lemma. If {vi1 , . . . , vik} ∩ {vj1 , . . . , vjm} = ∅, then
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)
22/27
-
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.
23/27
-
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 . . . vi . . . vn ucab cabcba cbabca bcacba cba
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.
23/27
-
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 . . . vi . . . vn ucab cabcba cbabca bcacba cba
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.
23/27
-
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 . . . vi . . . vn ucab cabcba cbabca bcacba cba
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.
23/27
-
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 . . . vi . . . vn ucab cabcba cbabca bcacba cba
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. 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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
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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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
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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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
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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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
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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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
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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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
-
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
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
24/27
-
Analyzing Arrow’s Theorem
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
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
25/27
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Analyzing Arrow’s Theorem
v1 v2 v3 uacb bac cab bcaabc acb bac abcbac bac cab acb
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
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
25/27
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Dependency strengthening theorems
v1 v2 v3 uabc bac cab bcaacb acb bac abcacb bac cab acb
xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
Dictatorship: θD :=∨n
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
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
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
26/27
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Dependency strengthening theorems
v1 v2 v3 uabc bac cab bcaacb acb bac abcacb bac cab acb
xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
Dictatorship: θD :=∨n
i=1 xyz = id(xi)id(yi)id(zi)
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
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
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
26/27
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Dependency strengthening theorems
v1 v2 v3 uabc bac cab bcaacb acb bac abcacb bac cab acb
xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
Dictatorship: θidD :=∨n
i=1 xyz = id(xi)id(yi)id(zi)
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)}
Excluded values and interactions
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
26/27
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Dependency strengthening theorems
v1 v2 v3 uabc bac cab bcaacb acb bac abcacb bac cab acb
xyz = F (x1y1z1, . . . , xnynzn)= f (x̄)g(ȳ)h(z̄)
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
Dictatorship: θexD :=∨n
i=1 xyz = ex(xi)ex(yi)ex(zi)ex(1) = 0ex(0) = 1
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)}
Excluded values and interactions
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
26/27
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Dependency strengthening theorems
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
Dictatorship: θexD :=∨n
i=1 xyz = ex(xi)ex(yi)ex(zi)ex(1) = 0ex(0) = 1
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)}
Excluded values and interactions
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
C.f. [Tang, Lin 2009]
26/27
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Dependency strengthening theorems
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
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
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)}
Excluded values and interactions
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
gD.
C.f. [Tang, Lin 2009]
26/27
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Dependency strengthening theorems
Functionality: θF := =(x1y1z1, . . . , xnynzn; xyz)
IIA: θIIA := =(x1, . . . , xn; x) ∧ =(y1, . . . , yn; y) ∧ =(z1, . . . , zn; z)
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
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)}
Excluded values and interactions
Thm. For any f,g ∈ {id,ex}, θF , θIIA, θfUD, θfCR, θgUN ` θ
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 .
26/27
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Thank you!
27/27
Arrow's Impossiblity TheoremFormalizing Arrow's Theorem in dependence and independence logicArrow's Theorem as a dependency strengthening theorem