1/22 Formalizing Arrow s Theorem in Logics of Dependence and Independence Fan Yang Delft University of Technology Logics for Social Behaviour ETH Zürich 8-11 February, 2016 Joint work with Eric Pacuit
Outline 2/22 1 Arrow s Theorem 2 Logics of Dependence and Independence 3 Formalizing Arrow s Theorem in Independence Logic
Arrow s Theorem 3/22 A (finite) set of alternatives: A = {a, b, c, d,... } A finite set of voters: V = {v 1,..., v n } A ranking R A A is a transitive and complete relation on A. A linear ranking R is a ranking that is a linear relation. a profile R year voter 1 voter 2 voter 3 group decision 2000 abc cab acb abc 2001 abc cab acb abc 2002 cab bca acb cba 2003 cab bca acb cba =F(R) Preference aggregation function F : P(A) n P(A), where P(A) = O(A) or L(A)
4/22 Arrow s Theorem Theorem (Arrow) If A 3 and V = n, then any preference aggregation function F : P(A) n P(A) satisfying Independence of Irrelevant Alternatives and Unanimity is a dictatorship. Functionality of Preference Aggregation Rule (dependence) Independence of Irrelevant Alternatives (dependence) Unanimity (dependence) Dictatorship (dependence) Universal Domain (independence) All Rankings Independence
Logics of Dependence and Independence 5/22
First Order Quantifiers: x 1 y 1 x 2 y 2 φ Henkin Quantifiers (Henkin, 1961): ( x1 y 1 x 2 y 2 ) φ Independence Friendly Logic (Hintikka, Sandu, 1989): Dependence Logic (Väänänen 2007): x 1 y 1 x 2 y 2 /{x 1 }φ x 1 y 1 x 2 y 2 ( =(x 2, y 2 ) φ) The value of y 2 is completely determined by the value of x 2. Independence Logic (Grädel, Väänänen 2013): x 1 y 1 x 2 y 2 (x 1 y 2 φ) The value of x 1 is completely independent of the value of y 2. 6/22
7/22 Dependence Logic: φ ::= α α φ φ φ first-order φ xφ logic xφ + =( x, y) Independence Logic: φ ::= α α φ φ φ first-order φ xφ logic xφ + x y The ranking of voter v i is completely independent of the rankings of the other voters v j (j i). v i v j j i The group decision u is completely determined by the rankings of the voters v 1,..., v n. =(v 1,..., v n, u)
Team semantics (Hodges 1997) The group decision u is completely determined by the rankings of the voters v 1, v 2, v 3. M = S =(v 1, v 2, v 3, u) { A team S a profile R year voter 1 voter 2 voter 3 group decision s 1 2000 abc cab acb abc s 2 2001 abc cab acb abc s 3 2002 cab bca acb cba s 4 2003 cab bca acb cba = F(R) M = S =( x, y) iff for any s, s S, s( x) = s ( x) = s(y) = s (y). (R i, F) = s R,F ; v 1 R 1,..., v n R n, u F(R). {(R i, F) i I} = team S F = {s Ri,F i I} Functionality of Preference Aggregation Rule θ F := =(v 1,..., v n, u) 8/22
Team semantics (Hodges 1997) The ranking of voter v 1 is completely independent of the rankings of the other voters v 2 and v 3. v 1 v 2, v 3 voter 1 voter 2 voter 3 group decision s 1 abc bac cab bca s 2 acb acb bac cba s 3 acb bac cab acb s 4 abc acb bac abc. M = S x y iff for all s, s S, there exists s S s.t. s ( x) = s( x) and s ( y) = s ( y). Independence: For any profiles (R 1,..., R n ), (R 1,..., R n) dom(f) and any voter x i V, there is a profile (R 1,..., R n ) dom(f ) such that R i = R i and R j = R j for all j i. θ I := { vi v j j i : 1 i n } 9/22
Fix a finite set A = {a, b, c,... } of alternatives containing at least three elements and a set V = {v 1,..., v n } of voters. Let v 1,..., v n, u be distinguished variables. The signature L A consists of unary predicate symbols R ab (x) for each pair (a, b) A A. Strict preference P ab (x) := R ab (x) R ba (x). An intended first-order model M: dom(m) = L(A) the set of all linear rankings of A = {a, b, c} R M ab = {abc, cab, acb}, i.e., RM ab (R ) iff ar b for all R L(A). Fact: There is a set Γ DM of first-order L A -formulas such that M is a model of Γ DM iff M is (isomorphic to) the intended model. 10/22
11/22 φ ::= α α φ φ φ φ xφ xφ =( x, y) (Flatness) For every formula φ of first-order logic, M = S φ s S : M = {s} φ. M = S α iff for all s S, M = s α; M = S α iff for all s S, M = s α; group voter 1 voter 2 decision s M = S 1 { 1 abc cab bca S1 R ab (v 1 ) s 2 acb cba cba M = S2 R ab (v 1 ) s S 2 { 3 bac cab abc s 4 bca acb bac M = S1 S 2 R ab (v 1 ) M = S1 S 2 R ab (v 1 )
12/22 For any first-order formulas φ and ψ, M = S φ ψ iff for all s S, if M = {s} φ = M = {s} ψ. voter 1 voter 2 voter 3 group decision s 1 abc acb acb acb s 2 bca abc bac abc s 3 cba acb bca acb.. P ab (v 2 ) P ab (u) Voter v i is a dictator: θ d (v i ) := a,b A (P ab(v i ) P ab (u)) n There is a dictator: θ D := θ d (v i ), i=1 where is the intuitionistic disjunction. M = S φ ψ iff M = S φ or M = S ψ.
Theorem (Väänänen, Grädel, Kontinen, Galliani) Over sentences, both dependence logic (D) and independence logic (I) have the same expressive power as Σ 1 1. In particular, on finite structures, both logics capture NP. D-formulas characterize all Σ 1 1 downward monotone properties. I-formulas characterize all Σ 1 1 properties. D < I φ ::= α α =(x 1,..., x k, y) x 1... x k y 1... y m φ φ φ φ xφ xφ x 1... x k y 1... y k Every value of x is also a value of y. Theorem (Galliani, Hella 2013) Inclusion logic (FO + x y) has the same expressive power as positive greatest fixed point logic. In particular, on ordered finite structures, inclusion logic captures PTIME. 13/22
Every value of the variable x is also a ranking of voter v 1 in the set of profiles. x v 1 x voter 1 voter 2 voter 3 group decision s 1 abc acb cab bac bca s 2 acb bac bac abc cba s 3 abc abc cab acb acb M = S x y iff for all s S, there exists s S such that s( x) = s ( y). θ AR := { x(x v i ): 1 i n} All rankings: For any voter v i and any ranking R, there is a profile (R 1,..., R n ) dom(f ) such that R i = R. (Recall: dom(m) = L(A).) 14/22
15/22 Independence logic φ ::= α α =( x, y) x y x y φ φ φ φ xφ xφ Independence logic Σ 1 1
16/22 (Partial) axiomatizations Definition: M = S φ iff M = S φ whenever S. Definition: φ is negatable iff φ is definable in the logic. Theorem There is a natural deduction system such that for any set Γ {φ} of formulas of independence logic with φ negatable, Γ φ Γ = φ. Example: First-order formulas are negatable. θ D is negatable. is negatable. Hence Γ (i.e., Γ is inconsistent) can be derived.
Formalizing Arrow s Theorem in Independence Logic 17/22
18/22 Theorem (Arrow) If A 3 and V = n, then any preference aggregation function F : P(A) n P(A) satisfying Independence of Irrelevant Alternatives and Unanimity is a dictatorship. Γ Arrow = θ D or Γ Arrow, θ D = = Γ Arrow θ D or Γ Arrow, θ D (by Completeness Theorem) Functionality of Preference Aggregation Rule [θ F := =(v 1,..., v n, u)] Independence of Irrelevant Alternatives Unanimity Dictatorship [θ D := n i=1 ( a,b A ) (P ab (v i ) P ab (u)) ] Universal Domain All Ranking [θ AR := { x(x v i ) : 1 i n}] Independence [θ I := { vi v j j i : 1 i n } ]
19/22 Independence of Irrelevant Alternatives (IIA) The social decision on the relative preference between two alternatives a, b depends only on the individual preferences between these alternatives. It is independent of their rankings with respect to other alternatives. voter 1 voter 2 voter 3 group decision s 1 abc bac bca cba s 2 cab bca bac bca s 3 abc bac bca bac.. θ IIA := { =(R ab (v 1 ), R ba (v 1 )..., R ab (v n ), R ba (v n ), R ab (u)) a, b A} M = S =(φ 1,..., φ k, ψ) iff for all s, s S, if s {φ1,...,φ k } s, then s {ψ} s, where s Γ s is defined as for all γ Γ : M = {s} γ M = {s } γ.
20/22 Unanimity If every voter prefers a to b, then so should the group decision. voter 1 voter 2 voter 3 group decision s 1 abc cab acb abc s 2 bca abc bac abc s 3 cba cba bca cba.. θ U := { ( P ab (v 1 ) P ab (v n ) ) P ab (u) a, b A}
21/22 Theorem (Arrow) If A 3 and V = n, then any preference aggregation function F : P(A) n P(A) satisfying Independence of Irrelevant Alternatives and Unanimity is a dictatorship. Theorem (Arrow s Theorem) Γ Arrow = θ D, where Γ Arrow = Γ DM {θ F, θ IIA, θ U, θ AR, θ I }. Γ Arrow θ D