Logic and Group Decision Making Eric Pacuit Department of - PowerPoint PPT Presentation

Logic and Group Decision Making Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu August 12, 2013 Eric Pacuit 1

An Email Eric Pacuit 2

An Email “Interesting Eric Pacuit 2

An Email “Interesting...but what does logic have to do with group decision making??? I’ve never seen logic prevail at any of our faculty meetings.” Eric Pacuit 2

Logic and Group Decision Making Group decision making from a logicians perspective... Eric Pacuit 3

Logic and Group Decision Making Group decision making from a logicians perspective... 1. Logical (and algebraic) methods can be used to prove various results (Eckert & Herzberg, Nehring & Pivato) 2. Two non-standard logics for reasoning about social choice 3. A challenge: probabilities in group decision making (Goranko & Bulling) 4. Logics for social epistemology (Rendsvig) Eric Pacuit 3

Arrow’s Theorem Theorem Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship. K. Arrow. Social Choice & Individual Values . 1951. Eric Pacuit 4

Broader Applications ◮ Is it possible to choose rationally among rival scientific theories on the basis of the accuracy, simplicity, scope and other relevant criteria? No Yes S. Okasha. Theory choice and social choice: Kuhn versus Arrow . Mind, 120, 477, pgs. 83 - 115, 2011. M. Moureau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice . FEW, 2012. Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence . Synthese, 2011. Eric Pacuit 5

Broader Applications ◮ Is it possible to choose rationally among rival scientific theories on the basis of the accuracy, simplicity, scope and other relevant criteria? //// No Yes S. Okasha. Theory choice and social choice: Kuhn versus Arrow . Mind, 120, 477, pgs. 83 - 115, 2011. M. Morreau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice . Erkenntnis, forthcoming. Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence . Synthese, 2011. Eric Pacuit 5

Broader Applications ◮ Is it possible to choose rationally among rival scientific theories on the basis of the accuracy, simplicity, scope and other relevant criteria? //// No Yes S. Okasha. Theory choice and social choice: Kuhn versus Arrow . Mind, 120, 477, pgs. 83 - 115, 2011. M. Morreau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice . Erkenntnis, forthcoming. ◮ Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence . Synthese, 2011. Eric Pacuit 5

Broader Applications ◮ Is it possible to merge classic AGM belief revision with the Ramsey test? P. G¨ ardenfors. Belief revisions and the Ramsey Test for conditionals . The Philo- sophical Review, 95, pp. 81 - 93, 1986. H. Leitgeb and K. Segerberg. Dynamic doxastic logic: why, how and where to? . Synthese, 2011. H. Leitgeb. A Dictator Theorem on Belief Revision Derived From Arrow’s The- orem . Manuscript, 2011. Eric Pacuit 5

Two non-standard logics for reasoning about social choice Eric Pacuit 6

D. Osherson and S. Weinstein. Preference based on reasons . Review of Symbolic Logic, 2012. Eric Pacuit 7

ϕ � X ψ “The agent considers ϕ at least as good as ψ for reason X ” Eric Pacuit 8

ϕ � X ψ “The agent considers ϕ at least as good as ψ for reason X ” The agent envisions a situation in which ϕ is true and that otherwise differs little from his actual situation. Likewise she envisions a world where ψ is true and otherwise differs little from his actual situation. Finally, the utility according to u X of the first imagined situation exceeds that of the second. Eric Pacuit 8

p : “ i purchases a fire alarm” Eric Pacuit 9

p : “ i purchases a fire alarm” p ≻ 1 ¬ p : u 1 measures safety Eric Pacuit 9

p : “ i purchases a fire alarm” p ≻ 1 ¬ p : u 1 measures safety p ≺ 2 ¬ p : u 2 measures finances Eric Pacuit 9

p : “ i purchases a fire alarm” p ≻ 1 ¬ p : u 1 measures safety p ≺ 2 ¬ p : u 2 measures finances What is the status of p ≻ 1 , 2 ¬ p ? p ≺ 1 , 2 ¬ p ? Eric Pacuit 9

At a set of atomic proposition, S a set of reasons . � W , s , u , V � ◮ W is a set of states ◮ s : W × ℘ � = ∅ ( W ) → W is a selection function ( s ( w , A ) ∈ A ) ◮ u : W × S → R is a utility function ◮ V : At → ℘ ( W ) is a valuation function Eric Pacuit 10

At a set of atomic proposition, S a set of reasons . � W , s , u , V � ◮ W is a set of states ◮ s : W × ℘ � = ∅ ( W ) → W is a selection function ( s ( w , A ) ∈ A ) ◮ u : W × S → R is a utility function ◮ V : At → ℘ ( W ) is a valuation function M , w | = θ � X ψ iff u X ( s ( w , [ [ θ ] ] M )) ≥ u X ( s ( w , [ [ ψ ] ] M )) provided [ [ θ ] ] M � = ∅ and [ [ ψ ] ] M � = ∅ Eric Pacuit 10

Universal Modality is Definable ♦ ϕ = def ϕ � X ϕ � ϕ = def ¬ ( ¬ ϕ � X ¬ ϕ ) Eric Pacuit 11

Reflexive : for all w if w ∈ A then s ( w , A ) = w . Eric Pacuit 12

Reflexive : for all w if w ∈ A then s ( w , A ) = w . M is reflexive implies ( p � X ⊤ ) ∨ ( ¬ p � X ⊤ ) is valid. Eric Pacuit 12

Regular : Suppose that A ⊆ B and w 1 ∈ A then, if s ( w , B ) = w 1 then s ( w , A ) = w 1 . Eric Pacuit 13

Regular : Suppose that A ⊆ B and w 1 ∈ A then, if s ( w , B ) = w 1 then s ( w , A ) = w 1 . M is regular implies (( p ∨ q ) ≻ X r ) → (( p ≻ X r ) ∨ ( q ≻ X r )) is valid. Eric Pacuit 13

Modeling Social Choice Problems The set of reasons: { 1 } , . . . , { k } , { 1 , 2 , . . . , k } . The signature contains a monadic predicate P . Px 1 ≻ i Px 2 : “agent i strictly prefers the object assigned to x 1 over the object assigned to x 2 ” Px 1 ≻ 1 ,..., k Px 2 : “society strictly prefers the object assigned to x 1 over the object assigned to x 2 ” Eric Pacuit 14

Universal Domain Fix a set of variables x 1 , x 2 , . . . , x m (with m ≥ 3). Let χ ( x 1 , x 2 , . . . , x m ) be the formula saying that each of x 1 , . . . , x m is equal to exactly one of the x 1 , . . . , x m . Eric Pacuit 15

Universal Domain Fix a set of variables x 1 , x 2 , . . . , x m (with m ≥ 3). Let χ ( x 1 , x 2 , . . . , x m ) be the formula saying that each of x 1 , . . . , x m is equal to exactly one of the x 1 , . . . , x m . Suppose that ψ is the conjunction of: χ ( x 1 , . . . , x m ) ∧ ( Px 1 ≻ 1 Px 2 ) ∧ · · · ∧ ( Px m − 1 ≻ 1 Px m ) . . . χ ( x 1 , . . . , x m ) ∧ ( Px 1 ≻ k Px 2 ) ∧ · · · ∧ ( Px m − 1 ≻ k Px m ) Let ϕ univ be the universal closure of ♦ ψ Eric Pacuit 15

Pareto Let ϕ pareto be the universal closure of the above formula � (( Px 1 ≻ 1 Px 2 ∧ · · · Px k ≻ 1 Px k ) → Px ≻ 1 ,..., k Py ) Eric Pacuit 16

IIA Fix two variables x , y . Let ψ ( x ′ , y ′ ) be the formula that says each of x ′ , y ′ is equal to exactly one of x , y . The formula ϕ iia is the universal closure of: ( ψ ( x 1 , y 1 ) ∧ · · · ∧ ψ ( x k , y k )) → ( ♦ (( Px 1 ≻ 1 Py 1 ∧ · · · ∧ ( Px k ≻ k Py k ) ∧ Px ≻ 1 ,..., k Py )) → � ((( Px 1 ≻ 1 Py 1 ) ∧ · · · ∧ ( Px k ≻ k Py k )) → Px ≻ 1 ,..., k Py )) Eric Pacuit 17

Dictator Let ϕ dictator be the disjunction of: ∀ x 1 · · · x m � ((( Px 1 ≻ 1 Px 2 ) ∧ ( Px 2 ≻ 1 Px 3 ) · · · ( Px m − 1 ≻ 1 Px m )) ↔ (( Px 1 ≻ 1 ,... k Px 2 ) ∧ ( Px 2 ≻ 1 ,... k Px 3 ) ∧ · · · ( Px m − 1 ≻ 1 ,... k Px m ))) . . . ∀ x 1 · · · x m � ((( Px 1 ≻ k Px 2 ) ∧ ( Px 2 ≻ k Px 3 ) · · · ( Px m − 1 ≻ k Px m )) ↔ (( Px 1 ≻ 1 ,... k Px 2 ) ∧ ( Px 2 ≻ 1 ,... k Px 3 ) ∧ · · · ( Px m − 1 ≻ 1 ,... k Px m ))) Eric Pacuit 18

Arrow’s Theorem { ϕ univ , ϕ pareto , ϕ iia } | = ϕ dictator Eric Pacuit 19

Dependence Logic J. V¨ a¨ an¨ anen. Dependence Logic . Cambridge University Press, 2007. E. Gr¨ adel and J. V¨ a¨ an¨ anen. Dependence and Independence . Studia Logica, vol. 101(2), pp. 399-410, 2013. Eric Pacuit 20

Let V be a set of variables and D a domain. A substitution is a function s : V → D . A team X is a set of substitutions. Eric Pacuit 21

Let V be a set of variables and D a domain. A substitution is a function s : V → D . A team X is a set of substitutions. = =( x 1 , . . . , x n , y ) iff for all s , s ′ ∈ X , X | ( s ( x 1 , . . . , x n ) = s ′ ( x 1 , . . . , x n )) → ( s ( y ) = s ( y )) Eric Pacuit 21

Let V be a set of variables and D a domain. A substitution is a function s : V → D . A team X is a set of substitutions. = =( x 1 , . . . , x n , y ) iff for all s , s ′ ∈ X , X | ( s ( x 1 , . . . , x n ) = s ′ ( x 1 , . . . , x n )) → ( s ( y ) = s ( y )) = ( x 1 , . . . x n ) ⊥ y iff for all s , s ′ ∈ X , there exists s ′′ ∈ X such X | that s ′′ ( x 1 , . . . , x n ) = s ( x 1 , . . . , x n ) and s ′′ ( y ) = s ′ ( y ) Eric Pacuit 21