Social Choice Theory for Logicians Lecture 5

Transcription

1 Social Choice Theory for Logicians Lecture 5 Eric Pacuit Department of Philosophy University of Maryland, College Park ai.stanford.edu/ epacuit epacuit@umd.edu June 22, 2012 Eric Pacuit: The Logic Behind Voting 1/77

2 Plan Arrow, Sen, Muller-Satterthwaite Characterizing Voting Methods: Majority (May, Asan & Sanver), Scoring Rules (Young), Borda Count (Farkas and Nitzan, Saari), Approval Voting (Fishburn) Voting to get things right (Distance-based measures, Condorcet and extensions) Strategizing (Gibbard-Satterthwaite) 1. Generalizations 1.1 Infinite Populations Judgement aggregation (List & Dietrich) 2. Logics 3. Applications Eric Pacuit: The Logic Behind Voting 2/77

3 Plan The logic of axiomatization results Logics for reasoning about aggregation methods Preference (modal) logics Applications Eric Pacuit: The Logic Behind Voting 3/77

4 Setting the Stage: Logic and Games M. Pauly and W. van der Hoek. Modal Logic form Games and Information. Handbook of Modal Logic (2006). G. Bonanno. Modal logic and game theory: Two alternative approaches. Risk Decision and Policy 7 (2002). J. van Benthem. Extensive games as process models. Journal of Logic, Language and Information 11 (2002). J. Halpern. A computer scientist looks at game theory. Games and Economic Behavior 45:1 (2003). R. Parikh. Social Software. Synthese 132: 3 (2002). Eric Pacuit: The Logic Behind Voting 4/77

5 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Eric Pacuit: The Logic Behind Voting 5/77

6 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Given a semantic domain D and a target class T D Eric Pacuit: The Logic Behind Voting 5/77

7 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Given a semantic domain D and a target class T D Fix a language L and a satisfaction relation = D L Eric Pacuit: The Logic Behind Voting 5/77

8 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Given a semantic domain D and a target class T D Fix a language L and a satisfaction relation = D L L be a set of axioms Eric Pacuit: The Logic Behind Voting 5/77

9 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Given a semantic domain D and a target class T D Fix a language L and a satisfaction relation = D L L be a set of axioms absolutely axiomatizes T iff for all M D, M T iff M = (i.e., defines T ) Eric Pacuit: The Logic Behind Voting 5/77

10 What do the (Im)possibility results say? M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , Given a semantic domain D and a target class T D Fix a language L and a satisfaction relation = D L L be a set of axioms absolutely axiomatizes T iff for all M D, M T iff M = (i.e., defines T ) relatively axiomatizes T iff for all ϕ L, T = ϕ iff = ϕ (i.e., axiomatizes the theory of T ) Eric Pacuit: The Logic Behind Voting 5/77

11 What do the (Im)possibility results say? May s Theorem: is the set of aggregation functions w.r.t. 2 candidates, T is majority rule, L is the language of set theory, is the properties of May s theorem, then absolutely axiomatizes T. Eric Pacuit: The Logic Behind Voting 6/77

12 What do the (Im)possibility results say? May s Theorem: is the set of aggregation functions w.r.t. 2 candidates, T is majority rule, L is the language of set theory, is the properties of May s theorem, then absolutely axiomatizes T. Arrow s Theorem: is the set of aggregation functions w.r.t. 3 or more candidates, T is a dictatorship, L is the language of set theory, is the properties of May s theorem, then absolutely axiomatizes T. Eric Pacuit: The Logic Behind Voting 6/77

13 A Minimal Language M. Pauly. Axiomatizing Collective Judgement Sets in a Minimal Logical Language Let Φ I be the set of individual formulas (standard propositional language) V I the set of individual valuations Eric Pacuit: The Logic Behind Voting 7/77

14 A Minimal Language M. Pauly. Axiomatizing Collective Judgement Sets in a Minimal Logical Language Let Φ I be the set of individual formulas (standard propositional language) V I the set of individual valuations Φ C the set of collective formulas: α ϕ ψ ϕ α: The group collectively accepts α. V C the set of collective valuations: v : Φ C {0, 1} Eric Pacuit: The Logic Behind Voting 7/77

15 A Minimal Language Let CON n = {v V C v( α) = 1 iff i n, v i (α) = 1} E. ϕ ψ provided ϕ ψ is a tautology M. (ϕ ψ) ( ϕ ψ) C. ( ϕ ψ) ( ϕ ψ) N. D. Theorem [Pauly, 2005] V C (KD) = CON n, provided n 2 Φ 0. (D = V C, T = CON n, = EMCND, then absolutely axiomatizes T.) Eric Pacuit: The Logic Behind Voting 8/77

16 A Minimal Language Let MAJ n = {v V C v([>]α) = 1 iff {i v i (α) = 1} > n 2 } STEM contains all instances of the following schemes S. [>]ϕ [>] ϕ T. ([ ]ϕ 1 [ ]ϕ k [ ]ψ 1 [ ]ψ k ) 1 i k ([= ]ϕ i [=]ψ i ) where v V I : {i v(ϕ i ) = 1} = {i v(ψ i ) = 1} E. [>]ϕ [>]ψ provided ϕ ψ is a tautology M. [>](ϕ ψ) ([>]ϕ [>]ψ) Theorem [Pauly, 2005] V C (STEM) = MAJ. (D = V C, T = MAJ n, = STEM, then absolutely axiomatizes T.) Eric Pacuit: The Logic Behind Voting 9/77

17 Compare principles in terms of the language used to express them M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , T. Daniëls. Social choice and logic of simple games. Journal of Logic and Computation, 21, 6, pgs , Eric Pacuit: The Logic Behind Voting 10/77

18 Compare principles in terms of the language used to express them M. Pauly. On the Role of Language in Social Choice Theory. Synthese, 163, 2, pgs , T. Daniëls. Social choice and logic of simple games. Journal of Logic and Computation, 21, 6, pgs , How much classical logic is needed for the judgement aggregation results? T. Daniëls and EP. A general approach to aggregation problems. Journal of Logic and Computation, 19, 3, pgs , F. Dietrich. A generalised model of judgment aggregation. Social Choice and Welfare 28(4): , Eric Pacuit: The Logic Behind Voting 10/77

19 Plan The logic of axiomatization results Logics for reasoning about aggregation methods Preference (modal) logics Applications Eric Pacuit: The Logic Behind Voting 11/77

20 Judgement Aggregation Logic T. Agotnes, W. van der Hoek, M. Wooldridge. On the logic of preference and judgement aggregation. Autonomous Agent and Multi-Agent Systems, 22, pgs. 4-30, Some Notation: N = {1,..., n} a set of agents A is the agenda (set of formulas of some logic L on the table satisfying certain fullness conditions ) Let J(A, L) is the set of judgements (eg. maximally consistent subsets of A) γ J(A, L) n is a judgement profile with γ i agent i s judgement set Eric Pacuit: The Logic Behind Voting 12/77

21 Judgement Aggregation Logic: Semantics Tables F, γ, p Eric Pacuit: The Logic Behind Voting 13/77

22 Judgement Aggregation Logic: Semantics Tables F, γ, p Example: P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F is an aggregations function F : J(A, L) n J(A, L) Eric Pacuit: The Logic Behind Voting 13/77

23 Judgement Aggregation Logic: Semantics Tables F, γ, p Example: P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} γ J(A, L) n (assuming consistency and completeness) Eric Pacuit: The Logic Behind Voting 13/77

24 Judgement Aggregation Logic: Semantics Tables F, γ, p Example: P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} p A Eric Pacuit: The Logic Behind Voting 13/77

25 Judgement Aggregation Logic: Language Atomic Formulas: At = {i, σ, h p p A, i N} Formulas: ϕ ::= α ϕ ϕ ϕ ϕ ϕ Eric Pacuit: The Logic Behind Voting 14/77

26 Judgement Aggregation Logic: Language Eric Pacuit: The Logic Behind Voting 14/77

27 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p F, γ, p = i iff p γ i F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

28 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p The current proposition on the table is q F, γ, p = i iff p γ i F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

29 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p The current proposition on the table is q F, γ, p = i iff p γ i Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

30 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p The current proposition on the table is q F, γ, p = i iff p γ i Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

31 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p The current proposition on the table is q F, γ, p = i iff p γ i Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ Quantification over the set of judgement profiles F, γ, p = ϕ iff p A, F, γ, p = ϕ Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

32 Judgement Aggregation Logic: Truth F, γ, p = h q iff q = p The current proposition on the table is q F, γ, p = i iff p γ i Voter i accepts the current proposition on the table F, γ, p = σ iff p F (γ) Society accepts the current proposition on the table F, γ, p = ϕ iff γ J(A, L) n, F, γ, p = ϕ Quantification over the set of judgement profiles F, γ, p = ϕ iff p A, F, γ, p = ϕ Quantification over the agenda Boolean connectives as usual Eric Pacuit: The Logic Behind Voting 15/77

33 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False Eric Pacuit: The Logic Behind Voting 16/77

34 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} Eric Pacuit: The Logic Behind Voting 16/77

35 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = Eric Pacuit: The Logic Behind Voting 17/77

36 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = σ Eric Pacuit: The Logic Behind Voting 18/77

37 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = (1 3) Eric Pacuit: The Logic Behind Voting 19/77

38 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3)) Eric Pacuit: The Logic Behind Voting 20/77

39 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 True True True F maj True True True A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3)) Eric Pacuit: The Logic Behind Voting 21/77

40 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 True True True F maj True True True A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3) }{{} ) All agents agree on P Eric Pacuit: The Logic Behind Voting 22/77

41 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True False False Individual 3 False True False F maj True True False A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3)) Eric Pacuit: The Logic Behind Voting 23/77

42 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True True True Individual 3 True True True F maj True True True A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3)) Eric Pacuit: The Logic Behind Voting 24/77

43 Judgement Aggregation Logic: Example P P Q Q Individual 1 True True True Individual 2 True True True Individual 3 True True True F maj True True True A = {P, Q, P Q, P, Q, (P Q)} F maj, γ, P = ((1 2) (2 3) (1 3)) }{{} All agents agree on all propositions in the agenda Eric Pacuit: The Logic Behind Voting 25/77

44 F maj, γ, P = (σ G {1,2,3}, G 2 i G i) Eric Pacuit: The Logic Behind Voting 26/77

45 Judgement Aggregation Logic: Results Eric Pacuit: The Logic Behind Voting 27/77

46 Judgement Aggregation Logic: Results Sound and complete axiomatization Eric Pacuit: The Logic Behind Voting 27/77

47 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Eric Pacuit: The Logic Behind Voting 27/77

48 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Eric Pacuit: The Logic Behind Voting 27/77

49 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Eric Pacuit: The Logic Behind Voting 27/77

50 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Eric Pacuit: The Logic Behind Voting 27/77

51 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Eric Pacuit: The Logic Behind Voting 27/77

52 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 27/77

53 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 28/77

54 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 29/77

55 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 30/77

56 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 31/77

57 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Eric Pacuit: The Logic Behind Voting 32/77

58 Judgement Aggregation Logic: Results Sound and complete axiomatization Model checking is decidable, but relatively difficult Expressivity: Discursive Dilemma: (( MV ) ), where MV := σ G N, G > n 2 i G i, Impossibility results: Nondictatorship: i N (σ i), Unanimity: ((1 n) σ) Independence: o O ((o σ) (o σ)) Given any judgement profile, any choice of the voters and any P A, if society accepts P then for any profile (if the choices are the same w.r.t. P then society should accept P) Complete axiomatization Eric Pacuit: The Logic Behind Voting 32/77

59 U. Endriss. Logic and Social Choice Eric Pacuit: The Logic Behind Voting 33/77

60 Plan The logic of axiomatization results Logics for reasoning about aggregation methods Preference (modal) logics Applications Eric Pacuit: The Logic Behind Voting 34/77

61 Preference (Modal) Logics x, y objects x y: x is at least as good as y Eric Pacuit: The Logic Behind Voting 35/77

62 Preference (Modal) Logics x, y objects x y: x is at least as good as y 1. x y and y x (x y) 2. x y and y x (y x) 3. x y and y x (x y) 4. x y and y x (x y) Eric Pacuit: The Logic Behind Voting 35/77

63 Preference (Modal) Logics x, y objects x y: x is at least as good as y 1. x y and y x (x y) 2. x y and y x (y x) 3. x y and y x (x y) 4. x y and y x (x y) Properties: transitivity, connectedness, etc. Eric Pacuit: The Logic Behind Voting 35/77

64 Preference (Modal) Logics Modal betterness model M = W,, V Eric Pacuit: The Logic Behind Voting 36/77

65 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ Eric Pacuit: The Logic Behind Voting 36/77

66 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ M, w = ϕ iff there is v w and w v such that M, v = ϕ Eric Pacuit: The Logic Behind Voting 36/77

67 Preference (Modal) Logics 1. ϕ ϕ 2. ϕ ϕ 3. ϕ ψ ( ψ (ψ ϕ)) 4. ϕ ϕ Theorem The above logic (with Necessitation and Modus Ponens) is sound and complete with respect to the class of preference models. J. van Benthem, O. Roy and P. Girard. Everything else being equal: A modal logic approach to ceteris paribus preferences. JPL, Eric Pacuit: The Logic Behind Voting 37/77

68 Preference Modalities ϕ ψ: the state of affairs ϕ is at least as good as ψ (ceteris paribus) G. von Wright. The logic of preference. Edinburgh University Press (1963). Eric Pacuit: The Logic Behind Voting 38/77

69 Preference Modalities ϕ ψ: the state of affairs ϕ is at least as good as ψ (ceteris paribus) G. von Wright. The logic of preference. Edinburgh University Press (1963). Γ ϕ: ϕ is true in better world, all things being equal. J. van Benthem, O. Roy and P. Girard. Everything else being equal: A modal logic approach to ceteris paribus preferences. JPL, Eric Pacuit: The Logic Behind Voting 38/77

70 All Things Being Equal... u r b, u b, r With boots (b), I prefer my raincoat (r) over my umbrella (u) Without boots ( b), I also prefer my raincoat (r) over my umbrella (u) But I do prefer an umbrella and boots over a raincoat and no boots Eric Pacuit: The Logic Behind Voting 39/77

71 All Things Being Equal... u r b, u b, r With boots (b), I prefer my raincoat (r) over my umbrella (u) Without boots ( b), I also prefer my raincoat (r) over my umbrella (u) But I do prefer an umbrella and boots over a raincoat and no boots Eric Pacuit: The Logic Behind Voting 39/77

72 All Things Being Equal... u r b, u b, r With boots (b), I prefer my raincoat (r) over my umbrella (u) Without boots ( b), I also prefer my raincoat (r) over my umbrella (u) But I do prefer an umbrella and boots over a raincoat and no boots Eric Pacuit: The Logic Behind Voting 39/77

73 All Things Being Equal... u r b, u b, r With boots (b), I prefer my raincoat (r) over my umbrella (u) Without boots ( b), I also prefer my raincoat (r) over my umbrella (u) But I do prefer an umbrella and boots over a raincoat and no boots Eric Pacuit: The Logic Behind Voting 39/77

74 All Things Being Equal... u r b, u b, r All things being equal, I prefer my raincoat over my umbrella Without boots ( b), I also prefer my raincoat (r) over my umbrella (u) But I do prefer an umbrella and boots over a raincoat and no boots Eric Pacuit: The Logic Behind Voting 39/77

75 All Things Being Equal... Let Γ be a set of (preference) formulas. Write w Γ v if for all ϕ Γ, w = ϕ iff v = ϕ. 1. M, w = Γ ϕ iff there is a v W such that w Γ v and M, v = ϕ. 2. M, w = Γ ϕ iff there is a v W such that w( Γ )v and M, v = ϕ. 3. M, w = Γ < ϕ iff there is a v W such that w( Γ <)v and M, v = ϕ. Key Principles: Γ ϕ Γ ϕ if Γ Γ ±ϕ Γ (α ±ϕ) Γ {ϕ} α Eric Pacuit: The Logic Behind Voting 40/77

76 All Things Being Equal... Let Γ be a set of (preference) formulas. Write w Γ v if for all ϕ Γ, w = ϕ iff v = ϕ. 1. M, w = Γ ϕ iff there is a v W such that w Γ v and M, v = ϕ. 2. M, w = Γ ϕ iff there is a v W such that w( Γ )v and M, v = ϕ. 3. M, w = Γ < ϕ iff there is a v W such that w( Γ <)v and M, v = ϕ. Key Principles: Γ ϕ Γ ϕ if Γ Γ ±ϕ Γ (α ±ϕ) Γ {ϕ} α Eric Pacuit: The Logic Behind Voting 40/77

77 All Things Being Equal... Let Γ be a set of (preference) formulas. Write w Γ v if for all ϕ Γ, w = ϕ iff v = ϕ. 1. M, w = Γ ϕ iff there is a v W such that w Γ v and M, v = ϕ. 2. M, w = Γ ϕ iff there is a v W such that w( Γ )v and M, v = ϕ. 3. M, w = Γ < ϕ iff there is a v W such that w( Γ <)v and M, v = ϕ. Key Principles: Γ ϕ Γ ϕ if Γ Γ ±ϕ Γ (α ±ϕ) Γ {ϕ} α Eric Pacuit: The Logic Behind Voting 40/77

78 Preference Lifting, I Given a preference ordering over a set of objects X, we want to lift this to an ordering ˆ over (X ). Given, what reasonable properties can we infer about ˆ? S. Barberá, W. Bossert, and P.K. Pattanaik. Ranking sets of objects. In Handbook of Utility Theory, volume 2. Kluwer Academic Publishers, Eric Pacuit: The Logic Behind Voting 41/77

79 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? Eric Pacuit: The Logic Behind Voting 42/77

80 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? Eric Pacuit: The Logic Behind Voting 42/77

81 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? Eric Pacuit: The Logic Behind Voting 42/77

82 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? You know that w x y z Can you infer that {w, x} ˆ {y, z}? Eric Pacuit: The Logic Behind Voting 42/77

83 Preference Lifting, III There are different interpretations of X ˆ Y : You will get one of the elements, but cannot control which. You can choose one of the elements. You will get the full set. Eric Pacuit: The Logic Behind Voting 43/77

84 Preference Lifting, IV Kelly Principle (EXT) {x} ˆ {y} provided x y (MAX) A ˆ Max(A) (MIN) Min(A) ˆ A J.S. Kelly. Strategy-Proofness and Social Choice Functions without Single- Valuedness. Econometrica, 45(2), pp , Eric Pacuit: The Logic Behind Voting 44/77

85 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , Eric Pacuit: The Logic Behind Voting 45/77

86 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , Independence (IND) A {x} ˆ B {x} if A ˆ B and x A B Eric Pacuit: The Logic Behind Voting 45/77

87 Preference Lifting, V Theorem (Kannai and Peleg). If X 6, then no weak order satisfies both the Gärdenfors principle and independence. Y. Kannai and B. Peleg. A Note on the Extension of an Order on a Set to the Power Set. Journal of Economic Theory, 32(1), pp , Eric Pacuit: The Logic Behind Voting 46/77

88 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t Eric Pacuit: The Logic Behind Voting 47/77

89 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t Eric Pacuit: The Logic Behind Voting 47/77

90 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t Eric Pacuit: The Logic Behind Voting 47/77

91 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t Eric Pacuit: The Logic Behind Voting 47/77

92 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) Eric Pacuit: The Logic Behind Voting 48/77

93 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) Eric Pacuit: The Logic Behind Voting 48/77

94 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) ϕ ψ := A(ϕ ψ) Eric Pacuit: The Logic Behind Voting 48/77

95 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) ϕ ψ := A(ϕ ψ) ϕ ψ := A(ϕ ψ) Eric Pacuit: The Logic Behind Voting 48/77

96 From Worlds to Sets, III M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t Eric Pacuit: The Logic Behind Voting 49/77

97 From Worlds to Sets, III M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t Eric Pacuit: The Logic Behind Voting 49/77

98 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) Eric Pacuit: The Logic Behind Voting 50/77

99 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) ϕ ψ := A(ψ ϕ) Eric Pacuit: The Logic Behind Voting 50/77

100 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) ϕ ψ := A(ψ ϕ) We must assume the ordering is total Eric Pacuit: The Logic Behind Voting 50/77

101 From Sets to Worlds P 1 >> P 2 >> P 3 >> >> P n x > y iff x and y differ in at least one P i and the first P i where this happens is one with P i x and P i y F. Liu and D. De Jongh. Optimality, belief and preference Eric Pacuit: The Logic Behind Voting 51/77

102 Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ Eric Pacuit: The Logic Behind Voting 52/77

103 Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ M = W,,, V Eric Pacuit: The Logic Behind Voting 52/77

104 Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ M = W,,, V J. van Eijck. Yet more modal logics of preference change and belief revision. manuscript, F. Liu. Changing for the Better: Preference Dynamics and Agent Diversity. PhD thesis, ILLC, Eric Pacuit: The Logic Behind Voting 52/77

105 A(ψ ϕ) vs. K(ψ ϕ) Eric Pacuit: The Logic Behind Voting 53/77

106 A(ψ ϕ) vs. K(ψ ϕ) Should preferences be restricted to information sets? Eric Pacuit: The Logic Behind Voting 53/77

107 A(ψ ϕ) vs. K(ψ ϕ) Should preferences be restricted to information sets? M, w = ϕ iff there is a v with w v and w v such that M, v = ϕ K(ψ ϕ) Eric Pacuit: The Logic Behind Voting 53/77

108 D. Osherson and S. Weinstein. Preference based on reasons. Review of Symbolic Logic, Eric Pacuit: The Logic Behind Voting 54/77

109 ϕ X ψ The agent considers ϕ at least as good as ψ for reason X Eric Pacuit: The Logic Behind Voting 55/77

110 ϕ X ψ The agent considers ϕ at least as good as ψ for reason X i envisions a situation in which ϕ is true and that otherwise differs little from his actual situation. Likewise i envisions a world where ψ is true and otherwise differs little from his actual situation. Finally, there utility according to u X of the first imagined situation exceeds that of the second. Eric Pacuit: The Logic Behind Voting 55/77

111 p: i purchases a fire alarm Eric Pacuit: The Logic Behind Voting 56/77

112 p: i purchases a fire alarm p 1 p: u 1 measures safety Eric Pacuit: The Logic Behind Voting 56/77

113 p: i purchases a fire alarm p 1 p: u 1 measures safety p 2 p: u 2 measures finances Eric Pacuit: The Logic Behind Voting 56/77

114 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: The Logic Behind Voting 56/77

115 (p 1 ) 2 : it s in your financial interest that your buying a low-power automobile is in you safety interesting which might well be true inasmuch as low-power vehicles are cheaper. Eric Pacuit: The Logic Behind Voting 57/77

116 (p 1 ) 2 : it s in your financial interest that your buying a low-power automobile is in you safety interesting which might well be true inasmuch as low-power vehicles are cheaper. q 1 (p 2 q): from the point of view of family pride, you d rather that your brother not run for mayor than that Miss Smith be the superior candidate. Eric Pacuit: The Logic Behind Voting 57/77

117 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: The Logic Behind Voting 58/77

118 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: The Logic Behind Voting 58/77

119 ϕ = def ϕ X ϕ ϕ = def ( ϕ X ϕ) Eric Pacuit: The Logic Behind Voting 59/77

120 Reflexive: for all w if w A then s(w, A) = w. Eric Pacuit: The Logic Behind Voting 60/77

121 Reflexive: for all w if w A then s(w, A) = w. M is reflexive implies (p X ) ( p X ) is valid. Eric Pacuit: The Logic Behind Voting 60/77

122 Reflexive: for all w if w A then s(w, A) = w. M is reflexive implies (p X ) ( p X ) is valid. (p (p X p)) ( p ( p X p)) Eric Pacuit: The Logic Behind Voting 60/77

123 Regular: if A B and w 1 A then If s(w, B) = w 1 then s(w, A) = w 1. Eric Pacuit: The Logic Behind Voting 61/77

124 Regular: if 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: The Logic Behind Voting 61/77

125 M is regular and reflexive then ((p 1 ) 2 (q 1 )) ( p 2 q) is valid. If it is ecologically better for p than for q to politically backfire the abstaining from p is ecologically better than abstaining from q. Eric Pacuit: The Logic Behind Voting 62/77

126 M is proximal if for all w and A, If s(w, A) = w 1 then there is no w 2 A such that V 1 (w) V 1 (w 2 ) V 1 (w) V 1 (w 1 ), where is the symmetric difference. Eric Pacuit: The Logic Behind Voting 63/77

127 M is proximal if for all w and A, If s(w, A) = w 1 then there is no w 2 A such that V 1 (w) V 1 (w 2 ) V 1 (w) V 1 (w 1 ), where is the symmetric difference. (((p r) X (q r)) ((p r) X (q r))) (p X q) is invalid in the class of regular and in the class of proximal models, but valid in the class of models that are both proximal and regular. Eric Pacuit: The Logic Behind Voting 63/77

128 M is proximal if for all w and A, If s(w, A) = w 1 then there is no w 2 A such that V 1 (w) V 1 (w 2 ) V 1 (w) V 1 (w 1 ), where is the symmetric difference. (((p r) X (q r)) ((p r) X (q r))) (p X q) is invalid in the class of regular and in the class of proximal models, but valid in the class of models that are both proximal and regular. (p ((p q) X r)) (q X r) Eric Pacuit: The Logic Behind Voting 63/77

129 Plan The logic of axiomatization results Logics for reasoning about aggregation methods Preference (modal) logics Applications Eric Pacuit: The Logic Behind Voting 64/77

130 Infinite Voting Populations Given an aggregation method F, let D = {C C is winning for F } Eric Pacuit: The Logic Behind Voting 65/77

131 Infinite Voting Populations Given an aggregation method F, let D = {C C is winning for F } Given a set of winning coalitions D, we can define F as follows: F (J) = {α {i i judges that α} D} Eric Pacuit: The Logic Behind Voting 65/77

132 Infinite Voting Populations Given an aggregation method F, let D = {C C is winning for F } Given a set of winning coalitions D, we can define F as follows: F (J) = {α {i i judges that α} D} What is the general relationship between sets of coalitions and aggregators? Eric Pacuit: The Logic Behind Voting 65/77

133 Infinite Voting Populations F. Herzberg and D. Eckert. Impossibility results for infinite-electorate abstract aggregation rules. Journal of Philosophical Logic, 41, pgs , F. Herzberg and D. Eckert. The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates. Mathematical Social Sciences, 64, pgs , L. Lauwers and L. van Liedekerke. Ultraproducts and aggregation. Journal of Mathematical Economics, 24, pgs , Eric Pacuit: The Logic Behind Voting 66/77

134 Infinite Voting Populations F. Herzberg and D. Eckert. Impossibility results for infinite-electorate abstract aggregation rules. Journal of Philosophical Logic, 41, pgs , F. Herzberg and D. Eckert. The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates. Mathematical Social Sciences, 64, pgs , L. Lauwers and L. van Liedekerke. Ultraproducts and aggregation. Journal of Mathematical Economics, 24, pgs , Theorem. Let D be a filter and suppose that F D preserves ψ and assume that there is some A Ω I with finite witness multiplicity with respect to ψ. Then, If D is an ultrafilter, then it is principal (whence F D is a dictatorship) If ϕ is free of negation, disjunction and universal quantification then D contains a finite coalition (whence F D is an oligarchy) Eric Pacuit: The Logic Behind Voting 66/77

135 May s Theorem: Notation Fix an infinite set W. Suppose that there are two alternatives, x and y, under consideration. We assume that each voter has a linear preference over x and y, so for each w W, either w prefers x to y or y to x, but not both. Assume that a subset X W, represents the set of all voters that prefer x to y. Thus X represents the outcome of a particular vote. Eric Pacuit: The Logic Behind Voting 67/77

136 May s Theorem: Notation There are three possible outcomes to consider: 0 means that alternative y was chosen, 1 2 means the vote was a tie, and 1 means that alternative x was chosen. An aggregation function is a function f : 2 W {0, 1 2, 1}. A set X W, f (X ) represents the social preference of the group W ( 1 2 is interpreted as a tie). Eric Pacuit: The Logic Behind Voting 68/77

137 Properties of f Consider f : 2 W {0, 1 2, 1} Eric Pacuit: The Logic Behind Voting 69/77

138 Properties of f Consider f : 2 W {0, 1 2, 1} Decisiveness f is a total function. Eric Pacuit: The Logic Behind Voting 69/77

139 Properties of f Consider f : 2 W {0, 1 2, 1} Decisiveness f is a total function. Neutrality for all X W, f (X C ) = 1 f (X ) Eric Pacuit: The Logic Behind Voting 69/77

140 Properties of f Consider f : 2 W {0, 1 2, 1} Decisiveness f is a total function. Neutrality for all X W, f (X C ) = 1 f (X ) Positive Responsiveness if, for all X, Y W, X Y and f (X ) 0 implies f (Y ) = 1. Eric Pacuit: The Logic Behind Voting 69/77

141 Anonymity Anonymity states that it is the number of votes that counts when determining the outcome, not who voted for what. Eric Pacuit: The Logic Behind Voting 70/77

142 Anonymity Anonymity states that it is the number of votes that counts when determining the outcome, not who voted for what. When W is finite, this condition is straightforward to impose: Fix an arbitrary order on W, then each subset of W can be represented by a finite sequence of 1s and 0s. Then f satisfies anonymity if f is symmetric in this sequence of 1s and 0s. Eric Pacuit: The Logic Behind Voting 70/77

143 Anonymity for an Infinite Population A permutation on a set X is a 1-1 map π : X X. f is anonymous iff for all π and X W, f (X ) = f (π[x ]). Eric Pacuit: The Logic Behind Voting 71/77

144 Anonymity for an Infinite Population A permutation on a set X is a 1-1 map π : X X. f is anonymous iff for all π and X W, f (X ) = f (π[x ]). Too strong! Let X, Y be any (countably) infinite subsets of W, then there is a π such that π[x ] = Y. Hence, for all X, Y W, f (X ) = f (Y ). Eric Pacuit: The Logic Behind Voting 71/77

145 Anonymity for an Infinite Population A finite permutation on a set X is a 1-1 map π : X X such that there is a finite set F X such that for all w W F, π(w) = w. f is finitely anonymous iff for all finite permutations π and X W, f (X ) = f (π[x ]). Eric Pacuit: The Logic Behind Voting 72/77

146 Digression: Bounded Anonymity and Density Let X N and n N, let X (n) = {m X m n} d(x ) = lim n X (n) n Eric Pacuit: The Logic Behind Voting 73/77

147 Digression: Bounded Anonymity and Density Let X N and n N, let X (n) = {m X m n} d(x ) = lim n X (n) n d(e) = 1 2 Eric Pacuit: The Logic Behind Voting 73/77

148 Digression: Bounded Anonymity and Density Let X N and n N, let X (n) = {m X m n} d(x ) = lim n X (n) n d(e) = 1 2 X (n) Unfortunately, lim n n does not always exist. π is a bounded permutation iff {k k n < π(k)} lim = 0 n n Eric Pacuit: The Logic Behind Voting 73/77

149 May s Theorem Generalized Bounded anonymity: F (A) = F (π[a]) for all bounded permutations Density positive responsiveness: f satisfies monotonicity and, if f (A) = 1/2 and all sets with density D with A D and d(a) > 1, we have f (A D) = 1. Theorem (Fey) If an aggregation rule f satisfies neutrality, density positive responsiveness and bounded anonymity, then f agrees with a density majority rule. M. Fey. May s Theorem with an Infinite Population. Social Choice and Welfare (2004). Eric Pacuit: The Logic Behind Voting 74/77

150 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 , M. Moureau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice. FEW, Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence. Synthese, Eric Pacuit: The Logic Behind Voting 75/77

151 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 , M. Moureau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice. FEW, Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence. Synthese, Eric Pacuit: The Logic Behind Voting 75/77

152 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 , M. Moureau. Mr. Accuracy, Mr. Simplicity and Mr. Scope: from social choice to theory choice. FEW, Is it possible to rationally merge evidence from multiple methods? J. Stegenga. An impossibility theorem for amalgamating evidence. Synthese, Eric Pacuit: The Logic Behind Voting 75/77

153 Broader Applications Is it possible to merge classic AGM belief revision with the Ramsey test? P. Gärdenfors. Belief revisions and the Ramsey Test for conditionals. The Philosophical Review, 95, pp , H. Leitgeb and K. Segerberg. Dynamic doxastic logic: why, how and where to?. Synthese, H. Leitgeb. A Dictator Theorem on Belief Revision Derived From Arrow s Theorem. Manuscript, Eric Pacuit: The Logic Behind Voting 75/77

154 Plan Arrow, Sen, Muller-Satterthwaite Characterizing Voting Methods: Majority (May, Asan & Sanver), Scoring Rules (Young), Borda Count (Farkas and Nitzan, Saari), Approval Voting (Fishburn) Voting to get things right (Distance-based measures, Condorcet and extensions) Strategizing (Gibbard-Satterthwaite) Generalizations Infinite Populations Judgement aggregation (List & Dietrich) Logics Applications Eric Pacuit: The Logic Behind Voting 76/77

155 Thank you! Eric Pacuit: The Logic Behind Voting 77/77