Dependence and Independence in Social Choice Theory

Transcription

1 Dependence and Independence in Social Choice Theory Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org March 4, 2014 Eric Pacuit 1

2 Competing desiderata for a group decision 1. The voters preferences should completely determine the group decision. 2. The group decision should depend in the right way on the voters opinions. 3. The voters are free to adopt any preference ordering and the voters opinions are independent of each other (unless there is good reason to think otherwise). Eric Pacuit 2

3 Competing desiderata for a group decision 1. The voters preferences should completely determine the group decision. (Dependence) 2. The group decision should depend in the right way on the voters opinions. (Dependence) 3. The voters are free to adopt any preference ordering and the voters opinions are independent of each other (unless there is good reason to think otherwise). (Independence) Eric Pacuit 2

4 Notation: Candidates, Voters N is a finite set of voters (assume that N = {1, 2, 3,..., n}) X is a (typically finite) set of alternatives: e.g., candidates, restaurants, social states, etc. Eric Pacuit 3

5 Notation: Preferences A relation on X is a linear order if it is transitive, irreflexive, and complete (hence, acyclic) L(X ) is the set of all linear orders over the set X O(X ) is the set of all reflexive, transitive and complete relations over the set X Given R O(X ), let the strict subrelation be P R = {(x, y) x R y and y R x} and the indifference subrelation be I R = {(x, y) x R y and y R x} Eric Pacuit 4

6 Notation: Profiles A profile for the set of voters N is a sequence of (linear) orders over X, denoted R = (R 1,..., R n ). L(X ) n is the set of all profiles for n voters (similarly for O(X ) n ) For a profile R = (R 1,..., R n ) O(X ) n, let N R (A P B) = {i A P i B} be the set of voters that rank A above B (similarly for N R (A I B) and N R (B P A)) Eric Pacuit 5

7 Group Decision Making Methods F : D R Eric Pacuit 6

8 Group Decision Making Methods F : D R Comments D is the domain of the function: the set of possible election scenarios (i.e., D L(X ) n, D O(X ) n, D U(X ) n, where U(X ) is the set of utility functions on X ) The group decision is completely determined by the voters opinions: every profile R D is associated with exactly one group decision. Eric Pacuit 6

9 Group Decision Making Methods F : D R Variants Social Welfare Functions: R = L(X ) or R = O(X ). Social Choice Function: R = (X ), where (X ) is the set of all subsets of X. Eric Pacuit 6

10 Group Decision Making Methods F : D R Variants D = J(X ) n and R = J(X ), where J(X ) is the set of complete, consistent subsets of a set X of propositional formulas D = U(X ) n and R = U(X ) where U(X ) is the set of utility functions D = (X ) n and R = (X ) where (X ) is the set of probability measures on X (p i (A) is the probability that i would choose A if i could act as a dictator.) Eric Pacuit 6

11 The setup (from Jouko s various talks) The set of variables is V = {x 1, x 2,..., x n } {y} (each x i is a voter and y is the social outcome) The domain D is a suitable description of the preferences (and perhaps a different description of the possible group decisions). E.g., D = L(X ) A substitution s : V D is an election scenario listing the voters preferences and the group decision. A team X is a set of possible election scenarios Eric Pacuit 7

12 The setup (from Jouko s various talks) The set of variables is V = {x 1, x 2,..., x n } {y} (each x i is a voter and y is the social outcome) The domain D is a suitable description of the preferences (and perhaps a different description of the possible group decisions). E.g., D = L(X ) A substitution s : V D is an election scenario listing the voters preferences and the group decision. A team X is a set of possible election scenarios Eric Pacuit 7

13 The setup (from Jouko s various talks) The set of variables is V = {x 1, x 2,..., x n } {y} (each x i is a voter and y is the social outcome) The domain D is a suitable description of the preferences (and perhaps a different description of the possible group decisions). E.g., D = L(X ) A substitution s : V D is an election scenario listing the voters preferences and the group decision. A team X is a set of possible election scenarios Eric Pacuit 7

14 The setup (from Jouko s various talks) The set of variables is V = {x 1, x 2,..., x n } {y} (each x i is a voter and y is the social outcome) The domain D is a suitable description of the preferences (and perhaps a different description of the possible group decisions). E.g., D = L(X ) A substitution s : V D is an election scenario listing the voters preferences and the group decision. A team S is a set of possible election scenarios Eric Pacuit 7

15 The setup (from Jouko s various talks) (Atomic) formulas describe the voters preferences and the group decision. P AB (x i ) is true if s(x i ) ranks A strictly above B, R AB (x i ) is true if A is weakly preferred to B (i.e., A s(x i ) B) P AB (y) is true when the group ranks A strictly above B C A (y) means that A was chosen by the group, C A (y) means that the group does not choose A An underlying theory describing (individual and group) rationality assumptions, E.g., transitivity: P AB (x) P BC (x) P AC (x) Resolute rules, C A (y) C B (y), P AB (x) (R AB (x) R AB (x)), etc. Eric Pacuit 7

16 Desiderata 1 The voters preferences should completely determine the group decision. =(x 1, x 2,..., x n, y) Eric Pacuit 8

17 Desiderata 2 The group decision should depend in the right way on the voters opinions. Eric Pacuit 9

18 Desiderata 2 The group decision should depend in the right way on the voters opinions. Single-profile conditions Multi-profile conditions Variable domain conditions Eric Pacuit 9

19 Single-Profile Conditions Condorcet: Elect the Condorcet winner whenever it exists. A Condorcet candidate in a profile R is a candidate A such that N R (A P B) > N R (B P A) for all other candidates B X Eric Pacuit 10

20 Single-Profile Conditions Condorcet: Elect the Condorcet winner whenever it exists. A Condorcet candidate in a profile R is a candidate A such that N R (A P B) > N R (B P A) for all other candidates B X Pareto: Never elect a candidate that is dominated. Weak Pareto: for all profiles R, if for all i N, A P i B, then A P F (R) B (recall that P i is the strict subrelation of R i ) Eric Pacuit 10

21 Single-Profile Conditions Condorcet: Elect the Condorcet winner whenever it exists. A Condorcet candidate in a profile R is a candidate A such that N R (A P B) > N R (B P A) for all other candidates B X Pareto: Never elect a candidate that is dominated. Weak Pareto: for all profiles R, if for all i N, A P i B, then A P F (R) B (recall that P i is the strict subrelation of R i ) ( i N P AB(x i )) P AB (y) (needs to be stated for every pair A, B of candidates) Eric Pacuit 10

22 Multi-Profile Conditions IIA: The group s ranking of A and B should only depend on the voter s rankings of A and B for all profiles R, R if for all i N, R i {A,B} = R i {A,B}, then F (R) {A,B} = F (R ) {A,B} =(R AB (x 1 ),..., R AB (x n ), R AB (y)) Eric Pacuit 11

23 Multi-Profile Conditions, continued Anonymity: The outcome does not depend on the names of the voters. If π is a permutation of the voters, for all profiles R = (R 1,..., R n ), R, if R = (R π(1),..., R π(n) ), then F (R) = F (R ). (not the same as =(x π(1),..., x π(n), y)) Eric Pacuit 12

24 Multi-Profile Conditions, continued Anonymity: The outcome does not depend on the names of the voters. If π is a permutation of the voters, for all profiles R = (R 1,..., R n ), R, if R = (R π(1),..., R π(n) ), then F (R) = F (R ). (not the same as =(x π(1),..., x π(n), y)) Eric Pacuit 12

25 Multi-Profile Conditions, continued Anonymity: The outcome does not depend on the names of the voters. If π is a permutation of the voters, for all profiles R = (R 1,..., R n ), R, if R = (R π(1),..., R π(n) ), then F (R) = F (R ). (not the same as =(x π(1),..., x π(n), y)) Neutrality: The outcome does not depend on the names of the candidates. Monotonicity: More support should never hurt a candidate. Eric Pacuit 12

26 Early Criticism of Multi-Profile Conditions If tastes change, we may expect a new ordering of all the conceivable states; but we do not require that the difference between the new and the old ordering should bear any particular relation to the changes of taste which have occurred. We have, so to speak, a new world and a new order, and we do not demand correspondence between the change in the world and the change in the order (pg ) I. Lilttle. Social Choice and Individual Values. Journal of Political Economy, 60:5, pgs , Eric Pacuit 13

27 Variable Domain Conditions Participation: It should never be in a voter s best interests not to vote. Multiple-Districts: If a candidate wins in each district, then that candidate should also win when the districts are merged. Eric Pacuit 14

28 Variable Population Model Let N be the set of potential voters. Let V = {V V N and V is finite} be the set of all voting blocks. For V V, a profile for V is a function π : V P where P is O(X ), L(X ) or some set B of ballots. Let Π P be the set of all profiles based on P. Eric Pacuit 15

29 Variable Population Model Two profiles π : V P and π : V P are disjoint if V V = If π : V P and π : V P are disjoint, then (π + π ) : (V V ) P is the profile where for all i V V, { (π + π π(i) if i V )(i) = π (i) if i V Eric Pacuit 16

30 Consistency: If F (π) F (π ), then F (π + π ) = F (π) F (π ) Eric Pacuit 17

31 What are the relationships between these principles? Is there a procedure that satisfies all of them? Eric Pacuit 18

32 What are the relationships between these principles? Is there a procedure that satisfies all of them? A few observations: Condorcet winners may not exist. No positional scoring method satisfies the Condorcet Principle. The Condorcet and Participation principles cannot be jointly satisfied (Moulin s Theorem). Eric Pacuit 18

33 Desiderata 3 The voters are free to adopt any preference ordering and the voters opinions are independent of each other (unless there is good reason to think otherwise). Eric Pacuit 19

34 Domain Conditions Universal Domain: The domain of the social welfare (choice) function is D = L(X ) n (or O(X ) n ) Eric Pacuit 20

35 Domain Conditions Universal Domain: The domain of the social welfare (choice) function is D = L(X ) n (or O(X ) n ) Epistemic Rationale: If we do not wish to require any prior knowledge of the tastes of individuals before specifying our social welfare function, that function will have to be defined for every logically possible set of individual orderings. (Arrow, 1963, pg. 24) Eric Pacuit 20

36 Domain Conditions No Restrictions: S = all(x i ) if for all R L(X ), there is an s S such that s(x i ) = R S = triple(x i ) if for all P L({A, B, C}) there is a R O(X ) and s X such that R {A,B,C} = P and s(x i ) = R n i=1 all(x i) Eric Pacuit 21

37 Domain Conditions No Restrictions: S = all(x i ) if for all R L(X ), there is an s S such that s(x i ) = R S = triple(x i ) if for all P L({A, B, C}) there is a R O(X ) and s X such that R {A,B,C} = P and s(x i ) = R n i=1 all(x i) Independence: {x j j i} x i Eric Pacuit 21

38 Domain Restrictions Single-Peaked preferences Sen s Value Restriction Assumptions about the distribution of preferences W. Gaertner. Domain Conditions in Social Choice Theory. Cambridge University Press, Eric Pacuit 22

39 1 1 1 A B C B C A C A B Eric Pacuit 23

40 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

41 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

42 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

43 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

44 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

45 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

46 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

47 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 24

48 1 1 1 A B C B C B C A A A B C Alternatives Eric Pacuit 25

49 1 1 1 C C B A B C B A A A B C Alternatives Eric Pacuit 25

50 1 1 1 C C B A B C B A A A C Alternatives B Eric Pacuit 25

51 1 1 1 A B C B C A C A B A B C Alternatives Eric Pacuit 25

52 1 1 1 A B C B C A C A B C B Alternatives A Eric Pacuit 25

53 1 1 1 A B C B C A C A B A C Alternatives B Eric Pacuit 25

54 1 1 1 A B C B C A C A B C A Alternatives B Eric Pacuit 25

55 D. Black. On the rationale of group decision-making. Journal of Political Economy, 56:1, pgs , Eric Pacuit 26

56 Single-Peakedness: the preferences of group members are said to be single-peaked if the alternatives under consideration can be represented as points on a line and each of the utility functions representing preferences over these alternatives has a maximum at some point on the line and slopes away from this maximum on either side. Eric Pacuit 27

57 Single-Peakedness: the preferences of group members are said to be single-peaked if the alternatives under consideration can be represented as points on a line and each of the utility functions representing preferences over these alternatives has a maximum at some point on the line and slopes away from this maximum on either side. Theorem. If there is an odd number of voters that display single-peaked preferences, then a Condorcet winner exists. Eric Pacuit 27

58 D. Miller. Deliberative Democracy and Social Choice. Political Studies, 40, pgs , C. List, R. Luskin, J. Fishkin and I. McLean. Deliberation, Single-Peakedness, and the Possibility of Meaningful Democracy: Evidence from Deliberative Polls. Journal of Politics, 75(1), pgs , Eric Pacuit 28

59 Sen s Value Restriciton A. Sen. A Possibility Theorem on Majority Decisions. Econometrica 34, 1966, pgs Eric Pacuit 29

60 Sen s Theorem Assume n voters (n is odd). Eric Pacuit 30

61 Sen s Theorem Assume n voters (n is odd). Triplewise value-restriction: For every triple of distinct candidates A, B, C there exists an x i {A, B, C} and r {1, 2, 3} such that no voter ranks x i has her rth preference among A, B, C. Eric Pacuit 30

62 Sen s Theorem Assume n voters (n is odd). Triplewise value-restriction: For every triple of distinct candidates A, B, C there exists an x i {A, B, C} and r {1, 2, 3} such that no voter ranks x i has her rth preference among A, B, C. Theorem (Sen, 1966). For every profile satisfying triplewise value-restriction, pairwise majority voting generates a transitive group preference ordering. Eric Pacuit 30

63 Restrict the distribution of preferences M. Regenwetter, B. Grofman, A.A.J. Marley and I. Tsetlin. Behavioral Social Choice. Cambridge University Press, Eric Pacuit 31

64 Let P be a probability on L(X ). Eric Pacuit 32

65 Let P be a probability on L(X ). For any pair A, B X, let P AB be the marginal pairwise ranking probability for A over B: P AB = P(P) R L(X ),ARB Eric Pacuit 32

66 Let P be a probability on L(X ). For any pair A, B X, let P AB be the marginal pairwise ranking probability for A over B: P AB = P(P) R L(X ),ARB For any triple A, B, C: P ABC = P(P) R L(X ),ARBRC Eric Pacuit 32

67 Let P be a probability on L(X ). For any pair A, B X, let P AB be the marginal pairwise ranking probability for A over B: P AB = P(P) R L(X ),ARB For any triple A, B, C: P ABC = P(P) R L(X ),ARBRC The net probability induced by P is: NP(R) = P(R) P(R 1 ), where R L(X ) and R 1 in the inverse of R (A R 1 B iff B R A). NP ABC = P ABC P CBA Eric Pacuit 32

68 Fix three candidates {A, B, C} NP satisfies NW (C) iff NP ABC 0 and NP BAC 0 Eric Pacuit 33

69 Fix three candidates {A, B, C} NP satisfies NW (C) iff NP ABC 0 and NP BAC 0 NP satisfies NM(C) iff NP ACB 0 and NP BCA 0 ( NP ACB = 0) Eric Pacuit 33

70 Fix three candidates {A, B, C} NP satisfies NW (C) iff NP ABC 0 and NP BAC 0 NP satisfies NM(C) iff NP ACB 0 and NP BCA 0 ( NP ACB = 0) NP satisfies NB(C) iff NP CAB 0 and NP CBA 0 Eric Pacuit 33

71 Fix three candidates {A, B, C} NP satisfies NW (C) iff NP ABC 0 and NP BAC 0 NP satisfies NM(C) iff NP ACB 0 and NP BCA 0 ( NP ACB = 0) NP satisfies NB(C) iff NP CAB 0 and NP CBA 0 NP is marginally value restricted for the triple {Y, Z, W } iff there is an element C {Y, Z, W } such that NP satisfies NW (c), NB(c) or NB(c). Net value restriction holds on X if marginal net value restrictions holds on each triple. Eric Pacuit 33

72 Consider a probability P on L(X ). A weak majority preference relation and a strict majority preference relation are defined as follows: A B iff P AB P BA A B iff P AB > P BA Eric Pacuit 34

73 Theorem (Regenwetter et al.). The weak majority preference relations is transitive iff for each triple {A, B, C} X at least one of the following two conditions holds: 1. NP is marginally value restricted on {A, B, C} Eric Pacuit 35

74 Theorem (Regenwetter et al.). The weak majority preference relations is transitive iff for each triple {A, B, C} X at least one of the following two conditions holds: 1. NP is marginally value restricted on {A, B, C} and, in addition, if at least one net preference is nonzero then the following implication is true NP ABC = 0 NP BAC NP ACB (with possible relabelings). Eric Pacuit 35

75 Theorem (Regenwetter et al.). The weak majority preference relations is transitive iff for each triple {A, B, C} X at least one of the following two conditions holds: 1. NP is marginally value restricted on {A, B, C} and, in addition, if at least one net preference is nonzero then the following implication is true NP ABC = 0 NP BAC NP ACB (with possible relabelings). 2. There is a R 0 {ABC, ACB, BAC, BCA, CAB, CBA} such that R 0 has marginal net preference majority. Eric Pacuit 35

76 Theorem (Regenwetter et al.). The weak majority preference relations is transitive iff for each triple {A, B, C} X at least one of the following two conditions holds: 1. NP is marginally value restricted on {A, B, C} and, in addition, if at least one net preference is nonzero then the following implication is true NP ABC = 0 NP BAC NP ACB (with possible relabelings). 2. There is a R 0 {ABC, ACB, BAC, BCA, CAB, CBA} such that R 0 has marginal net preference majority. We say CDE has net preference majority provided: NP CDE > R {CED,DEC,DCE,ECD,EDC},NP(R )>0 NP R Eric Pacuit 35

77 Decisiveness A voter i is decisive for A over B provided for all profiles R, if A P i B, then A P F (R) B. =(P AB (x i ), P AB (y)) (Here, note that we have P AB (x i ) P BA (x i )) Eric Pacuit 36

78 F : L(X ) n ( (X ) ) Eric Pacuit 37

79 F : L(X ) n ( (X ) ) Pareto: For all profiles R L(X ) n and alternatives A, B, if A R i B for all i N, then B F (R). Eric Pacuit 37

80 F : L(X ) n ( (X ) ) Pareto: For all profiles R L(X ) n and alternatives A, B, if A R i B for all i N, then B F (R). Liberalism: For all voters i N, there exists two alternatives A i and B i such that for all profiles R L(X ) n, if A i R i B i, then B F (R). That is, i is decisive over A i and B i. Eric Pacuit 37

81 F : L(X ) n ( (X ) ) Pareto: For all profiles R L(X ) n and alternatives A, B, if A R i B for all i N, then B F (R). Liberalism: For all voters i N, there exists two alternatives A i and B i such that for all profiles R L(X ) n, if A i R i B i, then B F (R). That is, i is decisive over A i and B i. Minimal Liberalism: There are two distinct voters i and j such that there are alternatives A i, B i, A j, and B j such that i is decisive over A i and B i and j is decisive over A j and B j. Eric Pacuit 37

82 Sen s Impossibility Theorem. Suppose that X contains at least three elements. No social choice function F : L(X ) n ( (X ) ) satisfies (universal domain) and both minimal liberalism and the Pareto condition. A. Sen. The Impossibility of a Paretian Liberal. Journal of Political Economy, 78:1, pp , Eric Pacuit 38

83 Suppose that X contains at least three elements and there are elements A, B, C and D such that 1. Voter 1 is decisive over A and B: for any profile R L(X ) n, if A R 1 B, then B F (R) 2. Voter 2 is decisive over C and D: for any profile R L(X ) n, if C R 2 D, then D F (R) Two cases: 1. B C and 2. B = C. Eric Pacuit 39

84 Suppose that X = {A, B, C, D} and Voter 1 is decisive over the pair A, B Voter 2 is decisive over the pair C, D Eric Pacuit 40

85 1 2 D A B C B C D A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Pareto implies C F (R) Eric Pacuit 40

86 1 2 D A B C B C D A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Pareto implies C F (R) Eric Pacuit 40

87 1 2 D A B C B C D A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Pareto implies C F (R) Eric Pacuit 40

88 1 2 D A B C B C D A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Pareto implies C F (R) Eric Pacuit 40

89 1 2 D A B C B C D A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Pareto implies C F (R) Eric Pacuit 40

90 Suppose that X = {A, B, C} and Voter 1 is decisive over the pair A, B Voter 2 is decisive over the pair B, C Voter 1 s preference R 1 L(X ) is C R 1 A R 1 B Voter 2 s preference R 2 L(X ) is B R 2 C R 2 A Eric Pacuit 41

91 1 2 C A B B C A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for C, D implies D F (R) Pareto implies A F (R) Eric Pacuit 41

92 1 2 C A B B C A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for B, C implies C F (R) Pareto implies A F (R) Eric Pacuit 41

93 1 2 C A B B C A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for B, C implies C F (R) Pareto implies A F (R) Eric Pacuit 41

94 1 2 C A B B C A Voter 1 is decisive for A, B implies B F (R) Voter 2 is decisive for B, C implies C F (R) Pareto implies A F (R) Eric Pacuit 41

95 What is the moral? Eric Pacuit 42

96 What is the moral? It is that in a very basic sense liberal values conflict with the Pareto principle. If someone takes the Pareto principle seriously, as economists seem to do, then he has to face problems of consistency in cherishing liberal values, even very mild ones... Eric Pacuit 42

97 What is the moral? It is that in a very basic sense liberal values conflict with the Pareto principle. If someone takes the Pareto principle seriously, as economists seem to do, then he has to face problems of consistency in cherishing liberal values, even very mild ones... While the Pareto criterion has been thought to be an expression of individual liberty, it appears that in choices involving more than two alternatives it can have consequences that are, in fact, deeply illiberal. (pg. 157) A. Sen. The Impossibility of a Paretian Liberal. Journal of Political Economy, 78:1, pp , Eric Pacuit 42

98 What s the moral? Eric Pacuit 43

99 What s the moral? all(x 1 ) all(x 2 ) x 1 x 2 (P YZ (x 1 ) P YZ (x 2 )) P YZ (y), for Y, Z distinct elements of {A, B, C, D} =(x 1, x 2, y) =(P AB (x 1 ), P AB (y)) =(P CD (x 2 ), P CD (y)) are inconsistent. Eric Pacuit 43

100 What s the moral? all(x 1 ) all(x 2 ) x 1 x 2 (P YZ (x 1 ) P YZ (x 2 )) P YZ (y), for Y, Z distinct elements of {A, B, C, D} =(x 1, x 2, y) =(P AB (x 1 ), P AB (y)) =(P CD (x 2 ), P CD (y)) are inconsistent. Eric Pacuit 43

101 Characterizing Majority Rule When there are only two candidates A and B, then all voting methods give the same results Eric Pacuit 44

102 Characterizing Majority Rule When there are only two candidates A and B, then all voting methods give the same results Majority Rule: A is ranked above (below) B if more (fewer) voters rank A above B than B above A, otherwise A and B are tied. Eric Pacuit 44

103 Characterizing Majority Rule When there are only two candidates A and B, then all voting methods give the same results Majority Rule: A is ranked above (below) B if more (fewer) voters rank A above B than B above A, otherwise A and B are tied. When there are only two options, can we argue that majority rule is the best procedure? K. May. A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision. Econometrica, Vol. 20 (1952). Eric Pacuit 44

104 May s Theorem: Details Let N = {1, 2, 3,..., n} be the set of n voters and X = {A, B} the set of candidates. Social Welfare Function: F : O(X ) n O(X ), where O(X ) is the set of orderings over X (there are only three possibilities: A P B, A I B, or B P A) A P B F Maj (R) = A I B B P A if N R (A P B) > N R (B P A) if N R (A P B) = N R (B P A) if N R (B P A) > N R (A P B) Eric Pacuit 45

105 May s Theorem: Details Let N = {1, 2, 3,..., n} be the set of n voters and X = {A, B} the set of candidates. Social Welfare Function: F : {1, 0, 1} n {1, 0, 1}, as df asdf add fasdfdfs where 1 means A P B, 0 means A I B, and 1 means B P A 1 if N R (1) > N R ( 1) F Maj (v) = 0 if N R (1) = N R ( 1) 1 if N R ( 1) > N R (1) Eric Pacuit 45

106 May s Theorem: Details Unanimity: unanimously supported alternatives must be the social outcome. If v = (v 1,..., v n ) with for all i N, v i = x then F (v) = x (for x {1, 0, 1}). Anonymity: all voters should be treated equally. F (v 1, v 2,..., v n ) = F (v π(1), v π(2),..., v π(n) ) where π is a permutation of the voters. Neutrality: all candidates should be treated equally. F ( v) = F (v) where v = ( v 1,..., v n ). Eric Pacuit 45

107 May s Theorem: Details Unanimity: unanimously supported alternatives must be the social outcome. If v = (v 1,..., v n ) with for all i N, v i = x then F (v) = x (for x {1, 0, 1}). Anonymity: all voters should be treated equally. F (v 1,..., v n ) = F (v π(1), v π(2),..., v π(n) ) where v i {1, 0, 1} and π is a permutation of the voters. Neutrality: all candidates should be treated equally. F ( v) = F (v) where v = ( v 1,..., v n ). Eric Pacuit 45

108 May s Theorem: Details Unanimity: unanimously supported alternatives must be the social outcome. If v = (v 1,..., v n ) with for all i N, v i = x then F (v) = x (for x {1, 0, 1}). Anonymity: all voters should be treated equally. F (v 1,..., v n ) = F (v π(1), v π(2),..., v π(n) ) where v i {1, 0, 1} and π is a permutation of the voters. Neutrality: all candidates should be treated equally. F ( v) = F (v) where v = ( v 1,..., v n ). Eric Pacuit 45

109 May s Theorem: Details Unanimity: unanimously supported alternatives must be the social outcome. If v = (v 1,..., v n ) with for all i N, v i = x then F (v) = x (for x {1, 0, 1}). Anonymity: all voters should be treated equally. F (v 1,..., v n ) = F (v π(1), v π(2),..., v π(n) ) where v i {1, 0, 1} and π is a permutation of the voters. Neutrality: all candidates should be treated equally. F ( v) = F (v) where v = ( v 1,..., v n ). Eric Pacuit 45

110 May s Theorem: Details Positive Responsiveness (Monotonicity): unidirectional shift in the voters opinions should help the alternative toward which this shift occurs If F (v) = 0 or F (v) = 1 and v v, then F (v ) = 1 where v v means for all i N v i v i and there is some i N with v i < v i. Eric Pacuit 45

111 May s Theorem: Details May s Theorem (1952) A social decision method F satisfies unanimity, neutrality, anonymity and positive responsiveness iff F is majority rule. Eric Pacuit 45

112 Proof Idea If (1, 0, 1) is assigned 1 or 1 then Eric Pacuit 46

113 Proof Idea If (1, 0, 1) is assigned 1 or 1 then Anonymity implies ( 1, 0, 1) is assigned 1 or 1 Eric Pacuit 46

114 Proof Idea If (1, 0, 1) is assigned 1 or 1 then Anonymity implies ( 1, 0, 1) is assigned 1 or 1 Neutrality implies (1, 0, 1) is assigned 1 or 1 Contradiction. Eric Pacuit 46

115 Proof Idea If (1, 1, 1) is assigned 0 or 1 then Eric Pacuit 47

116 Proof Idea If (1, 1, 1) is assigned 0 or 1 then Neutrality implies ( 1, 1, 1) is assigned 0 or 1 Eric Pacuit 47

117 Proof Idea If (1, 1, 1) is assigned 0 or 1 then Neutrality implies ( 1, 1, 1) is assigned 0 or 1 Anonymity implies (1, 1, 1) is assigned 0 or 1 Eric Pacuit 47

118 Proof Idea If (1, 1, 1) is assigned 0 or 1 then Neutrality implies ( 1, 1, 1) is assigned 0 or 1 Anonymity implies (1, 1, 1) is assigned 0 or 1 Positive Responsiveness implies (1, 0, 1) is assigned 1 Eric Pacuit 47

119 Proof Idea If (1, 1, 1) is assigned 0 or 1 then Neutrality implies ( 1, 1, 1) is assigned 0 or 1 Anonymity implies (1, 1, 1) is assigned 0 or 1 Positive Responsiveness implies (1, 0, 1) is assigned 1 Positive Responsiveness implies (1, 1, 1) is assigned 1 Contradiction. Eric Pacuit 47

120 Other characterizations G. Asan and R. Sanver. Another Characterization of the Majority Rule. Economics Letters, 75 (3), , E. Maskin. Majority rule, social welfare functions and game forms. in Choice, Welfare and Development, The Clarendon Press, pgs , G. Woeginger. A new characterization of the majority rule. Economic Letters, 81, pgs , Eric Pacuit 48

121 May s Theorem in Dependence Logic Eric Pacuit 49

122 Let D = { 1, 0, 1} V = {x 1, x 2,..., x n, y} Profiles are substitutions: s : V D An election scenario is a set of profiles (i.e., a team). Let T (x) mean A and B are tied for x, A(x) mean x ranks A above B and B(x) mean x ranks B above A. Eric Pacuit 50

123 Function: =(x 1,..., x n, y) Unanimity: The conjunction of ( n i=1 x i = 0) y = 0 ( n i=1 x i = 1) y = 1 ( n i=1 x i = 1) y = 1 Eric Pacuit 51

124 Function: =(x 1,..., x n, y) Unanimity: The conjunction of ( n i=1 x i = 0) y = 0 ( n i=1 x i = 1) y = 1 ( n i=1 x i = 1) y = 1 Neutrality: For all s, s X, if s (x i ) = s(x i ) for all i = 1,..., n, then s (y) = s(y) Eric Pacuit 51

125 Function: =(x 1,..., x n, y) Unanimity: The conjunction of ( n i=1 x i = 0) y = 0 ( n i=1 x i = 1) y = 1 ( n i=1 x i = 1) y = 1 Neutrality: For all s, s X, if s (x i ) = s(x i ) for all i = 1,..., n, then s (y) = s(y) Positive Responsiveness: For all s, s X, if n i=1 s(x i) s (x i ) and n i=1 s(x i) < s(x i ), then (s(y) = 0 or s(y) = 1) implies s (y) = 1. Eric Pacuit 51

126 Neutrality and Positive Responsiveness are generalized version of dependency conditions: Eric Pacuit 52

127 Neutrality and Positive Responsiveness are generalized version of dependency conditions: Suppose that R, R are relations on the domain M, M, X = [R, R ](x, y) iff for all s, s X, if R(s(x), s (x)) then R(s(y), s (y)) [=, =](x, y) is =(x, y) Eric Pacuit 52

128 R Neut (z 1,..., z n, z 1,..., z n) is n i=1 ((T (z i) T (z i )) (A(z i) B(z i )) (B(z i) A(z i )). R Neut (z, z ) is (T (z) T (z )) (A(z) B(z )) (B(z) A(z )) Eric Pacuit 53

129 R Neut (z 1,..., z n, z 1,..., z n) is n i=1 ((T (z i) T (z i )) (A(z i) B(z i )) (B(z i) A(z i )). R Neut (z, z ) is (T (z) T (z )) (A(z) B(z )) (B(z) A(z )) R Mon (z 1, z 2,..., z n, z 1, z 2,..., z n) is n i=1 z i z i n i=1 (z i < z i ) R Mon (z) is (T (z) A(z)) A(z) Eric Pacuit 53

130 Arrow s Theorem K. Arrow. Social Choice and Individual Values. John Wiley & Sons, Eric Pacuit 54

131 Arrovian Dictator A voter d N is a dictator if society strictly prefers A over B whenever d strictly prefers A over B. Eric Pacuit 55

132 Arrovian Dictator A voter d N is a dictator if society strictly prefers A over B whenever d strictly prefers A over B. There is a d N such that for each profile R = (R 1,..., R d,..., R n ), if A P d B, then A P F (R) B Eric Pacuit 55

133 Arrovian Dictator A voter d N is a dictator if society strictly prefers A over B whenever d strictly prefers A over B. There is a d N such that for each profile R = (R 1,..., R d,..., R n ), if A P d B, then A P F (R) B (Moreau s Zelig example) M. Morreau. Arrow s Theorem. Stanford Encyclopedia of Philosophy, forthcoming, Eric Pacuit 55

134 Arrow s Theorem There are at least three candidates and finitely many voters i all(x i) i ({x j j i} x i ) ( i P AB(x i )) P AB (y) (for all pairs A, B) =(R AB (x 1 ),..., R AB (x n ), R AB (y)) (for all pairs A, B) There exists a d such that =(P AB (x d ), P AB (y)) for all A, B Theorem (Arrow, 1951). Suppose that there are at least three candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship. Eric Pacuit 56

135 Arrow s Theorem D. Campbell and J. Kelly. Impossibility Theorems in the Arrovian Framework. Handbook of Social Choice and Welfare Volume 1, pgs , J. Geanakoplos. Three Brief Proofs of Arrow s Impossibility Theorem. Economic Theory, 26, P. Suppes. The pre-history of Kenneth Arrow s social choice and individual values. Social Choice and Welfare, 25, pgs , Eric Pacuit 57

136 Weakening IIA Given a profile and a set of candidates S X, let R S denote the restriction of the profile to candidates in S. Eric Pacuit 58

137 Weakening IIA Given a profile and a set of candidates S X, let R S denote the restriction of the profile to candidates in S. Binary Independence: For all profiles R, R and candidates A, B X : If R {A,B} = R {A,B}, then F (R) {A,B} = F (R ) {A,B} Eric Pacuit 58

138 Weakening IIA Given a profile and a set of candidates S X, let R S denote the restriction of the profile to candidates in S. Binary Independence: For all profiles R, R and candidates A, B X : If R {A,B} = R {A,B}, then F (R) {A,B} = F (R ) {A,B} m-ary Independence: For all profiles R, R and for all S X with S = m: If R S = R S, then F (R) S = F (R ) S Eric Pacuit 58

139 Weakening IIA Theorem. (Blau) Suppose that m = 2,..., X 1. If a social welfare function F satisfies m-ary independence, then it also satisfies binary independence. J. Blau. Arrow s theorem with weak independence. Economica, 38, pgs , S. Cato. Independence of Irrelevant Alternatives Revisited. Theory and Decision, Eric Pacuit 59

140 Let S (X ). F is S-independent if for all profiles R, R, and all S S, if R S = R S, then F (R) S = F (R ) S S (X ) is connected provided for all x, y X there is a finite set S 1,..., S k S such that {x, y} = S j j {1,...,k} Eric Pacuit 60

141 Let S (X ). F is S-independent if for all profiles R, R, and all S S, if R S = R S, then F (R) S = F (R ) S S (X ) is connected provided for all x, y X there is a finite set S 1,..., S k S such that {x, y} = S j j {1,...,k} Theorem (Sato). (i) Suppose that S (X ) is connected. If a collective choice rule F satisfies S-independence, then it also satisfies binary independence. (ii) Suppose that S (X ) is not connected. Then, there exists a social welfare function F that satisfies S-independence and weak Pareto but does not satisfy binary independence. Eric Pacuit 60

142 Arrow s Theorem Theorem (Arrow, 1951). Suppose that there are at least three candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship. Eric Pacuit 61

143 Weakening Unanimity F : D O(X ) Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then A P F (R) B Inversely Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then B P F (R) A Eric Pacuit 62

144 Weakening Unanimity F : D O(X ) Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then A P F (R) B Inversely Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then B P F (R) A Null: For all A, B X and for all R D: A I F (R) B Eric Pacuit 62

145 Weakening Unanimity F : D O(X ) Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then A P F (R) B Inversely Dictatorial: there is a d N such that for all A, B X and all profiles R: if A P d B, then B P F (R) A Null: For all A, B X and for all R D: A I F (R) B Non-Imposition: For all A, B X, there is a R D such that A F (R) B Eric Pacuit 62

146 Weakening Unanimity Theorem (Wilson) Suppose that N is a finite set. If a social welfare function satisfies universal domain, independence of irrelevant alternatives and non-imposition, then it is either null, dictatorial or inversely dictatorial. R. Wilson. Social Choice Theory without the Pareto principle. Journal of Economic Theory, 5, pgs , Y. Murakami. Logic and Social Choice. Routledge, S. Cato. Social choice without the Pareto principle: A comprehensive analysis. Social Choice and Welfare, 39, pgs , Eric Pacuit 63

147 Arrow s Theorem Theorem (Arrow, 1951). Suppose that there are at least three candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship. Eric Pacuit 64

148 Social Choice Functions F : D (X ) Resolute: For all profiles R D, F (R) = 1 Non-Imposed: For all candidates A X, there is a R D such that F (R) = {A}. Monotonicity: For all profiles R and R, if A F (R) and for all i N, N R (A P i B) N R (A P i B) for all B X {A}, then A F (R ). Dictator: A voter d is a dictator if for all R D, F (R) = {A}, where A is d s top choice. Eric Pacuit 65

149 Social Choice Functions Muller-Satterthwaite Theorem. Suppose that there are more than three alternatives and finitely many voters. Every resolute social choice function F : L(X ) n X that is monotonic and non-imposed is a dictatorship. E. Muller and M.A. Satterthwaite. The Equivalence of Strong Positive Association and Strategy-Proofness. Journal of Economic Theory, 14(2), pgs , Eric Pacuit 66

150 Arrow s Theorem Theorem (Arrow, 1951). Suppose that there are at least three candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship. Eric Pacuit 67

151 Conclusions Many generalizations and variants of Arrow s Theorem (e.g., infinite voters), many characterization results A number of different logics have been used to formalize various aspects of Arrow s Theorem and related impossibility results Dependence logic and social choice: Social choice may suggest new dependence atoms. DL is good match for Social Choice: it focuses on reasoning about dependence (i.e., IIA) and independence (i.e., freedom of choice). Eric Pacuit 68