Dependence and Independence
in Social Choice Theory
Eric Pacuit
Department of Philosophy
University of Maryland, College Park
pacuit.org
[email protected]
March 4, 2014
Eric Pacuit
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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
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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)
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Notation: Candidates, Voters
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N is a finite set of voters (assume that N = {1, 2, 3, . . . , n})
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X is a (typically finite) set of alternatives: e.g., candidates,
restaurants, social states, etc.
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Notation: Preferences
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A relation on X is a linear order if it is transitive, irreflexive,
and complete (hence, acyclic)
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L(X ) is the set of all linear orders over the set X
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O(X ) is the set of all reflexive, transitive and complete
relations over the set X
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Given R ∈ O(X ), let the strict subrelation be
PR = {(x, y ) | x R y and y 6R x} and the indifference
subrelation be IR = {(x, y ) | x R y and y R x}
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Notation: Profiles
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A profile for the set of voters N is a sequence of (linear)
orders over X , denoted R = (R1 , . . . , Rn ).
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L(X )n is the set of all profiles for n voters (similarly for
O(X )n )
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For a profile R = (R1 , . . . , Rn ) ∈ O(X )n , let
NR (A P B) = {i | A Pi B} be the set of voters that rank A
above B (similarly for NR (A I B) and NR (B P A))
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Group Decision Making Methods
F :D→R
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Group Decision Making Methods
F :D→R
Comments
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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 )
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The group decision is completely determined by the voters’
opinions: every profile R ∈ D is associated with exactly one
“group decision”.
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Group Decision Making Methods
F :D→R
Variants
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Social Welfare Functions: R = L(X ) or R = O(X ).
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Social Choice Function: R = ℘(X ) − ∅, where ℘(X ) is the set
of all subsets of X .
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Group Decision Making Methods
F :D→R
Variants
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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
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D = U(X )n and R = U(X ) where U(X ) is the set of utility
functions
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D = ∆(X )n and R = ∆(X ) where ∆(X ) is the set of
probability measures on X (pi (A) is the probability that i
would choose A if i could act as a dictator.)
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The setup (from Jouko’s various talks)
The set of variables is V = {x1 , x2 , . . . , xn } ∪ {y } (each xi 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
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The setup (from Jouko’s various talks)
The set of variables is V = {x1 , x2 , . . . , xn } ∪ {y } (each xi 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
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The setup (from Jouko’s various talks)
The set of variables is V = {x1 , x2 , . . . , xn } ∪ {y } (each xi 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
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The setup (from Jouko’s various talks)
The set of variables is V = {x1 , x2 , . . . , xn } ∪ {y } (each xi 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
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The setup (from Jouko’s various talks)
(Atomic) formulas describe the voters’ preferences and the group
decision.
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PAB (xi ) is true if s(xi ) ranks A strictly above B,
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RAB (xi ) is true if A is weakly preferred to B (i.e., A s(xi ) B)
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PAB (y ) is true when the group ranks A strictly above B
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CA (y ) means that A was chosen by the group,
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CA (y ) means that the group does not choose A
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···
An underlying theory describing (individual and group) rationality
assumptions,
E.g., transitivity: PAB (x) ∧ PBC (x) ⊃ PAC (x)
Resolute rules, CA (y ) ⊃ CB (y ),
PAB (x) ⊃ (RAB (x) ∧ ¬RAB (x)), etc.
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Desiderata 1
The voters’ preferences should completely determine the group
decision.
=(x1 , x2 , . . . , xn , y )
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Desiderata 2
The group decision should depend in the right way on the voters’
opinions.
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Desiderata 2
The group decision should depend in the right way on the voters’
opinions.
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Single-profile conditions
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Multi-profile conditions
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Variable domain conditions
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Single-Profile Conditions
Condorcet: Elect the Condorcet winner whenever it exists.
A Condorcet candidate in a profile R is a candidate A such that
|NR (A P B)| > |NR (B P A)| for all other candidates B ∈ X
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Single-Profile Conditions
Condorcet: Elect the Condorcet winner whenever it exists.
A Condorcet candidate in a profile R is a candidate A such that
|NR (A P B)| > |NR (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 Pi B, then
A PF (R) B (recall that Pi is the strict subrelation of Ri )
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Single-Profile Conditions
Condorcet: Elect the Condorcet winner whenever it exists.
A Condorcet candidate in a profile R is a candidate A such that
|NR (A P B)| > |NR (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 Pi B, then
A PF (R) B (recall that Pi is the strict subrelation of Ri )
(
V
i∈N
PAB (xi )) ⊃ PAB (y )
(needs to be stated for every pair A, B of candidates)
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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, R0 if for all i ∈ N, Ri |{A,B} = R0 i |{A,B} , then
F (R)|{A,B} = F (R0 )|{A,B}
=(RAB (x1 ), . . . , RAB (xn ), RAB (y ))
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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 = (R1 , . . . , Rn ), R0 , if R0 = (Rπ(1) , . . . , Rπ(n) ), then
F (R) = F (R0 ). (not the same as =(xπ(1) , . . . , xπ(n) , y ))
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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 = (R1 , . . . , Rn ), R0 , if R0 = (Rπ(1) , . . . , Rπ(n) ), then
F (R) = F (R0 ). (not the same as =(xπ(1) , . . . , xπ(n) , y ))
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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 = (R1 , . . . , Rn ), R0 , if R0 = (Rπ(1) , . . . , Rπ(n) ), then
F (R) = F (R0 ). (not the same as =(xπ(1) , . . . , xπ(n) , y ))
Neutrality: The outcome does not depend on the names of the
candidates.
Monotonicity: More support should never hurt a candidate.
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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. 423-424)
I. Lilttle. Social Choice and Individual Values. Journal of Political Economy,
60:5, pgs. 422 - 432, 1952.
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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.
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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 ΠP be the set of all profiles based on P.
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Variable Population Model
Two profiles π : V → P and π 0 : V 0 → P are disjoint if
V ∩ V0 = ∅
If π : V → P and π 0 : V 0 → P are disjoint, then
(π + π 0 ) : (V ∪ V 0 ) → P is the profile where for all i ∈ V ∪ V 0 ,
(
π(i)
if i ∈ V
(π + π 0 )(i) =
0
π (i) if i ∈ V 0
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Consistency: If F (π) ∩ F (π 0 ) 6= ∅, then F (π + π 0 ) = F (π) ∩ F (π 0 )
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What are the relationships between these principles? Is there a
procedure that satisfies all of them?
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What are the relationships between these principles? Is there a
procedure that satisfies all of them?
A few observations:
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Condorcet winners may not exist.
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No positional scoring method satisfies the Condorcet Principle.
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The Condorcet and Participation principles cannot be jointly
satisfied (Moulin’s Theorem).
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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).
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Domain Conditions
Universal Domain: The domain of the social welfare (choice)
function is D = L(X )n (or O(X )n )
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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)
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Domain Conditions
No Restrictions:
S |= all(xi ) if for all R ∈ L(X ), there is an s ∈ S such that
s(xi ) = R
S |= triple(xi ) if for all P ∈ L({A, B, C }) there is a R 0 ∈ O(X ) and
s ∈ X such that R 0 |{A,B,C } = P and s(xi ) = R 0
Vn
i=1 all(xi )
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Domain Conditions
No Restrictions:
S |= all(xi ) if for all R ∈ L(X ), there is an s ∈ S such that
s(xi ) = R
S |= triple(xi ) if for all P ∈ L({A, B, C }) there is a R 0 ∈ O(X ) and
s ∈ X such that R 0 |{A,B,C } = P and s(xi ) = R 0
Vn
i=1 all(xi )
Independence:
{xj | j 6= i} ⊥ xi
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Domain Restrictions
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Single-Peaked preferences
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Sen’s Value Restriction
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Assumptions about the distribution of preferences
W. Gaertner. Domain Conditions in Social Choice Theory. Cambridge University
Press, 2001.
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D. Black. On the rationale of group decision-making. Journal of Political
Economy, 56:1, pgs. 23 - 34, 1948.
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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.
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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.
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D. Miller. Deliberative Democracy and Social Choice. Political Studies, 40, pgs.
54 - 67, 1992.
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. 80 - 95, 2013.
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Sen’s Value Restriciton
A. Sen. A Possibility Theorem on Majority Decisions. Econometrica 34, 1966,
pgs. 491 - 499.
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Sen’s Theorem
Assume n voters (n is odd).
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Sen’s Theorem
Assume n voters (n is odd).
Triplewise value-restriction: For every triple of distinct
candidates A, B, C there exists an xi ∈ {A, B, C } and r ∈ {1, 2, 3}
such that no voter ranks xi has her r th preference among A, B, C .
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Sen’s Theorem
Assume n voters (n is odd).
Triplewise value-restriction: For every triple of distinct
candidates A, B, C there exists an xi ∈ {A, B, C } and r ∈ {1, 2, 3}
such that no voter ranks xi has her r th preference among A, B, C .
Theorem (Sen, 1966). For every profile satisfying triplewise
value-restriction, pairwise majority voting generates a transitive
group preference ordering.
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Restrict the distribution of preferences
M. Regenwetter, B. Grofman, A.A.J. Marley and I. Tsetlin. Behavioral Social
Choice. Cambridge University Press, 2006.
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Let P be a probability on L(X ).
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Let P be a probability on L(X ). For any pair A, B ∈ X , let PAB be
the marginal pairwise ranking probability for A over B:
X
PAB =
P(P)
R∈L(X ),ARB
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Let P be a probability on L(X ). For any pair A, B ∈ X , let PAB be
the marginal pairwise ranking probability for A over B:
X
PAB =
P(P)
R∈L(X ),ARB
For any triple A, B, C :
PABC =
X
P(P)
R∈L(X ),ARBRC
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Let P be a probability on L(X ). For any pair A, B ∈ X , let PAB be
the marginal pairwise ranking probability for A over B:
X
PAB =
P(P)
R∈L(X ),ARB
For any triple A, B, C :
PABC =
X
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).
NPABC = PABC − PCBA
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Fix three candidates {A, B, C }
NP satisfies NW (C ) iff NPABC ≤ 0 and NPBAC ≤ 0
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Fix three candidates {A, B, C }
NP satisfies NW (C ) iff NPABC ≤ 0 and NPBAC ≤ 0
NP satisfies NM(C ) iff NPAC B ≤ 0 and NPBC A ≤ 0
(⇔ NPACB = 0)
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Fix three candidates {A, B, C }
NP satisfies NW (C ) iff NPABC ≤ 0 and NPBAC ≤ 0
NP satisfies NM(C ) iff NPAC B ≤ 0 and NPBC A ≤ 0
(⇔ NPACB = 0)
NP satisfies NB(C ) iff NPC AB ≤ 0 and NPC BA ≤ 0
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Fix three candidates {A, B, C }
NP satisfies NW (C ) iff NPABC ≤ 0 and NPBAC ≤ 0
NP satisfies NM(C ) iff NPAC B ≤ 0 and NPBC A ≤ 0
(⇔ NPACB = 0)
NP satisfies NB(C ) iff NPC AB ≤ 0 and NPC BA ≤ 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.
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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 PAB ≥ PBA
A B iff PAB > PBA
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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 }
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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 NPABC = 0 ⇒ NPBAC 6= NPACB
(with possible relabelings).
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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 NPABC = 0 ⇒ NPBAC 6= NPACB
(with possible relabelings).
2. There is a R0 ∈ {ABC , ACB, BAC , BCA, CAB, CBA} such
that R0 has marginal net preference majority.
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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 NPABC = 0 ⇒ NPBAC 6= NPACB
(with possible relabelings).
2. There is a R0 ∈ {ABC , ACB, BAC , BCA, CAB, CBA} such
that R0 has marginal net preference majority.
We say CDE has net preference majority provided:
NPCDE >
X
NPR 0
R 0 ∈{CED,DEC ,DCE ,ECD,EDC },NP(R 0 )>0
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Decisiveness
A voter i is decisive for A over B provided for all profiles R, if
A Pi B, then A PF (R) B.
=(PAB (xi ), PAB (y ))
(Here, note that we have ¬PAB (xi ) ↔ PBA (xi ))
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F : L(X )n → (℘(X ) − ∅)
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F : L(X )n → (℘(X ) − ∅)
Pareto: For all profiles R ∈ L(X )n and alternatives A, B, if A Ri B
for all i ∈ N, then B 6∈ F (R).
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F : L(X )n → (℘(X ) − ∅)
Pareto: For all profiles R ∈ L(X )n and alternatives A, B, if A Ri B
for all i ∈ N, then B 6∈ F (R).
Liberalism: For all voters i ∈ N, there exists two alternatives Ai
and Bi such that for all profiles R ∈ L(X )n , if Ai Ri Bi , then
B 6∈ F (R). That is, i is decisive over Ai and Bi .
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F : L(X )n → (℘(X ) − ∅)
Pareto: For all profiles R ∈ L(X )n and alternatives A, B, if A Ri B
for all i ∈ N, then B 6∈ F (R).
Liberalism: For all voters i ∈ N, there exists two alternatives Ai
and Bi such that for all profiles R ∈ L(X )n , if Ai Ri Bi , then
B 6∈ F (R). That is, i is decisive over Ai and Bi .
Minimal Liberalism: There are two distinct voters i and j such
that there are alternatives Ai , Bi , Aj , and Bj such that i is decisive
over Ai and Bi and j is decisive over Aj and Bj .
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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. 152 - 157, 1970.
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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 R1 B, then B 6∈ F (R)
2. Voter 2 is decisive over C and D: for any profile R ∈ L(X )n , if
C R2 D, then D 6∈ F (R)
Two cases: 1. B 6= C and 2. B = C .
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Suppose that X = {A, B, C , D} and
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Voter 1 is decisive over the pair A, B
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Voter 2 is decisive over the pair C , D
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Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Pareto implies C 6∈ F (R)
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B
A
C
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D
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Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Pareto implies C 6∈ F (R)
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C
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Pareto implies C 6∈ F (R)
Eric Pacuit
40
1
2
D
B
A
C
B
D
C
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Pareto implies C 6∈ F (R)
Eric Pacuit
40
1
2
D
B
A
C
B
D
C
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Pareto implies C 6∈ F (R)
Eric Pacuit
40
Suppose that X = {A, B, C } and
Eric Pacuit
I
Voter 1 is decisive over the pair A, B
I
Voter 2 is decisive over the pair B, C
I
Voter 1’s preference R1 ∈ L(X ) is C R1 A R1 B
I
Voter 2’s preference R2 ∈ L(X ) is B R2 C R2 A
41
1
2
C
B
A
C
B
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for C , D implies D 6∈ F (R)
Pareto implies A 6∈ F (R)
Eric Pacuit
41
1
2
C
B
A
C
B
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for B, C implies C 6∈ F (R)
Pareto implies A 6∈ F (R)
Eric Pacuit
41
1
2
C
B
A
C
B
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for B, C implies C 6∈ F (R)
Pareto implies A 6∈ F (R)
Eric Pacuit
41
1
2
C
B
A
C
B
A
Voter 1 is decisive for A, B implies B 6∈ F (R)
Voter 2 is decisive for B, C implies C 6∈ F (R)
Pareto implies A 6∈ F (R)
Eric Pacuit
41
“What is the moral?
Eric Pacuit
42
“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
“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. 152 - 157, 1970.
Eric Pacuit
42
What’s the moral?
Eric Pacuit
43
What’s the moral?
I
all(x1 ) ∧ all(x2 )
I
x1 ⊥ x2
I
(PYZ (x1 ) ∧ PYZ (x2 )) ⊃ PYZ (y ), for Y , Z distinct elements of
{A, B, C , D}
I
=(x1 , x2 , y )
I
=(PAB (x1 ), PAB (y ))
I
=(PCD (x2 ), PCD (y ))
are inconsistent.
Eric Pacuit
43
What’s the moral?
I
all(x1 ) ∧ all(x2 )
I
x1 ⊥ x2
I
(PYZ (x1 ) ∧ PYZ (x2 )) ⊃ PYZ (y ), for Y , Z distinct elements of
{A, B, C , D}
I
=(x1 , x2 , y )
I
=(PAB (x1 ), PAB (y ))
I
=(PCD (x2 ), PCD (y ))
are inconsistent.
Eric Pacuit
43
Characterizing Majority Rule
When there are only two candidates A and B, then all voting
methods give the same results
Eric Pacuit
44
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
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
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
FMaj (R) = A I B


BPA
Eric Pacuit
if |NR (A P B)| > |NR (B P A)|
if |NR (A P B)| = |NR (B P A)|
if |NR (B P A)| > |NR (A P B)|
45
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
FMaj (v) = 0


−1
Eric Pacuit
if |NR (1)| > |NR (−1)|
if |NR (1)| = |NR (−1)|
if |NR (−1)| > |NR (1)|
45
May’s Theorem: Details
I
Unanimity: unanimously supported alternatives must be the
social outcome.
If v = (v1 , . . . , vn ) with for all i ∈ N, vi = x then F (v) = x
(for x ∈ {1, 0, −1}).
I
Anonymity: all voters should be treated equally.
F (v1 , v2 , . . . , vn ) = F (vπ(1) , vπ(2) , . . . , vπ(n) ) where π is a
permutation of the voters.
I
Neutrality: all candidates should be treated equally.
F (−v ) = −F (v ) where −v = (−v1 , . . . , −vn ).
Eric Pacuit
45
May’s Theorem: Details
I
Unanimity: unanimously supported alternatives must be the
social outcome.
If v = (v1 , . . . , vn ) with for all i ∈ N, vi = x then F (v) = x
(for x ∈ {1, 0, −1}).
I
Anonymity: all voters should be treated equally.
F (v1 , . . . , vn ) = F (vπ(1) , vπ(2) , . . . , vπ(n) ) where vi
{1, 0, −1} and π is a permutation of the voters.
I
∈
Neutrality: all candidates should be treated equally.
F (−v ) = −F (v ) where −v = (−v1 , . . . , −vn ).
Eric Pacuit
45
May’s Theorem: Details
I
Unanimity: unanimously supported alternatives must be the
social outcome.
If v = (v1 , . . . , vn ) with for all i ∈ N, vi = x then F (v) = x
(for x ∈ {1, 0, −1}).
I
Anonymity: all voters should be treated equally.
F (v1 , . . . , vn ) = F (vπ(1) , vπ(2) , . . . , vπ(n) ) where vi
{1, 0, −1} and π is a permutation of the voters.
I
∈
Neutrality: all candidates should be treated equally.
F (−v ) = −F (v ) where −v = (−v1 , . . . , −vn ).
Eric Pacuit
45
May’s Theorem: Details
I
Unanimity: unanimously supported alternatives must be the
social outcome.
If v = (v1 , . . . , vn ) with for all i ∈ N, vi = x then F (v) = x
(for x ∈ {1, 0, −1}).
I
Anonymity: all voters should be treated equally.
F (v1 , . . . , vn ) = F (vπ(1) , vπ(2) , . . . , vπ(n) ) where vi
{1, 0, −1} and π is a permutation of the voters.
I
∈
Neutrality: all candidates should be treated equally.
F (−v) = −F (v) where −v = (−v1 , . . . , −vn ).
Eric Pacuit
45
May’s Theorem: Details
I
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 ≺ v0 , then F (v0 ) = 1
where v ≺ v0 means for all i ∈ N vi ≤ vi0 and there is some
i ∈ N with vi < vi0 .
Eric Pacuit
45
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
Proof Idea
If (1, 0, −1) is assigned 1 or −1 then
Eric Pacuit
46
Proof Idea
If (1, 0, −1) is assigned 1 or −1 then
X Anonymity implies (−1, 0, 1) is assigned 1 or −1
Eric Pacuit
46
Proof Idea
If (1, 0, −1) is assigned 1 or −1 then
X Anonymity implies (−1, 0, 1) is assigned 1 or −1
X Neutrality implies (1, 0, −1) is assigned −1 or 1
Contradiction.
Eric Pacuit
46
Proof Idea
If (1, 1, −1) is assigned 0 or −1 then
Eric Pacuit
47
Proof Idea
If (1, 1, −1) is assigned 0 or −1 then
X Neutrality implies (−1, −1, 1) is assigned 0 or 1
Eric Pacuit
47
Proof Idea
If (1, 1, −1) is assigned 0 or −1 then
X Neutrality implies (−1, −1, 1) is assigned 0 or 1
X Anonymity implies (1, −1, −1) is assigned 0 or 1
Eric Pacuit
47
Proof Idea
If (1, 1, −1) is assigned 0 or −1 then
X Neutrality implies (−1, −1, 1) is assigned 0 or 1
X Anonymity implies (1, −1, −1) is assigned 0 or 1
X Positive Responsiveness implies (1, 0, −1) is assigned 1
Eric Pacuit
47
Proof Idea
If (1, 1, −1) is assigned 0 or −1 then
X Neutrality implies (−1, −1, 1) is assigned 0 or 1
X Anonymity implies (1, −1, −1) is assigned 0 or 1
X Positive Responsiveness implies (1, 0, −1) is assigned 1
X Positive Responsiveness implies (1, 1, −1) is assigned 1
Contradiction.
Eric Pacuit
47
Other characterizations
G. Asan and R. Sanver. Another Characterization of the Majority Rule.
Economics Letters, 75 (3), 409-413, 2002.
E. Maskin. Majority rule, social welfare functions and game forms. in
Choice, Welfare and Development, The Clarendon Press, pgs. 100 - 109,
1995.
G. Woeginger. A new characterization of the majority rule. Economic
Letters, 81, pgs. 89 - 94, 2003.
Eric Pacuit
48
May’s Theorem in Dependence Logic
Eric Pacuit
49
Let D = {−1, 0, 1}
V = {x1 , x2 , . . . , xn , 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
Function: =(x1 , . . . , xn , y )
Unanimity: The conjunction of
V
I ( n xi = 0) ⊃ y = 0
Vi=1
I ( n xi = 1) ⊃ y = 1
Vi=1
I ( n xi = −1) ⊃ y = −1
i=1
Eric Pacuit
51
Function: =(x1 , . . . , xn , y )
Unanimity: The conjunction of
V
I ( n xi = 0) ⊃ y = 0
Vi=1
I ( n xi = 1) ⊃ y = 1
Vi=1
I ( n xi = −1) ⊃ y = −1
i=1
Neutrality: For all s, s 0 ∈ X , if s 0 (xi ) = −s(xi ) for all i = 1, . . . , n,
then s 0 (y ) = −s(y )
Eric Pacuit
51
Function: =(x1 , . . . , xn , y )
Unanimity: The conjunction of
V
I ( n xi = 0) ⊃ y = 0
Vi=1
I ( n xi = 1) ⊃ y = 1
Vi=1
I ( n xi = −1) ⊃ y = −1
i=1
Neutrality: For all s, s 0 ∈ X , if s 0 (xi ) = −s(xi ) for all i = 1, . . . , n,
then s 0 (y ) = −s(y )
Vn
0
0
Positive
Wn Responsiveness: For all s, s ∈ X , if i=1 s(xi ) ≤ s (xi )
and i=1 s(xi ) < s(xi ), then (s(y ) = 0 or s(y ) = 1) implies
s 0 (y ) = 1.
Eric Pacuit
51
Neutrality and Positive Responsiveness are generalized version of
dependency conditions:
Eric Pacuit
52
Neutrality and Positive Responsiveness are generalized version of
dependency conditions:
Suppose that R, R 0 are relations on the domain M,
M, X |= [R, R 0 ](x, y ) iff for all s, s 0 ∈ X , if R(s(x), s 0 (x)) then
R(s(y ), s 0 (y ))
[=, =](x, y ) is =(x, y )
Eric Pacuit
52
R
(z1 , . . . , zn , z10 , . . . , zn0 ) is
VNeut
n
0
0
0
i=1 ((T (zi ) ∧ T (zi )) ∨ (A(zi ) ∧ B(zi )) ∨ (B(zi ) ∧ A(zi )).
0
RNeut
(z, z 0 ) is (T (z) ∧ T (z 0 )) ∨ (A(z) ∧ B(z 0 )) ∨ (B(z) ∧ A(z 0 ))
Eric Pacuit
53
R
(z1 , . . . , zn , z10 , . . . , zn0 ) is
VNeut
n
0
0
0
i=1 ((T (zi ) ∧ T (zi )) ∨ (A(zi ) ∧ B(zi )) ∨ (B(zi ) ∧ A(zi )).
0
RNeut
(z, z 0 ) is (T (z) ∧ T (z 0 )) ∨ (A(z) ∧ B(z 0 )) ∨ (B(z) ∧ A(z 0 ))
RMon (z1 , z2 , . . . , zn , z10 , z20 , . . . , zn0 ) is
Vn
i=1 zi
≤ zi0 ∧
Wn
i=1 (zi
< zi0 )
0
RMon
(z) is (T (z) ∨ A(z)) ⊃ A(z)
Eric Pacuit
53
Arrow’s Theorem
K. Arrow. Social Choice and Individual Values. John Wiley & Sons, 1951.
Eric Pacuit
54
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
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 = (R1 , . . . , Rd , . . . , Rn ), if A Pd B, then A PF (R) B
Eric Pacuit
55
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 = (R1 , . . . , Rd , . . . , Rn ), if A Pd B, then A PF (R) B
(Moreau’s Zelig example)
M. Morreau. Arrow’s Theorem. Stanford Encyclopedia of Philosophy, forthcoming, 2014.
Eric Pacuit
55
Arrow’s Theorem
I
There are at least three candidates and finitely many voters
V
all(xi )
Vi
({xj | j 6= i} ⊥ xi )
Vi
( i PAB (xi )) ⊃ PAB (y ) (for all pairs A, B)
I
=(RAB (x1 ), . . . , RAB (xn ), RAB (y )) (for all pairs A, B)
I
There exists a d such that =(PAB (xd ), PAB (y )) for all A, B
I
I
I
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
Arrow’s Theorem
D. Campbell and J. Kelly. Impossibility Theorems in the Arrovian Framework. Handbook of Social Choice and Welfare Volume 1, pgs. 35 - 94,
2002.
J. Geanakoplos. Three Brief Proofs of Arrow’s Impossibility Theorem. Economic Theory, 26, 2005.
P. Suppes. The pre-history of Kenneth Arrow’s social choice and individual
values. Social Choice and Welfare, 25, pgs. 319 - 326, 2005.
Eric Pacuit
57
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
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, R0 and candidates
A, B ∈ X :
If R|{A,B} = R0 |{A,B} , then F (R)|{A,B} = F (R0 )|{A,B}
Eric Pacuit
58
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, R0 and candidates
A, B ∈ X :
If R|{A,B} = R0 |{A,B} , then F (R)|{A,B} = F (R0 )|{A,B}
m-Ary Independence: For all profiles R, R0 and for all S ⊆ X
with |S| = m:
If R|S = R0 |S , then F (R)|S = F (R0 )|S
Eric Pacuit
58
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. 413 420, 1971.
S. Cato. Independence of Irrelevant Alternatives Revisited. Theory and Decision,
2013.
Eric Pacuit
59
Let S ⊆ ℘(X ). F is S-independent if for all profiles R, R0 , and all
S ∈ S,
if R|S = R0 |S, then F (R)|S = F (R0 )|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 } =
Sj
j∈{1,...,k}
Eric Pacuit
60
Let S ⊆ ℘(X ). F is S-independent if for all profiles R, R0 , and all
S ∈ S,
if R|S = R0 |S, then F (R)|S = F (R0 )|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 } =
Sj
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
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
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 Pd B, then A PF (R) B
Inversely Dictatorial: there is a d ∈ N such that for all A, B ∈ X
and all profiles R: if A Pd B, then B PF (R) A
Eric Pacuit
62
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 Pd B, then A PF (R) B
Inversely Dictatorial: there is a d ∈ N such that for all A, B ∈ X
and all profiles R: if A Pd B, then B PF (R) A
Null: For all A, B ∈ X and for all R ∈ D: A IF (R) B
Eric Pacuit
62
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 Pd B, then A PF (R) B
Inversely Dictatorial: there is a d ∈ N such that for all A, B ∈ X
and all profiles R: if A Pd B, then B PF (R) A
Null: For all A, B ∈ X and for all R ∈ D: A IF (R) B
Non-Imposition: For all A, B ∈ X , there is a R ∈ D such that
A F (R) B
Eric Pacuit
62
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. 478 - 486, 1972.
Y. Murakami. Logic and Social Choice. Routledge, 1968.
S. Cato. Social choice without the Pareto principle: A comprehensive analysis.
Social Choice and Welfare, 39, pgs. 869 - 889, 2012.
Eric Pacuit
63
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
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 R0 , if A ∈ F (R) and for all
i ∈ N, NR (A Pi B) ⊆ NR0 (A Pi0 B) for all B ∈ X − {A}, then
A ∈ F (R0 ).
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
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. 412 418, 1977.
Eric Pacuit
66
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
Conclusions
Eric Pacuit
I
Many generalizations and variants of Arrow’s Theorem (e.g.,
infinite voters), many characterization results
I
A number of different logics have been used to formalize
various aspects of Arrow’s Theorem and related impossibility
results
I
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).
68