II-8.
Voting
Outline
Dusko Pavlovic
Introduction
(Part II: Security of Economics)
Lecture 8: Social choice
Introduction
Welfare
Elections
II-8.
Voting
Dusko Pavlovic
Welfare
Introduction
Elections
Social welfare and preference aggregation
Dusko Pavlovic
Social choice and manipulability
Spring 2013
Outline
II-8.
Voting
Social choice
Dusko Pavlovic
Dusko Pavlovic
Introduction
Welfare
Introduction
Introduction
Kenneth Arrow’s Thesis (1948, 1951)
Elections
Elections
!
voting, typically used to make "political" decisions,
and
!
the market mechanism, typically used to make
"economic" decisions.’
Social choice and manipulability
II-8.
Voting
Problem of voting
Dusko Pavlovic
Kenneth Arrow’s Thesis (1948, 1951)
II-8.
Voting
Dusko Pavlovic
Introduction
Introduction
Welfare
Welfare
Elections
’. . . In the emerging democracies with mixed economic
systems, Great Britain, France, and Scandinavia, the
same two modes of making social choices prevail, though
more scope is given to the method of voting and
decisions based directly or indirectly on it and less to the
rule of the price mechanism. Elsewhere in the world, and
even in smaller social units within the democracies, social
decisions are sometimes made by single individuals or
small groups.’
Welfare
’In a capitalist democracy there are essentially two
methods by which social choices can be made:
Social welfare and preference aggregation
Social choice
II-8.
Voting
Elections
There are 11 voters and 3 candidates: a, b and c. The
voters need to elect one candidate. They have different
preferences.
Describe a method to elect the candidate which satisfies
most voters.
II-8.
Voting
Voting problem
II-8.
Voting
Voting problem
Dusko Pavlovic
Suppose the preferences are as follows:
Dusko Pavlovic
Suppose the preferences are as follows:
Introduction
voters
3
2
2
4
preference
a!b!c
a!c!b
b!c!a
c!b!a
Introduction
Welfare
Elections
!
II-8.
Voting
Voting problem
voters
preference
3
2
2
4
a!b!c
a!c!b
b!c!a
c!b!a
II-8.
Voting
Voting problem
Dusko Pavlovic
Suppose the preferences are as follows:
Introduction
voters
3
2
2
4
preference
a!b!c
a!c!b
b!c!a
c!b!a
Introduction
Welfare
Elections
voters
preference
3
2
2
4
a!b!c
a!c!b
b!c!a
c!b!a
!
If each voter casts 1 vote, then the tally is
5:4:2 for a ! c ! b.
!
If each voter casts 1 vote, then the tally is
5:4:2 for a ! c ! b.
!
If each voter casts (1,1) votes, then the tally is
9:8:5 for b ! c ! a.
!
If each voter casts (1,1) votes, then the tally is
9:8:5 for b ! c ! a.
!
If each voter casts (2,1) votes, then the tally is
12:11:10 for c ! b ! a
Outline
II-8.
Voting
Preference relation
Dusko Pavlovic
Social welfare and preference aggregation
Social choice and manipulability
Elections
II-8.
Voting
Introduction
Welfare
Elections
Welfare
Dusko Pavlovic
Introduction
Introduction
Elections
If each voter casts 1 vote, then the tally is
5:4:2 for a ! c ! b.
Dusko Pavlovic
Suppose the preferences are as follows:
Welfare
Welfare
Definition
Elections
A preference over a set S is a binary relation ! on S such
that for all X , Y , Z ∈ S holds
X ! Y ∧ Y ! Z =⇒ X ! Z
(X ! Y ∨ Y ! X ) ∧ X ! X
II-8.
Voting
Preference relation
II-8.
Voting
Recall: Utility function
Dusko Pavlovic
Definition
Dusko Pavlovic
Introduction
Introduction
Welfare
Welfare
Elections
Elections
Definition
A preference over a set S is a binary relation ! on S such
that for all X , Y , Z ∈ S holds
A utility function corresponding to a preference preorder
!⊆ S × S is a function u : S → R such that
X ! Y ∧ Y ! Z =⇒ X ! Z
u(X ) > u(Y )
(X ! Y ∨ Y ! X ) ∧ X ! X
X !Y
⇐⇒
We write x ∼ y when x ! y ∧ y ! x holds.
II-8.
Voting
Recall: Utility function
II-8.
Voting
Preference space
Dusko Pavlovic
Dusko Pavlovic
Introduction
Introduction
Welfare
Elections
Remark
Elections
The preference space over a set S is the set P of all
preference relations ! over S
When the preferences involve random events, then the
argument X in a utility function u(X ) is a random variable.
P =
II-8.
Voting
Social welfare function
Welfare
Definition
!
!⊆ S × S | X ! Y ! Z =⇒ X ! Z
"
∧ (X ! Y ∨ Y ! X )
II-8.
Voting
Social welfare function
Dusko Pavlovic
Dusko Pavlovic
Introduction
Definition
Welfare
Introduction
Definition
Welfare
Elections
For a society consisting of the players i = 1, 2, . . . n, a
social welfare function (swf) is a mapping
!−"w : Pn
→
! ,−→
1 2
n
where != -!, !, . . . , !.
Elections
For a society consisting of the players i = 1, 2, . . . n, a
social welfare function (swf) is a mapping
!−"w : Pn
P
!!"w
→
! ,−→
1 2
P
!!"w
n
where != -!, !, . . . , !.
The relation !!"w is the aggregate preference, or social
welfare| induced by the profile !∈ Pn .
II-8.
Voting
Social welfare function
II-8.
Voting
Social welfare function
Dusko Pavlovic
Dusko Pavlovic
Introduction
There are many different ways to aggregate preferences.
Welfare
Introduction
There are many different ways to aggregate preferences.
Elections
!
If A = {a, b, c}, then P has 6 elements:
Welfare
Elections
!
If A = {a, b, c}, then P has 6 elements:
a!b!c
b!c!a
c!b!a
a!b!c
b!c!a
c!b!a
a!c!b
b!a!c
c!a!b
a!c!b
b!a!c
c!a!b
!
For a society of n = 2 members, the number of swfs
P2 → P is
636 ≈ 1028
Example: Utilitarianism of Jeremy Bentham
II-8.
Voting
Example: Utilitarianism of Jeremy Bentham
Dusko Pavlovic
"The Greatest Pleasure Principle"
Introduction
Dusko Pavlovic
"The Greatest Pleasure Principle"
Introduction
Welfare
!
given utilities ui : A → R for i = 1, 2, . . . n with
i
a!b
!
Welfare
Elections
!
i
a!b
derive u : A → R as
!
u(x) =
given utilities ui : A → R for i = 1, 2, . . . n with
ui (a) > ui (b)
⇐⇒
ui (a) > ui (b)
⇐⇒
u(x) =
n
n
#
ui (x)
i=1
and set
n
and set
a !!"w b
a !!"w b
u(a) > u(b)
⇐⇒
II-8.
Voting
Example: Borda Ranking
⇐⇒
u(a) > u(b)
II-8.
Voting
Example: Borda Ranking
Dusko Pavlovic
!
Suppose that there are ! candidates in A.
Introduction
Dusko Pavlovic
!
Then derive u : A → R as
Welfare
!
Elections
derive u : A → R as
n
#
ui (x)
i=1
II-8.
Voting
For each i, rename the candidates
$
%
(i) (i) (i)
(i)
A = a0 , a1 , a2 , . . . a!−1
Elections
u(x) =
Introduction
n
#
Welfare
ui (x)
i=1
and set
so that
(i)
i
(i)
i
(i)
i
i
(i)
a!−1 ! a!−2 ! a!−3 ! · · · ! a0
and set
& '
(i)
ui ak
= k
a !!"w b
⇐⇒
u(a) > u(b)
Elections
II-8.
Voting
General Ranking
II-8.
Voting
General Ranking
Dusko Pavlovic
Introduction
Dusko Pavlovic
!
Suppose that there are ! candidates in A.
!
Let (c!−1 , c!−2 , . . . , c0 ) be a voting vector.
!
For each i, rename the candidates
$
%
(i) (i) (i)
(i)
A = a0 , a1 , a2 , . . . a!−1
Welfare
Elections
Introduction
Welfare
Elections
Definition
A voting vector (or a procedure) for ! candidates is an
!-tuple
(c!−1 , c!−2 , . . . , c0 )
so that
which is descending, i.e. ci+1 ≥ ci for all i.
(i)
i
(i)
i
(i)
i
i
(i)
a!−1 ! a!−2 ! a!−3 ! · · · ! a0
and set
& '
(i)
ui ak
= ck
II-8.
Voting
General Ranking
II-8.
Voting
General Ranking
Dusko Pavlovic
!
Then derive u : A → R as
u(x) =
n
#
ui (x)
Dusko Pavlovic
Introduction
Introduction
Welfare
Welfare
Elections
Elections
Instances
i=1
and set
a !!"w b
⇐⇒
u(a) > u(b)
II-8.
Voting
Ranking problem
!
Borda ranking: (! − 1, ! − 2, . . . , 0)
!
plurality vote: (1, 0, . . . , 0)
!
antiplurality vote: (1, 1, . . . , 1, 0)
II-8.
Voting
Ranking problem
Dusko Pavlovic
Exercise
Introduction
Dusko Pavlovic
Solution
Introduction
Welfare
Consider again the preferences
voters
preference
3
2
2
4
a!b!c
a!c!b
b!c!a
c!b!a
Compute the aggregate rankings for the voting vectors:
(1,0,0), (4,1,0), (7,2,0), (7,3,0), (2,1,0), (3,2,0), (1,1,0).
Elections
Welfare
voting vector
ranking
(1,0,0)
(4,1,0)
(7,2,0)
(7,3,0)
(2,1,0)
(3,2,0)
(1,1,0)
a!c!b
a∼c!b
c!a!b
c!a∼b
c!b!a
b∼c!a
b!c!a
Elections
II-8.
Voting
Dictatorship Theorem
Dictatorship Theorem
Dusko Pavlovic
Introduction
Theorem (K. Arrow)
Introduction
Welfare
Elections
II-8.
Voting
Dusko Pavlovic
Welfare
Suppose that !−"w : Pn → P satisfies
Elections
Notation
!
Pareto or unanimity Principle (UP):
i
∀i.a ! b =⇒ a !!"w b
i
a{!}b = {i | a ! b}
!
Independence of Irrelevant Alternatives (IIA):
a{!}b = a{!}b ∧ a !!"w b =⇒ a !!"w b
holds for every two profiles !, !∈ Pn .
II-8.
Voting
Dictatorship Theorem
Dictatorship Theorem
Dusko Pavlovic
Theorem (K. Arrow)
Welfare
i
a!b
Dusko Pavlovic
Introduction
Then as soon as there are more than 2 candidates in A,
there must exist a dictator, i.e. a voter i such that
⇐⇒
Elections
II-8.
Voting
Introduction
Welfare
Remark
Elections
Note that the theorem does not say that the dictator has
to actively impose his preferences.
a !!"w b
holds for every preference profile !∈ Pn .
Dictatorship Theorem
II-8.
Voting
Condorcet requirement
Dusko Pavlovic
Remark
Note that the theorem does not say that the dictator has
to actively impose his preferences.
The theorem says that
!
for every swf satisfying UP and IIA
!
there is a voter who agrees with the social welfare for
every preference profile of the society.
II-8.
Voting
Dusko Pavlovic
Introduction
Introduction
Welfare
Welfare
Elections
Elections
Definition
A swf !−"w : Pn → P satisfies the Condorcet requirement
if
a !!"w b =⇒ #a{!}b > #b{!}a
Borda count violates Condorcet requirement
II-8.
Voting
Borda count violates Condorcet requirement
Dusko Pavlovic
Example
Introduction
Example
Introduction
Welfare
Consider the preferences
Elections
II-8.
Voting
Dusko Pavlovic
Welfare
Consider the preferences
Elections
voters
preference
voters
preference
30
1
29
10
10
1
a!b!c
a!c!b
b!a!c
b!c!a
c!a!b
c!b!a
30
1
29
10
10
1
a!b!c
a!c!b
b!a!c
b!c!a
c!a!b
c!b!a
Then b(109) !!"w a(101) !!"w c(33)
but a(41) ! b(40) and a(60) ! c(21).
II-8.
Voting
Condorcet ranking
Condorcet ranking allows cycles
Dusko Pavlovic
Introduction
Welfare
Elections
Dusko Pavlovic
Example
Given a preference profile !∈ Pn , the Condorcet ranking
2 is defined by setting
⇐⇒
#a{!}b > #b{!}a
Condorcet ranking allows cycles
II-8.
Voting
Introduction
Welfare
Consider the preferences
Definition
a2b
Elections
voters
preference
23
2
17
10
8
a!b!c
b!a!c
b!c!a
c!a!b
c!b!a
Condorcet ranking allows cycles
Dusko Pavlovic
Example
Welfare
Corollary
preference
23
2
17
10
8
a!b!c
b!a!c
b!c!a
c!a!b
c!b!a
Proof
If Condorcet ranking were transitive, then a 2 b and
b 2 c and c 2 a would imply a 2 a.
But by the definition of Condorcet ranking, this would
mean that #a{!}a > #a{!}a, which is impossible.
b(42) 2 c(18)
c(35) 2 a(25)
Welfare
Elections
Condorcet ranking may not be transitive.
Then
a(33) 2 b(27)
Introduction
Elections
voters
II-8.
Voting
Dusko Pavlovic
Introduction
Consider the preferences
II-8.
Voting
II-8.
Voting
Outline
II-8.
Voting
Social choice
Dusko Pavlovic
Dusko Pavlovic
Introduction
Introduction
Welfare
Introduction
Elections
Welfare
!
Social welfare and preference aggregation
Elections
It is often
not necessary to aggregate the individual
i
preferences ! into a full social preference relation
!!"w , but it is
Social choice and manipulability
!
Social choice function and relation
II-8.
Voting
sufficient to elect the best candidate c, i.e. such that
c !!"w x for all x ∈ A.
Social choice function and relation
Dusko Pavlovic
Dusko Pavlovic
Introduction
Introduction
Welfare
Elections
Definition
II-8.
Voting
Welfare
Elections
Example 1
A swf !−"w always induces a scr
A social choice function (scf) is a mapping !:"f Pn → A.
c ∈ !!"r
⇐⇒
∀x. c !!"w x
A social choice relation (scr) is a mapping !:"r Pn → ℘A.
It induces a scf if the aggregate preferences have top
elements.
Social choice function and relation
II-8.
Voting
II-8.
Voting
Manipulability
Dusko Pavlovic
Introduction
Example 2
Welfare
If the space of alternative choices A can be presented in
the form
A =
n
(
Introduction
A social function !:"f Pn → A is manipulable if there is a
voter i and a preference profile !∈ Pn such that
!!"f
i=1
= {σ ∈ A | σ BR σ}
i.e. the social choices are the equilibria of the game.
where
! =
)i
*
i
!
Ai
where each Ai is controlled by the player i, then the scr
can be defined to be
!!"r
Elections
Dusko Pavlovic
Definition
1 2
!!"f
)i
i
n
!, !, . . . , !, . . . !
)i
+
Welfare
Elections
II-8.
Voting
Manipulability
II-8.
Voting
Manipulability
Dusko Pavlovic
Definition
A social function !:"f Pn → A is manipulable if there is a
voter i and a preference profile !∈ Pn such that
i
!!"f
where
! =
)i
*
!
Introduction
Introduction
Welfare
Welfare
Elections
Elections
Terminology
!!"f
)i
1 2
i
n
!, !, . . . , !, . . . !
)i
i
Dusko Pavlovic
A social choice function that is not manipulable is said to
be incentive compatible.
+
i.e., i can induce an !-preferred social choice
i
if she does not vote honestly, according to !,
i
but dishonestly, according to some !.
)i
II-8.
Voting
Manipulability
II-8.
Voting
Manipulability
Dusko Pavlovic
Introduction
Dusko Pavlovic
Introduction
Exercise 1 (easy)
Welfare
Elections
Theorem (Gibbard-Satterthwaite)
Welfare
Prove that a scf is incentive compatible if and only if it is
monotone, i.e. satisfies
i
a0 ! a1
A surjective scf !:"f Pn → A between more than 2
candidates in A is either manipulable, or a dictatorship, or
both.
0
and
Elections
i
a1 ! a0
1
whenever
1
i
n
a0 = !!, . . . , !, . . . , !"f
0
1
i
n
a1 = !!, . . . , !, . . . , !"f
1
II-8.
Voting
Manipulability
Manipulability
Dusko Pavlovic
Comment
The monotonicity of a scf !−"f means that
!
i
i
0
1
if changing only ! to ! causes the social choice to
change from a0 to a1
!
then the change must have been
i
i
0
1
from a0 ! a1 to a1 ! a0 .
II-8.
Voting
Dusko Pavlovic
Introduction
Introduction
Welfare
Welfare
Elections
Elections
Exercise 2 (hard)
Derive the Gibbard-Satterthwaite Theorem from Arrow’s
Theorem.