Transcript Document

Social Choice Theory
By Shiyan Li
History


The theory of social choice and voting has
had a long history in the social sciences,
dating back to early work of Marquis de
Condorcet (the 1st rigorous mathematical
treatment of voting) and others in the
18th century.
Now it is a branch of discrete mathematics.
Purpose
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
Social Choice Theory is the study of
systems and institution for making
collective choice, choices that affect
a group of people.
Be used in multi-agent planning,
collective decision, computerized
election and so on.
Voters
Alternatives
Simple Majority Voting
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Choose one from two possible
alternatives by a group of voters.
Consider a democratic voting
situation.
Preferences and Outcome
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Alternatives: x or y
Every voter has a preferences.
Three possible situations of each voter’s
preference:
i) x is strictly better than y: +1
ii) y is strictly better than x: -1
iii) x and y are equivalent: 0
After the voting:
i) x is winner: +1
ii) y is winner: -1
iii) x and y tie: 0
General List
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Use a list to describe a collection of n
voters’ preferences
e.g. (-1, +1, 0, 0, -1, …, +1, -1)
n entries
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General List:
D = (d1, d2, d3, …, dn-1, dn)
di is +1, -1 or 0 depending on whether
individual i strictly prefers x to y, y to x or
is indifferent between them.
General List
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Consider the sum of list D:
When d1+d2+d3+…+dn-1+dn > 0,
x is to be chosen, simple majority
voting assigns +1.
When d1+d2+d3+…+dn-1+dn < 0,
y is to be chosen, simple majority
voting assigns -1.
When d1+d2+d3+…+dn-1+dn = 0,
x and y tie, simple majority voting
assigns 0.
Formal Definition of Simple Majority Voting
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Use the sign function to formally define
the simple majority voting:
(d1, d2, …, dn)
sgn(d1+d2+…+dn)
Function N+1 and N-1:
N+1: associates with a list D the number of
di‘s that are strictly positive
N-1: associates with a list D the number of
di‘s that are strictly negative
Formal Definition of Simple Majority Voting
Absolute Majority Voting
+1 if N+1(D)>N-1(D)

if N+1(d1, d2, d3, …, dn) > n/2
g (d1, d2, d3, …, dn) =
-1 if N-1(D)>N+1(D)
if N-1(d1, d2, d3, …, dn) > n/2
0 otherwise

E.g. for absolute majority voting:
for list D = (+1, -1, -1 ,0, +1, +1),
∵ n = 6, n/2 = 3,
N+1 (+1, -1, -1 ,0, +1, +1) = 3 > n/2
N-1 (+1, -1, -1 ,0, +1, +1) = 2 <n/2
∴ g(+1, -1, -1 ,0, +1, +1) = +1
Rule of Simple Majority Voting
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
Social Choice Rule:
is a function f(d1, d2 , …, dn ), the domain
of the function is the set of all list to which
f assigns some unambiguous outcome: +1,
-1 or 0.
A social choice rule of simple majority
voting can be characterized by 4
properties (Kenneth O. May, 1952).
Property 1 of Rule f

Property 1 – Universal Domain:
f satisfies universal domain if it has a
domain equal to all logically possible
lists (i.e. any combination of the
individual voters’ preferences) of n
entries of +1, -1 or 0.
Property 2 of Rule f
One-to-one Correspondence:
is a function s from the set {1, 2, …, n} to itself
such that s is defined on every integer from 1 to
n and distinct outcomes are assigned to two
different integers:
s(i) = s(j) implies i = j.

i
S(i)
one-to-one correspondence
i
S(i)
S(i)
not one-to-one correspondence
i
S(i)
i
Property 2 of Rule f
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Permutation:
Given two lists
and
D = (d1, d2 , …, dn)
D’ = (d1 ’, d2 ’ , …, dn ’)
say that D and D’ are permutation of one another if there is a one-to-one
correspondence s on {1, 2, …, n} such that
ds(i)’ = di.

E.g.:
and
voter: 1 2 3 4 5 6 7
(+1, +1, +1, 0, 0, -1, -1)
voter: 1 2 3 4 5 6 7
(-1, 0, +1, +1, 0, -1, +1)
are permutation of one another via the one-to-one correspondence:
1->3, 2->4, 3->7, 4->2, 5->5, 6->1, 7->6.
Property 2 of Rule f
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Property 2 – Anonymity:
A social choice rule will satisfy this property if it does not make
any difference who votes in which way as long as the numbers of
each type are the same (i.e. equal treatment of each voter).
Formal Definition:
A social choice rule f satisfies anonymity if whenever (d1, d2, …,
dn) and (d1’, d2’, …, dn’) in the domain of f are permutations of one
another then
f(d1, d2, …, dn) = f(d1’, d2’, …, dn’)
E.g.:
if
and
D = (+1, +1, +1, 0, 0, -1, -1)
D’ = (-1, 0, +1, +1, 0, -1, +1)
so D and D’ are permutations of each other,
and if
f(d1, d2, …, dn) = f(d1’, d2’, …, dn’)
then social choice rule f satisfies anonymity.
Property 3 of Rule f
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Property 3 – Neutrality:
A social choice rule satisifies neutrality if
whenever (d1, d2 , …, dn ) and (-d1, -d2 , …,
-dn ) are both the domain of f then
f(d1, d2 , …, dn )=-f(-d1, -d2 , …, -dn )
Note:
The condition of anonymity is a way of
treating individuals equally, the condition
of neutrality is a way of treating
alternatives x and y equally.
Property 4 of Rule f
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i-Variants:
Suppose there are
and
D = (d1, d2 , …, dn )
D’ = (d1’, d2’ , …, dn’ );
D and D’ are i-variants if for all j≠i, dj=dj’. Thus two i-variants
differ in at most the ith entry. (Note: It has not strictly stipulated
the relationship of di and di’, i.e., it is possible that di=di’, di>di’, or
di<di’.)

E.g.:
Two lists
and
D = (+1, -1, -1, 0, +1, -1, +1)
D’ = (+1, -1, 0, 0, +1, -1, +1)
are 3-variants since they differ only at the third place
Property 4 of Rule f
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Purpose:
Simple majority voting can not be strictly characterized by
property 1~3 yet (unresponsiveness).
E.g.:
Assume a constant rule (function) const0(D) that always
generates result 0 for any point in its domain.
i.e.
const0(D)
0
This constant rule satisfies all 3 properties mentioned above.
D contains all logically possible lists.
– Property 1
For all permutations D’, const0(D) = const0(D) = 0. – Property 2
For all lists in D, const0(D) = -const0(-D) = 0.
– Property 3
So, we still need a property to constrain rule f to simple majority
more strictly.
Property 4 of Rule f
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Property 4 – Positive Responsiveness:
f satisfies positive responsiveness if
for all i, whenever (d1, d2 , …, dn )
and (d1’, d2’ , …, dn’) are i-variants
with di’ > di, then
implies
f(d1, d2 , …, dn ) ≥ 0
f(d1’, d2’ , …, dn’) = +1.
Property 4 of Rule f

Positive responsiveness can be inferred by indirect i-variants.
E.g.:
Suppose to apply lists #1 below to f which is a rule satisfies
positive responsiveness:
f(+1, 0, -1, 0, 0, +1, -1) = 0.
First find a 3-variant list #2 of #1: (+1, 0, 0, 0, 0, +1, -1),
so f(+1, 0, 0, 0, 0, +1, -1) = +1.
Second find a 4-variant list #3 of #2: (+1, 0, 0, +1, 0, +1, -1),
so f(+1, 0, 0, +1, 0, +1, -1) = +1.
Then it can be concluded that f(+1, 0, -1, 0, 0, +1, -1) = 0
implies f(+1, 0, 0, +1, 0, +1, -1) = +1, although list #1 and #3
are not direct i-variants.
Property 4 of Rule f
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“Negative Responsiveness”:
Suppose rule f satisfies property 1~4.
For all i, whenever D = (d1, d2 , …, dn ) and D’ = (d1‘, d2‘, …, dn‘ ) are ivariants with di‘ < di (i.e. -di‘ > -di ).
If f(D) ≤ 0 then f(-D) = -f(D) ≥ 0 by neutrality.
So f(-D) ≥ 0.
There is a list -D’ which together with –D are i-variants with -di‘ > -di.
Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness.
So f(D’) = -f(-D’) = -1
Summary:
If f satisfies positive responsiveness and neutrality then for all i, whenever
D = (d1, d2 , …, dn ) and D’ = (d1‘, d2‘, …, dn‘ ) are i-variants with di‘ < di,
such that
f(D) ≤ 0 implies f(D’) = -1
May’s Theorem
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Simple majority voting is the only
rule that satisfies all four properties
(or conditions) simultaneously.
May’s Theorem

May’s Theorem:
If a social choice rule f satisfies all of
i)
ii)
iii)
iv)
universal domain
anonymity
neutrality
positive responsiveness
then f is simple majority voting.
Proof of May’s Theory

Step 1:
If rule f satisfies conditions i), ii), iii) and iv).
So the value of f(D) only depends on the number
of +1’s, 0’s and -1’s by anonymity.
Suppose there are n elements in D, N+1(D) and
N-1(D) is the number of +1’s and -1’s in D
correspondingly.
So the number of 0’s is n - N+1(D) - N-1(D).
Therefore, f(D) is entirely determined by N+1(D)
and N-1(D) by anonymity.
Proof of May’s Theory

Step 2:
Suppose N+1(D) = N-1(D) and f(D) = r.
Obviously
N+1(D) = N-1(D) = N+1(-D)
N-1(D) = N+1(D) = N-1(-D).
#1
#2
And because f satisfies universal domain, so f is also
defined at –D.
Since
f(-D) = -f(D) = -r by neutrality,
and
f(-D) = f(D) = r by #1 and #2.
Combining above results, –r = r so r = 0.
That is N+1(D) = N-1(D) implies f(D) = 0.

Step 3:
Proof of May’s Theory
Suppose N+1(D) > N-1(D) where there are n elements in D,
so that N+1(D) = N-1(D) + m where 0 < m ≤ n - N-1(D).
It will be proved that f(D) = +1 by mathematical induction below:
D = (d1, d2, …, dn).
Basis: m = 1.
∴ N+1(D) = N-1(D) + 1
∴ There is at least one di = 1.
Suppose
D’=(d1’, d2’, …, dn’), an i-variant determined by
dj’=dj if j≠i, and di’=0.
#1
f is defined at D and D’ by universal domain.
Obviously N+1(D’) = N-1(D’).
∴ f(D’) = 0 by step 2.
#2
∴ f(D) = +1 by #1, #2 and positive responsiveness.
Induction: Suppose
N+1(D)=N-1(D)+1 implies f(D)=+1.
It has to be shown that
N+1(D)=N-1(D)+(m+1) implies f(D)=+1.
So suppose
N+1(D)=N-1(D)+(m+1).
∴ There is at least one di = 1.
Suppose
D’=(d1’, d2’, …, dn’), an i-variant determined by
dj’=dj if j≠i, and di’=0.
#3
f is defined at D and D’ by universal domain.
Obviously N+1(D’) = N-1(D’)+m.
∴ f(D’) = 0 by induction hypothesis.
#4
∴ f(D) = +1 by #1, #2 and positive responsiveness.
Summary:
Follow an analogous derivation, an assertion “when N+1(D) < N-1(D), f(D) = -1” can be proved.
So: If N+1(D) > N-1(D), then f(D) = +1
If N+1(D) < N-1(D), then f(D) = -1
Proof of May’s Theory
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Summary of Proof:
From step 1, 2, and 3:
If N+1(D)=N-1(D), then f(D)=0.
If N+1(D)>N-1(D), then f(D)=+1.
If N+1(D)<N-1(D), then f(D)=-1.
These results just satisfy the formal
definition of simple majority voting.
So May’s theory is proved.
General Social Choice Rules
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X: a nonempty set of alternatives.
The elements of X must only be
mutually incompatible.
v: agenda, v ≠ Ø and v ⊆ X, a set of
alternatives that are currently available.
N: a set of individuals.
General Social Choice Rules
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xRiy: i ∈ N; x, y ∈ X; individual i determines
alternative x to be at least as good as alternative
y; or i weakly prefers x to y.
1. Ri is reflexive: xRix for all x ∈ X.
2. Ri is complete: xRiy or yRix (or both) for all x,
y ∈ X.
3. Ri is transitive: For all x, y, z ∈ X, if both xRiy
and yRiz then xRiz.
General Social Choice Rules
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xPiy: xRiy and not yRix; i strongly prefers x
to y.
yPix: yRix and not xRiy; i strongly prefers y
to x.
xIiy: xRiy and also yRix; i is indifferent
between x and y.
General Social Choice Rules
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Profile: an assignment of one
preference relation to each individual.
C(v): the elements chosen from
agenda v by choice function C.
(i) C(v) ⊂ v;
(ii) C(v) ≠ Ø.
General Social Choice Rules
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Social Choice Rule:
A social choice rule assigns to each
of a collection of profiles a
corresponding choice function.
agenda,
v
profile of preferences,
u
social choice rule,
f
choice function,
C = f(u)
chosen set,
Cu(v)
Standard Domain Constraint
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Standard domain constraint includes:
i) there are at least three alternatives in X;
ii) there are at least three individuals in N;
iii) the social choice rule has as domain all
logically possible profiles of preference orderings
on X;
iv) each choice function that is an output of the
rule has in its domain all finite nonempty
agendas.
Pareto Condition
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Weak Pareto Condition:
Let the social choice rule select
choice function Cu at profile u.
Suppose at u everyone unanimously
strictly prefers one alternative, say x,
to another, say y; then if x is
available (i.e., x ∈ v), y won’t be
chosen (i.e., y ∉ Cu(v))
Pareto Condition
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Strong Pareto Condition:
Let the social choice rule select
choice function Cu at profile u.
Suppose at u everyone unanimously
find one alternative, x, to be at least
as good as another, y, and at least
one individual strictly prefers x to y.
Then if x is available (i.e., x ∈ v),
ywon’t be chosen (i.e., y ∉ Cu(v))
Pareto Condition

Example:
For agenda: 1: (x y1) y2
2: x y1 y2
3: x (y1 y2)
In Weak Pareto Condition:
y2 ∉ Cu(v)
In Strong Pareto Condition:
y1, y2 ∉ Cu(v)
Pareto Condition
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X is Pareto-superior to y at profile u = (R1,
R2, …, Rn) if:
(i) xRiy for all individuals i in N;
(ii) xPiy for at least one individual i in N.

Alternatives for which there are no
available Pareto-superior alternatives are
called Pareto optimal.
Dictator
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Weak Dictator
Individual i is a weak dictator if for
every pair of alternatives, x and y,
every profile u = (R1, R2, …, Rn) and
every agenda v, if xPiy then
y ∈ Cu(v) implies x ∈ Cu(v).
Dictator
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Coalition
A subset S of the set N of all individuals is called a coalition.

Decisive Coalition
For a social choice rule that maps u to Cu, A coalition S is
called decisive for alternative x against alternative y if:
for ∀i: i ∈ S • xRiy;
∃j: j ∈ S • xPjy;
then ∀v: v ⊂ X, x ∈ v • y ∉ Cu(v).
If ∀x, y: x, y ∈ X • S is decisive for alternative x against
alternative y, then we simply say S is decisive.
Dictator
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Dictator
If a decisive coalition S = {i}, then i
is a dictator.
Borda Rules
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Borda Count
Assume that X is finite. Then associated
with any preference ordering Ri there is a
ranking function ri that associates an
integer with each alternative: ri(x) is the
number of alternatives stictly preferred to
x. Given a profile u = (R1, R2, …, Rn),
there is a ranking function r given by
r(x) = ∑iri(x).
The value of r(x) is called Borda count of x.
Borda Rules
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Global Borda Rule
Cu(v) = {x|r(x) ≤ r(y) for all y ∈ v}.
This rule has us choose from v those
alternatives with minimal Borda
count.
Independence of Irrelevant Alternatives

If two profiles u, u’, restricted to an
agenda v are identical, then the
choices made from that agenda
should be the same:
Cu(v) = Cu’(v).
Local Borda Rules
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Local Borda Count
Given a profile u = (R1, R2, …, Rn), there
is for each v:
rv(x) = ∑iriv(x).

Local Borda Rule
Cu(v)= {x|rv (x) ≤ rv (y) for all y ∈ v}.
Transitive Explanation
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Explanation:
A choice function C is explainable if there exists a relation Ω
such that
C(v) = {x ∈ v | xΩy for all y ∈ v}.

Transitive Explanation:
A choice function C has transitive explainable if there is a
reflexive, complete and transitive relation Ω such that
C(v) = {x ∈ v | xΩy for all y ∈ v}.

We say a social choice rule has transitive explainable if at
every admissible profile u the associated Cu has a transitive
explainable.
Arrow’s Impossibility Theorem

There does not exist any social choice rule
satisfying all of:
1. the standard domain constraint;
2. the strong Pareto condition;
3. independence of irrelevant alternatives;
4. has transitive explanations;
5. absence of a dictator.
Mechanism Design
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

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Implementing a social choice
function f(u1, …, un) using a game.
Center (auctioneer) does not know
the agents’ preferences.
Agents may lie.
Goal is to design the rules of the
game so that in equilibrium (s1, …,
sn), the outcome of the game is f(u1,
…, un).
Mechanism Design
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
Mechanism designer specifies the strategy sets Si and how
outcome is determined as a function of (s1, …, sn)  (S1, …,
Sn).
Variants
Strongest: There exists exactly one equilibrium. Its
outcome is f(u1, …, un).
Medium: In every equilibrium the outcome is f(u1, …, un).
Weakest: In at least one equilibrium the outcome is f(u1, …,
un).
References

Kelly, Jerry S., 1988, Social Choice
Theory An Introduction, SpringerVerlag, Berlin Heidelberg.