COLLECTIVE DECISION MAKING
Pierre Dehez
CORE
University of Louvain
pierre.dehez@uclouvain.be
Outline
1. Preferences, utility and choices
2. Cardinal welfarism
distributive justice
utilitarism vs egalitarism
Nash bargaining
social welfare orderings
transferable utility games
3. Ordinal welfarism
the case of two alternatives
social choice procedures
impossibility theorems
possibility theorems
2
References
Austen-Smith D. and J. Banks, Positive political theory I: Collective preferences,
University of Michigan Press, 1999.
Austen-Smith D. and J. Banks, Positive political theory II: Strategy and structure,
University of Michigan Press, 2005.
Brams S., Game theory and politics, Dover, 2004.
Brams S., Mathematics and democracy, Princeton University Press, 2008.
Moulin H., Axioms of cooperative decision making, Cambridge University Press, 1998.*
Moulin H., Fair division and collective welfare, MIT Press, 2003.*
Taylor A., Mathematics and politics, Springer-Verlag, 1995.
Peyton Young H., Equity. In theory and Practice, Princeton University Press, 1995.
Handbook of social choice and welfare, Elsevier, 2002.
* Moulin's monographies have inspired some of the material presented here.
3
1. Preferences, utility and choices
4
Preferences
Preferences over a set A of alternatives are defined by a (binary)
relation over A:
a
b  b is not preferred to a
from which the strict preference and indifference relations are deduced:
a
b  a is preferred to b  [a
a
b  indifference between a and b  [a
b and b  a]
b and b
a]
5
Preferences are rational if they verify the following properties:
- completeness: [a
b or b
a]
- reflexivity: a
a for all a  A
- transitivity: [a
b and b
for all a, b  A
c]  a
c
Completeness is by far the most demanding assumption!
A relation satisfying reflexivity and transitivity is a preorder.
We denote by L(A) the set of preorders on a set A.
6
Ordinal utilities
A preference preorder carries no information on the intensity of
preferences:
if a is preferred to b and c is preferred to d, we don't know
whether a is "more preferred" to b than c is preferred to d
Under minimal assumptions, preferences can be represented by a
utility function u : A  : a  A  u (a) which associates a real
number to each alternative, such that:
u(a)  u(b)  a
b
7
As such, this is an ordinal representation of preferences:
only the sign of the difference u(a)  u(b) provides
an information on the preferences between a and b
u (a)  u (b)  0  a
b
u (a)  u (b)  0  a
b
As a consequence, u and v  T (u ) where T :  is an arbitrary
increasing transformation, both represent the same preferences.
T(u) = u3 is the simplest nonlinear transformation with range .
8
Choices
Given a set of alternatives A and preferences
element in A:
 L( A), a choice is a best
a*  A
a*
a for all a  A
or
a * maximizes u(a) on A
There may be several best elements. The set of solutions is called the
choice set.
9
Let C ( A, ) denote the choice set associated to a set A of alternatives
and a preference relation . Then:
a, b  C ( A, )  a b
There is indifference between the elements of a choice set.
In the multivalued case, a neutral mechanism is necessary to eventually
retain a unique alternative.
For instance a random mechanism.
10
Cardinal utilities
Utilities are cardinal if utility difference have a meaning:
u(a)  u(b)  u(c)  u(d )  0
means that a is preferred to b more intensely than c is preferred to d.
Cardinal utility function are defined up to an increasing affine
transformation:
u and v  au  b, a  0
are utility functions representing the same preferences.
11
Collectivity: preference profiles
Consider a set A of alternatives and n individuals indexed by i running
from 1 to n, each having a preference relation i  L( A).
A preference profile P specifies a preference relation for each member
of the group:
P  ( 1,...,
n
)

L
( A)
n
A utility profile can be associated to any alternative a  A:
u(a)  u1 (a),..., un (a)  
n
12
One of the questions addressed by social choice is the determination
of a collective preference ordering for comparing utility profiles.
There are several levels of independence that collective preferences
may satisfy:
1. Ordinal, non-comparable: full independence
2. Ordinal, comparable: independence of common utility space
3. Cardinal, non-comparable: independence of utility scales
4. Cardinal, partially comparable: independence of zero utilities
5. Cardinal, comparable: independence of utility scales and zero utilities
13
Given a set of alternative A and a preference profile on A
( 1,...,
n
)

L
( A)
n
represented by utility functions u1,…,un we define the attainable
utility set
U ( A)  u 
n
u   u1 (a ),..., un (a)  , a  A
The problem is then to pick up a point in this set, possibly given the
specification of a disagreement point d in U(A).
14
1. Ordinal, non-comparable: full independence
This is the situation where each individual utility level is defined up to
an arbitrary increasing transformation:
(u1 ,..., un )
(v1 ,..., vn )

(T1 (u1 ),..., Tn (un ))
(T1 (v1 ),..., Tn (vn ))
where the Ti's are arbitrary increasing transformation from
into .
15
2. Ordinal, comparable: independence of common utility space
This is the situation where individual utility levels are defined up to an
arbitrary and common increasing transformation:
(u1 ,..., un )
(v1 ,..., vn )

(T (u1 ),..., T (un ))
(T (v1 ),..., T (vn ))
where T is an arbitrary increasing transformation from
into .
16
3. Cardinal, non-comparable: independence of utility scales
This is the situation where each individual utility level is defined up to
an increasing and affine transformation:
(u1 ,..., un )
(v1 ,..., vn )

(a1u1  b1 ,..., anun  bn ) (a1v1  b1 ,..., an vn  bn )
for all ai , bi  , ai  0.
17
4. Cardinal, partially comparable: independence of zero utilities
This is the situation where each individual utility level is defined up to
an increasing and affine transformation:
(u1 ,..., un )
(v1 ,..., vn )

(u1  b1 ,..., un  bn ) (v1  b1 ,..., vn  bn )
for all b1 ,..., bn  . Alternatively:
(u1 ,..., un )
(v1 ,..., vn )  (u1  v1,..., un  vn )
(0,...,0)
18
5. Cardinal, comparable: independence of utility scales and zero utilities
This is the situation where individual utility levels are defined up to an
increasing and affine common transformation:
(u1 ,..., un )
(v1 ,..., vn )

(a u1  b,..., a un  b) (a v1  b,..., a vn  b)
for all a, b  , a  0.
19
2. Cardinal welfarism
20
2.1 Distributive justice
21
"Equal treatment of equals"
is the basic principle of distributive justice. It is a minimal and clear
requirement of fairness.
"Unequal treatment of unequals" instead is a vague principle.
"Equals should be treated equally and unequals unequally,
in proportion to the relevant similarities and differences"
(Aristoste)
22
Liberalism: the social order emerges from the interaction of free wills.
Methodological individualism is at the root of liberalism.
Individuals are characterized by values, rights and obligations.
Distributive justice has two sides:
- procedural justice: is the distribution of rights fair ?
- end-state justice: is the outcome fair ?
We start with a simple problem of sharing a resource.
23
We assume that a utility index can be associated to each individual:
vi  ui ( xi )
where xi denotes the share of individual i in the resource.
It is a cardinal utility and utility levels can be compared. Its definition
depends upon the context.
It is an "objective" index and the individual is not responsible for its
shape.
It is the information that a benevolent dictator needs to decide on the
allocation of resources in a particular context.
24
Principle 1: ex ante equality
There are basic rights like freedom of speech, access to education,
freedom of religion, equal political rights (one person, one vote),…
They induce ex ante equality: equal claim to the basic resources.
Private ownership or differences in status (for instance seniority)
are instances of unequal exogenous rights which justify unequal
treatment.
25
Principle 2: ex post equality
… justifies unequal shares of resources to compensate for
involuntary differences in individuals' primary characteristics
like nutritional needs, health,…
If ui is an objective utility index for individual i resulting from his/her
primary characteristics, this principle allows for
or equivalently
ui ( x)  u j ( x)
ui ( xi )  u j ( x j )  xi  x j
This principle amounts to equalization of utilities.
26
Principle 3: reward or penalize
… justifies unequal shares yi's of resources to compensate
for voluntary differences in individuals' characteristics:
- past sacrifies justify a larger share
- past abuses justify a lesser share
How to reward individual contributions ?
The answer if difficult when there are externalities (extraction of
exhaustible resources, division of joint costs or surpluses).
27
Principle 4: best use of the resources (fitness)
…resources must go to those that can make the best use them.
Fitness justifies unequal treatment by differences in talent,
independently of basic rights, needs or merits.
Two definitions:
sum-fitness: maximization of the sum of the individual utilities
efficiency-fitness: Pareto optimality
Sum-fitness implies efficiency fitness.
28
How should the benevolent dictator use these four partially conflicting
principles very much depends upon the context.
Examples:
- access to the lifeboat,
- allocation of organs for transplant,
- seat rationing,
- political rights.
29
Lifeboat
exogenous rights: strict equality (lottery) or priority ranking
based on social status or wealth
compensation:
priority to the weak ones (equality of ex-post
survival chances)
reward:
exclude those responsible for the sinking
ship…
fitness:
keep the crew, the women, the children,…
30
Transplants
exogeneous rights: strict equality (lottery) or priority ranking
based on social status or wealth
compensation:
priority to those suffering most or whose life
expectancy is the shortest
reward:
priority to seniority on the waiting list
fitness:
maximization of the chances of success
31
Seats: auctioning or queuing
exogenous rights: only a lottery would induce a strict equality
reward:
queuing reward efforts while auctioning
does not
fitness:
queuing meets sum-fitness but involves a
waste of time  auctioning is better if
individuals are comparable, because otherwise
it favors the rich
32
Political rights
fitness and reward: justify unequal voting rights which were
commonplace in the past
exogenous rights: justifies equal rights (beyond some obvious
limitations justified by fitness)
compensation:
there are many examples of situations where
voting rights are not equal (EU distribution of
votes among countries supposed to take into
differences in population sizes)
33
Allocation methods
A given amount of some commodity has to be divided between
a given number of individuals and each individual has a claim.
The commodity could be a "good" or a "bad":
- for a good, individuals express demands
- for a bad, individuals have liabilities
There may be an excess or a deficit.
34
Data:
a set N = {1,…n} of "players"
an amount E > 0 to be allocated
players' claims: d1,…,dn > 0
Problem: find an allocation x = (x1,…,xn) sucth that x(N) = E.
Two cases: deficit: d ( N )  E
surplus: d ( N )  E
Notation:
for all S  N : x( S )   xi
iS
35
Examples:
joint venture: E is the revenue generated by the cooperation
and the di's are the stand-alone revenues (surplus)
bankcruptcy: E is the firm's liquidation value and the di's are the
creditors' claims (deficit)
inheritance:
E is the value of the deceased's estate and the di's
are the heirs' deeds (deficit or surplus)
taxation:
E is the tax to be levied and the di's are the taxable
incomes (deficit)
36
Assumption:
equal exogenous rights

the allocation depends only on the
distribution of claims or liabilities
An allocation method is a rule  that associates an allocation
( x1 ,..., xn )   ( E, d1,..., dn )
to any given allocation problem ( E, d1 ,..., dn ) such that x( N )  E.
37
Proportional rule (in the case of a surplus or a deficit)
di
xi 
E
d (N )
satisfies: xi  di for all i in case of a surplus
xi  di for all i in case of a deficit
and
E
xi  x j 
(di  d j ) for all i and j
d (N )
38
x2
x1 
d1
E
d1  d 2
x2 
d2
E
d1  d 2
x1  x2  E
x2 
PROP
d2
0
d2
x1
d1
d1
x1
39
Equal surplus rule (in case of a surplus)
1
xi  di  ( E  d ( N ))
n
satisfies
E  d (N )  0
xi  di for all i
xi  x j  di  d j for all i and j
40
x2
SURPLUS: d1+d2 < E and d1 > d2
x2  x1
x2  x1  (d2  d1 )
E  d1  d 2
2
E  d 2  d1
x2 
2
x1 
ES
d2
0
d1–d2
d1
x1
41
x2
SURPLUS: d1+d2 < E and d1 < d2
x2  x1  (d2  d1 )
x2  x1
ES
E  d1  d 2
2
E  d 2  d1
x2 
2
d2
x1 
d2–d1
0
d1
x1
42
Uniform gain rule (in case of a surplus)
xi  Max ( z, di )
n
where z satisfies E   Max ( z, di )
i 1
satisfies xi  di for all i.
Here z can be interpreted as the common gain.
This rule is also called "constrained" egalitarian.
43
y  2z
y = f(z)
y  d1  z
d1  d2
2d1
f ( z)  Max( z, d1 )  Max( z, d2 )
d1+d2
d1
0
d2
d1
z
44
x2
x2  x1
UG
d2
0
d1
x1
45
x2
SURPLUS: d1+d2 < E
x2  x1
UG
x2  x1  (d2  d1 )
ES
PROP
x2 
d2
x1
d1
d2
0
d1–d2
d1
x1
46
Uniform gain rule (in case of a deficit)
xi  Min ( z, di )
n
where z satisfies E   Min ( z, di )
i 1
satisfies xi  di for all i.
Here z can be interpreted as the common gain.
This rule is also called "constrained" egalitarian.
47
y  2z
y = f(z)
y  d2  z
d1+d2
d1  d2
2d2
f ( z)  Min( z, d1 )  Min( z, d2 )
d2
0
d2
d1
z
48
x2
DEFICIT: d1+d2 > E
x2  x1
d2
UG
0
d1
x1
49
Uniform loss rule (in case of a deficit)
xi  Max (di  z, 0) or di  xi  Min( z, di )
n
where z satisfies E   Max (di  z , 0)
i 1
satisfies xi  di for all i.
The idea is to substract the same amount from the claims subject
to the non-negativity constraint: z is the common loss (ex post
deficits are equalized).
This rule is also called levelling.
50
y = f(z)
DEFICIT: d1+d2 > E
d1+d2
d1  d2
d1
f ( z)  Max (d1  z,0)  Max (d2  z,0)
d1–d2
0
d2
y  d1  d2  2 z
d1
z
y  d1  z
51
x2
DEFICIT: d1+d2 > E
x2  x1
x2  x1  (d2  d1 )
E  d1  d 2
2
E  d 2  d1
x2 
2
x1 
d2
UL
0
d1–d2
d1
x1
52
x2
DEFICIT: d1+d2 > E
d2
UG
PROP
UL
0
d1–d2
d1
x1
53
x2
UG
ES
PROP
d2
UG
PROP
UL
0
d1–d2
d1
x1
54
Proportional (surplus/deficit)
Equal surplus (surplus)
Uniform gain (surplus)
Uniform gain (deficit)
Uniform loss (deficit)
xi 
E
di
d (N )
xi  di 
1
( E  d ( N ))
n
xi  Max ( z , d i )
where z is such that  xi  E
xi  Min ( z, di )
where z is such that
xi  Max (di  z, 0)
where z is such that
x
E
x
E
i
i
55
Algorithm for computing the uniform gain solution
- divide E in equal parts
xi  Max ( z, di )
xi  Min ( z, di )
- identify the individuals with the "wrong" share:
E
i such that di 
in case of a deficit
n
 xi  di
E
i such that di 
in case of a surplus
n
- reduce E accordingly and repeat the procedure with
the remaining individuals…
56
Deficit: n = 5, E = 40 and d = (20, 16, 10, 8, 6)
d ( N )  60  E
E
 8  x4  8 and x5  6
5
E  14
3
8.7  x1  x2  x3  8.7
57
Surplus: n = 5, E = 80 and d = (20, 16, 10, 8, 6)
d ( N )  60  E
E
 16  x1  20 and x2  16
5
E  36
3
14.7  x3  x4  x5  14.7
58
Algorithm for computing the uniform loss solution
xi  Max (di  z, 0)
- apply the equal surplus solution
- identify the individuals with a negative share:
1
i such that di  ( E  d ( N ))  0
n
 xi  0
- repeat the procedure with the remaining individuals…
59
Deficit: n = 5, E = 20 and d = (20, 16, 10, 8, 6)
1
zi  di  (40)
5
 z  (12, 8, 2, 0,  2)
 x4  x5  0
1
zi  di  (26)
3
 z  (11.3, 10.7, 1.3)
 x  (11.3, 10.7, 1.3, 0, 0)
60
Deficit: n = 5, E = 50 and d = (20, 16, 10, 8, 6)
1
zi  di  (10)
5
 z  (18,14,8, 6, 4)
 x  (18,14,8, 6, 4)
61
deficit
E=
di =
20
16
10
8
6
20
PRO
6.7
5.3
3.3
2.7
2
UG
4
4
4
4
4
UL
11.3
7.3
1.3
0
0
PRO
13.3
10.7
6.7
5.3
4
UG
8.7
8.7
8.7
8
6
UL
16
12
6
4
2
PRO
16.7
13.3
8.3
6.7
5
UG
13
13
10
8
6
UL
18
14
8
6
4
PRO
26.7
21.3
13.3
10.7
8
UG
20
16
14.7
14.7
14.7
ES
24
20
14
12
10
PRO
40
32
20
16
12
UG
24
24
24
24
24
ES
32
28
22
20
18
40
50
surplus
80
120
d
i
 60
62
Bankcruptcy: deficit
If creditors have equal exogenous rights, it is the proportional solution
that emerges. In reality, there are priorities that may be implied by
exogenous rights.
Medical supplies: deficit (di stands for the need of patient i)
The proportional solution is hardly acceptable in this context.
The uniform loss solution imposes itself if reducing the quantity of the
drug is equally bad for all (ex: insuline).
The uniform gain solution is appropriate if the drug is not essential (ex:
sleeping pills).
63
Fund raising: surplus (di stands for the contribution of donor i)
The proportional solution is definitely unfair: it penalizes the generous
donors. Nor is the equal surplus solution.
Uniform gain is the most naturel solution: it requires the less generous
donors to contribute first.
Fund raising: deficit
Uniform loss is not acceptable because it gives a uniform rebate
irrespectively of the contributions.
The proportional solution is definitely more appropriate.
64
The uniform gain may be acceptable as well. Indeed, it can be computed
through the following alternative algorithm, assuming that claims are
ordered as follows:
d1  d2  ...  dn
- decrease 1's claim by (d1 – d2)  x = (d2, d2, d3,… dn)
- decrease 1 and 2's claims by (d2 – d3)  x = (d3, d3, d3, d4, … dn)
- decrease 1, 2 and 3's claims by (d3 – d4)  x = (d4, d4, d4, d4, … dn)
… until arriving below E. The difference E isthen
xi returned
uniformly to the players whose claims have been decreased.
65
Deficit: n = 5, E = 40 and d = (20, 16, 10, 8, 6)
- decrease 1's claim by (d1 – d2) = 4  x = (16, 16, 10, 8, 6)
- decrease 1 and 2's claims by (d2 – d3) = 6  x = (10, 10, 10, 8, 6)
- decrease 1, 2 and 3's claims by (d3 – d4) = 2  x = (8, 8, 8, 8, 6)
The total is then 38. So add 2/3 to the first three players:
x  (8.7, 8.7, 8.7, 8, 6)
66
Four desirable properties
1. invariance with respect to transfers
2. truncation property
3. concession property
4. consistency
67
1. Invariance to transfers
if i and j "merge" into a single individual, is the resulting
share equal to the sum of the individuals' shares ?
Only the proportional rule is invariant to transfers.
The uniform gain rule is not: merging leads to a smaller
or equal share.
The uniform loss rule is not: merging leads to a higher
or equal share.
68
2. Truncation property
In case of a deficit, a solution satisfies the truncation property
if truncating the claims to E
di  di  Min  di , E 
does not affect the resulting allocation.
The uniform gain rule satisfies the truncation property.
The uniform loss rule and the proportional rule do not.
69
3. Concession property
In case of a deficit, we define the concession of N\i to individual i by:
zi  Max  0, E  d ( N \ i) 
Given an allocation rule, consider the following 2-step procedure:
- allocate zi to individual i
- apply the allocation rule to the problem of dividing what remains
E  E   zi
according to the reduced claims di  di  zi .
70
An allocation rule has the concession property if this 2-step procedure
reaches the same allocation.
The uniform loss rule satisfies the concession property.
The uniform gain rule and the proportional rule do not.
71
4. Consistency
An allocation rule  is consistent if for all problem (E,d) and all
subsets S in N:
x  (E, d )  x
where x
S
S
 ( x(S ), d S )
 ( xi | i  S ).
Pairwise consistency requires that condition to hold for any pair of
individuals.
For continuous and symmetric rules, pairwise consistency implies
consistency.
72
Example: n = 5, E = 50 and d = (20,16,10,8,6).
The uniform gain solution is x = (13,13,10,8,6).
Looking at S = {1,2,3} and applying the solution to the problem
defined by E = 36 and d = (20,16,10), we get x = (13,13,10).
The uniform loss solution is x = (18,14,8,6,4).
Looking at S = {1,2,3} and applying the solution to the problem
defined by E = 40 and d = (20,16,10), we get x = (18,14,8).
73
Application: taxation
Here E = T is the tax to be levied and di = yi represents i's taxable
income. It is assumed that we are in the deficit case: y( N )  T .
A taxation method is a function  which associates taxes
t   (T , y )
to any taxation problem (T,y) such that
t
i
 T and 0  ti  yi for all i
There are three classical taxation methods: flat tax, head (or poll) tax
and levelling tax.
74
Flat tax is the proportional solution:
tj
ti
ti   yi where  


yi y j
 yi
T
Head tax is the uniform gain solution:
ti  Min ( z, yi ) where z is such that
t
i
T
Levelling tax is the uniform loss solution:
ti  Max ( yi  z,0) where z is such that
t
i
T
75
y1  y2
t2
t2  t1
t2  t1  ( y1  y2 )
t2 
y2
t1
y1
y2
Head
Flat
Levelling
0
y1–y2
y1
t1
76
y1  y2
t2
t2  t1
t2  t1  ( y1  y2 )
t2 
y2
t1
y1
y2
Head
Flat
Levelling
0
y1–y2
y1
t1
77
Principles
1. Fair ranking
A higher income justifies both a higher tax burden and a higher aftertax income :

ti  t j
yi  y j  

 yi  ti  y j  t j
Under this principle, equal incomes are taxed equally.
78
y1  y2  Max  0, t1  ( y1  y2 )   t2  Min  y2 , t1 
t2
t2  t1  ( y1  y2 )
t2  t1
y2
Head
fair
Levelling
0
y1–y2
y1
t1
79
2. Progressive tax
A higher income justifies a higher the tax rate:
tj
ti
ti
yi
yi  y j 

or

yi y j
tj yj
3. Regressive tax
A higher income justifies a lower the tax rate:
tj
ti
yi  y j 

yi y j
80
t2
regressive region: head tax is
the most regressive method
T
t2 y2

t1
y1
y2
0
y1–y2
T
y1
t1
81
t2
progressive region: levelling tax
is the most progressive method
T
y2
t2 y2

t1 y1
0
y1–y2
T
y1
t1
82
The exponential method is defined by:
ti  Min   yip , yi  for some p  0
where  is choosen such that
t
i
 T.
It is progressive for p > 1 and regressive for p < 1.
It is the flat tax for p = 1.
It is the head tax for p = 0.
83
t2
y2
0
y1–y2
y1
t1
84
Equal sacrifice
"Equality of taxation means equality of sacrifice.
It means apportioning the contribution of each
person towards the expenses of the government
so that he shall feel neither more nor less inconvenience from his share of the payment than
every other person experiences from his."
John Stuart Mill, Principle of Economics, 1848
85
Let ui(y) be the utility associated to income y by individual i.
Equal sacrifice means choosing taxes in such a way that differences of
utilities are equalized:
ui ( yi )  ui ( yi  ti )  z for all i
To avoid interpersonal utility comparisons, we postulate a common
utility function u (a kind of social norm):
u( yi )  u( yi  ti )  z for all i
Mill proposed to use the Bernoulli utility function log y.
86
u ( y)  log y yields the proportional tax:
yj
tj
yi
ti



yi  ti y j  t j
yi y j
Equal relative sacrifice means choosing taxes in such a way that ratios
of utilities are equalized:
u ( yi  ti )
 z for all i
u ( yi )
It is merely equivalent to equal absolute sacrifice: the log of the ratio
equals the difference of the log.
87
Proposition Equal sacrifice implies fair ranking if and only if
u is increasing and concave.
u ( yi  ti )  u ( y j  t j )  u ( yi )  u ( y j )

yi  y j  yi  ti  y j  t j
u ( yi )  u ( yi  ti )  u ( y j )  u ( y j  t j )

yi  ti  y j  t j  ti  t j
88
Proposition The equal sacrifice method is progressive if and only if
the function y.u'(y) is non-increasing in y (i.e. u is more concave than
the log function).
Proposition The equal sacrifice method is regressive if and only if
the function y.u'(y) is non-decreasing in y (i.e. u is less concave than
the log function).
89
The contested garment rule
"Two people cling to a garment. The decision is that one
takes as much as his grasp reaches, the other takes as
much as his grasp reaches, and the rest is divided equally
among them." (Talmud)
Hence the "contested garment rule" for n = 2 is given by:
where
1
xi  zi   E  z1  z2  i  1, 2
2
zi  Max  0, E  d ( N \ i)   E  Min  E, di  i  1, 2
90
The solution can then be alternatively written as:
x1 
1
E  Min  E , d1   Min  E , d 2  

2
1
x2   E  Min  E , d 2   Min  E , d1  
2
The contested garment rule satisfies both the truncation property and
the concession property. Actually, it is the only 2-person rule satisfying
these two properties. They define it.
An allocation rule has the contested garment property if, when
applied to a 2-person problem, it coincides with the contested garment
solution.
91
2.2 Egalitarism vs utilitarism
92
Egalitarian vs utilitarian solutions
egalitarian solution (compensation):
find ( x1,..., xn ) such that ui ( xi )  u j ( x j ) for all i, j and x( N )  E
utilitarian solution (sum-fitness):
find ( x1,..., xn ) that maximizes
u (x )
i
i
subject to x( N )  E
93
The egalitarian solution solution may not be defined. The proper
formulation should instead be the following:
find ( x1 ,..., xn )  0 such that x( N )  E
and
xi  0  ui ( xi )  Min j u j ( x j )
Whether or not the utility function are concave (decreasing marginal
utility) impacts the comparison of the two solutions.
In the concave case, the two solutions are in some sense identical.
Furthermore, the three solutions studied earlier turns out to be special
cases.
94
The link between the two solutions when utility functions are
increasing and concave (and differentiable) appears by comparing
the revised definition of the egalitarian solution and the first order
condition associated to the utilitarian solution:
xi  0  ui ( xi )  Min j u j ( x j )
xi  0  ui( xi )  Max j uj ( x j )
Hence, the utilitarian solution with utility functions ui corresponds to
the egalitarian solution with utility functions – ui'.
If concavity is quite natural in a context of income distribution,
convexity may be adequate in other context e.g. medical rationing.
95
The following example illustrates the role of concavity.
A quantity E of some resource has to be divided between n individuals.
Each individual i is initially endowed with a quantity i and his/her
preferences are represented by
ui ( xi )  u(i  xi )
where u is some common (base) and strictly concave utility function.
Mill has shown that the egalitarian and utilitarian solutions coincide:
they both equalize the final outcome i  xi .
96
The egalitarian solution reads:
xi  0  u (i  xi )  Min jN u ( j  x j )
 i  xi  Min jN ( j  x j )
because u is increasing. Hence i  xi   j  x j for all i, j.
We observe that this solution is equivalent to the uniform gain solution
applied to the problem of dividing the amount
E '  E  ( N )
with claims di  i .
97
The utilitarian solution is the solution of the following maximization
problem:
Max  u ( j  x j )
subject to:
x
j
E
xi  0 i  1,..., n
Using the 1st order conditions, we have:
xi  0  u(i  xi )  Max jN u( j  x j )
 i  xi  Max jN ( j  x j )
because u' is decreasing. Hence i  xi   j  x j for all i, j.
98
If now u is a strictly convex function, the egalitarian solution
is unchanged. Being a uniform gain solution, it is independent
of the choice of the base utility function that only needs to be
increasing.
The utilitarian solution instead allocates all the resource to the
richest individual !
Indeed if positive amounts xi and xj are allocated to the i and j
such that i   j strict convexity implies:
u(i  xi )  u( j  x j )  u(i  xi  x j )  u( j )
i.e. transferring xj to i increases the sum of the utilities.
99
Another example
Assume the utility functions are of the form
ui ( xi )  i u( xi )
where the i's are positive "productivities" and u is some base
and strictly concave utility function such that u(0) = 0.
Here the two principles give opposite recommendations.
100
The egalitarian solution simply equalizes utilities:
iu( xi )   j u( x j ) for all i, j
The utility function u being strictly increasing, shares and
productivities are negatively correlated:
i   j  xi  x j
The utilitarian solution is defined by the 1st order conditions
iu( xi )   j u( x j ) for all i, j
By strict concavity, shares and productivities are now positively
correlated:
i   j  xi  x j
101
2.3 Nash bargaining
102
Consider a game in strategic (normal) form (S1, S2, u1, u2) involving two
players.
We denote by A the set of consequences, allowing for correlated
strategies and we work directly on the expected utility set
U  u 
2
u   u1 (a ), u2 (a )  , a  A
Players may agree on a choice of strategies, knowing that in
case of disagreement, they find themself in some situation that
corresponds to a pair of utilities d = (d1,d2)  U(A).
One possibility is to refer to prudent (MaxMin) strategies in
which case di is the security level of player i.
103
A bargaining problem is defined by pair (U,d) where
U is a subset of 2
that is closed, convex and bounded above
d is a point in U such that there exists some u  U, u >> d
Example: the battle of sexes
strategic form
correlated strategies
a2
b2
a1
2,1
0,0
b1
0,0
1,2
a2
b2
a1
p1
p2
b1
p3
p4
0  pi  1 and
p
i
1
104
u2
(1,2)
U
(0,0)
battle of sexes
a2
b2
a1
2,1
0,0
b1
0,0
1,2
(2,1)
u1
105
In general, U is the convex hull of the utility pairs corresponding
to pure strategies:
u2
U
C7
U C8
u1
106
Bargaining problem need not result from a game situation. This is the
case of allocation problems like the bankcruptcy problem.
u2
L
C1 + C2 > L
C2
U
d = (0,0)
C1
L
u1
107
A solution to a bargaining problem (U,d) is a point u* in U satisfying the
following minimal properties:
collective rationality: u U such that u  u * and u  u *
individual rationality:u*  d
We look for a rule  associating a solution to any bargaining problem
(U,d).
A bargaining problem (U,d) is symmetric if d1 = d2 and inter-changing
the players results in the same set U i.e. the 45° line
is a symmetry axis of U.
108
individual + collective rationality
u2
d2
d
U
d1
u1
109
symmetric bargaining problem
45°
d
110
Three natural axioms to be imposed on a rule :
Efficiency (collective rationality):
there is no u  U such that u > (U,d)
Individual rationality :
1(U,d)  d1 and 2(U,d)  d2
Symmetry:
if (U,d) is symmetric then 1(U,d) = 2(U,d)
111
These three axioms determine the solution of symmetric
bargaining problems:
(U,d)
d
112
u2
battle of sexes
(1,2)
3 3
( , )
2 2
U
(0,0)
(2,1)
u1
113
Because utilities are expected utilities we need the following
further axiom.
Independence with respect to preference representation (covariance)
(U,d)  (V,c) wherevi = ai + bi ui (bi > 0)
ci = ai + bi di
 i(V,c) = ai + bi i(U,d) (i = 1,2)
Indeed we are in a cardinal framework with non comparable utilities
(independence of utility scales).
114
ui  di
vi 
bi  di
b2=5
(4.5, 4)
d2=3
b2-d2=2
(2.5, 1)
1
(0.5, 0.5)
1
d1=2
b1-d1=5
b1=7
115
This additional axiom determines the solution to bargaining
problems whose individually rational boundary is a line segment:
(U,d)
d
It is indeed just the middle of that segment.
116
The following axiom extends the solution to all bargaining problems.
Independence with respect to irrelevant alternatives:
If (U,d) and (V,d) are such that
U  V and (V,d)  U
then (U,d) = (V,d).
117
(V,d )  U

(U,d ) = (V,d )
UV
118
Here the bargaining problem (V,d) is defined by the line tangent to U
at its mid point:
u2
this is the Nash solution
(U,d)
U
d
u1
119
contour curves are
rectangular hyperbolas
u2
u*
(u1 – d1)(u2 – d2) = constant
U
d
u1
120
Indeed the line segment tangent to a rectangular hyperbola and restricted
to the axis is divided in its middle at the tangency point.
Hence, the Nash solution is nothing but the solution of the maximization
of the product of the gains on U:
MaxuU (u1 – d1)(u2 – d2)
121
u2
Ci > L /2
L

u1* = u2* = L/2
C2
L/2
0
u*
L/2
C1
L
u1
122
u2
C2 < L /2
L

u1* = L – C2
u2* = C2
L/2
C2
0
u*
L/2
C1
L
u1
123
Problem with the Nash solution: truncating the U set leaves
the solution unchanged !
u2
u*
0
u1
124
Relative utilitarism (Kalai et Smorodinsky)
Each individual has an aspiration level bi defined as the maximum utility
level compatible with individual rationality.
Nash solution does not depend on individual aspirations.
Relative utilitarism consists in satisfying individual in proportion
to their aspirations.
This solution relies on an monotonocity axiom replacing Nash's axiom of
independence with respect to irrelevant alternatives.
125
u2
b
u*
"idéal" point
d2
d
U
d1
u1
126
u2
0
u1
127
u2
bi = Ci
L
ui* =
Ci
L
C1 + C2
C2
u*
0
C1
L
u1
128
2.4 Social welfare orderings
129
Welfarism postulates that the welfare of individuals is the only
ingredient to be used to compare states of the world.
Cardinal welfarism assumes that
- individual welfare utilities are measured by a utility index
- utilities can be "compared"
Because it concentrates exclusively on utility profiles, welfarism
has no ethical content. For instance, the "non-envy" criterion does
not enter into account.
The task of the benevolent dictator is to compare utility profiles
(u1 ,..., un ) and to identify the best profile.
130
Efficiency-fitness is one of the basic concept of welfarism. It underlies
utilitarism.
Let A denote the set of feasible states. A state x in A Pareto-dominates
a state x' in A if
ui ( x)  ui ( x ') for all i, with strict inequality for at least one i
i.e. there is unanimity to move from state x to state x'.
A feasible state x is Pareto optimal if it is not dominated by any
feasible state.
The other principle is compensation. It underlies egalitarism.
131
The preferences of the benevolent dictator are denoted by . It is
called social welfare ordering and it is assumed to be complete and
transitive. The most widely used are:
- utilitarian: (u1,..., un )
- Nash: (u1,..., un )
(u1,..., un ) 
(u1,..., un ) 
u  u
i
i
u u
i
i
- egalitarian (leximin):
(u1 ,..., un )
where
L
(u1,..., un )  (v1,..., vn )
L
(v1,..., vn )
is the lexicographic ordering applied to a reordering
of the utility profiles in an increasing way.
132
Lifeboat Consider the case of 5 individuals and the following feasible
arrangments:
A  {{1, 2},{1,3},{1, 4},{2,3,5}{3, 4,5},{2, 4,5}}
Assume first that all individuals value equally being (10) and not being
onboard (1).
Utilitarism recommends choosing one of the 3-person arrangments.
Egalitarism recommends the same solution:
(1,1, 10,10,10)
L
(1,1, 1,10,10)
133
Assume now that utilities differ:
in
1
10
2
6
3
6
4
5
5
3
out
0
1
1
1
0
Utilitarism now recommends choosing either {1,2} or {1,3}. The
ranking is given by:
{1, 2} {1,3} {1, 4} {2,3,5} {2, 4,5} {3, 4,5}
134
in
1
10
2
6
3
6
4
5
5
3
out
0
1
1
1
0
Egalitarism recommends {2,3,5}.
Indeed the corresponding ranking is:
{2,3,5} {2, 4,5} {3, 4,5} {1, 2} {1,3} {1, 4}
obtained from:
(0,1, 3, 6, 6)
L
(0,1, 3, 5, 6)
L
(0,1,1, 6,10)
L
(0,1,1, 5,10)
135
Collective utility function
Most social welfare orderings can be represented by a collective utility
function W(u1,…,un).
A collective utility function W is additive if there exists some
increasing function f such that:
W (u1,..., un )   f (ui )
for all (u1,…,un).
136
Additive collective utility functions
Social welfare orderings are assumed to be complete and transitive.
Five additional assumptions
1. Monotonicity:
ui  ui for all i  j and u j  uj  (u1,..., un )
(u1,..., un )
2. Symmetry:
if (u1,..., un ) is obtained from (u1 ,..., un ) by permuting
individuals, then (u1 ,..., un ) (u1,..., un )
137
Monotonicity is compatible with Pareto optimality:
if (u1 ,..., un ) Pareto-dominates (u1,..., un )
then (u1 ,..., un ) (u1,..., un )
Hence, maximal elements on the set of feasible states A of a monotonic
social welfare ordering are Pareto-optimal.
Symmetry is equivalent to "equal treatment of equals": only differences
in utilities may justify discrimination.
138
3. Ignoring unconcerned individuals:
(ui , a)
(u i , a)  (ui , b)
(u i , b) for all a, b
where ui  (u j | j  i).
Hence social welfare orderings depends only on the welfare of the
individuals who are affected.
Proposition Any social welfare ordering represented by an
additive collective utility function satisfies the above property.
Under continuity, the converse is true: ignoring unconcerned
individual implies additivity.
139
4. Pigou-Dalton transfer principle: aversion for inequality
If the utility profiles (u1 ,..., un ) and (u1,..., un ) are such that:
u1  u2
ui  ui for all i  1, 2
u1  u1  a and u2  u2  a
then (u1,..., un )
(u1,..., un ).
i.e. operating a transfer that reduces the inequality between any two
individuals does not lead to a less preferred utility profile.
140
5. Independence of common scale
A common rescaling of every individual utility function leaves
the social welfare ordering unaffected:
(u1,..., un )
(u1,..., un )  (u1,..., un )
(u1,..., un )
whenever   0 and uiui  0 for all i.
Applied to an additive collective function, this property reads:
 f (u )  f (u)  0
i
i

 f (u )  f (u)  0
i
i
Restricting to increasing and continuous functions f leads to…
141
Proposition Any additive, increasing and continuous social welfare
ordering satisfying the invariance property (5) can be represented by
a collective utility function of one of the following three types:
W (u1 ,..., un )   uip for some p  0
 f (u )  u p
W (u1 ,..., un )   log ui
 f (u )  log u
1
W (u1 ,..., un )   p for some p  0  f (u )  u  p
ui
142
Maximizing
 log u is equivalent to maximizing u . Indeed,
i
i
log is an increasing function and we have:
log u
i
 log ui
Hence,  log uiis called the Nash collective utility function. It is the
limit of the other two families of utility function for p  0.
The classical utilitarian utility function
W (u1,..., un )  ui
is obtained by setting p = 1 in the first family.
143
Proposition An additive utility function
W (u1,..., un )   f (ui )
meets the Pigou-Dalton transfer principle if and only if the function f
is concave.
For instance, the quadratic utility function
W (u1,..., un )  ui2
2
promotes inequality. Indeed, because  ui    ui  transferring
2
utility to one individual is always preferable.
144
Conclusion: if we impose the five requirements
- monotonicity and symmetry
- ignoring unconcerned individuals
- aversion for inequality
- independance of common scale
we are left with the following family of utility functions:
W (u1 ,..., un )   uip for some p, 0  p  1
W (u1 ,..., un )   ui p for some p  0
including their limits for p  0.
It is a one dimensional family defined by a single parameter p  .
145
Leximin – egalitarian social welfare ordering
Equalization of utilities may not be possible because the ranges of
the utility functions differ.
Equalization of utilities may be incompatible with Pareto efficiency.
The leximin social welfare ordering selects the most egalitarian among
the Pareto optimal allocations.
146
The leximin welfare ordering cannot be represented by a collective
utility function.
However, it belongs to the family of additive concave collective utility
functions in a limit sense.
Proposition The social welfare ordering represented by the
collective utility function
W (u1,..., un )  ui p
converges to the leximin welfare ordering for p  .
147
no equality – efficiency trade-off
u2
u1 = u2
UT
U(A)
EG = LEX
u1 + u2 = constant
u1
148
no equality – efficiency trade-off
u2
u1 = u2
u1 + u2 = constant
U(A)
EG = LEX
UT
u1
149
equality – efficiency trade-off
u2
UT
u1 = u2
LEX
U(A)
u1 + u2 = constant
u1
150
u2
u1 = u2
NASH
U(A)
u1 u2 = constant
u1
151
Independence of the common utility space
The leximin ordering is invariant with respect to a common
transformation of the utilities:
(u1 ,..., un )
(T (u1 ),..., T (un ))
L
(v1 ,..., vn )
L
(T (v1 ),..., T (vn ))

Proposition Leximin is the only social welfare ordering satisfying the
Pigou-Dalton transfer principle and the independence of the common
utility space.
152
Independence of zero utilities
The utilitarian social welfare ordering is invariant of zero utilities:
(u1 ,..., un )
(v1 ,..., vn )

(u1  w1 ,..., un  wn ) (v1  w1 ,..., vn  wn )
for all (w1,..., wn ), or
(u1 ,..., un )
(v1,..., vn )  (u1  v1,..., un  vn )
(0,...,0)
Proposition The utilitarian social welfare ordering is the only social
welfare ordering satisfying independence of zero utilities.
153
Independence of utility scales
The Nash social welfare ordering is independent of utility scales:
(u1 ,..., un )
(v1 ,..., vn )

(a1u1  b1 ,..., anun  bn ) (a1v1  b1 ,..., an vn  bn )
for all ai  0 and bi .
Proposition The Nash social welfare ordering is the only social welfare
ordering satisfying independence of utility scales.
154
Example: location of a facility
Consider the "linear" city represented by the interval [0,1] along which
individuals are located:
individual i is located at ti [0,1]
If x denotes the location of the facility, the disutility of agent i is
measured by its distance to the x:
ui ( x)   x  ti
155
If there are agents located at 0 or 1, the egalitarian solution consists in
placing the facility in the middle: x  1 / 2. The corresponding ordered
utility vector is of the form (–1 /2, –1/2,….).
It differs from the utilitarian solution which picks the median
xˆ defined by:
1
{i | ti  xˆ} 
and
2
1
{i | ti  xˆ} 
2
This is indeed the point where total disutility is minimum: moving
away – in any direction – increases the disutility of at least 1/2 of
the individuals.
156
Both solutions coincide when the individuals are uniformly distributed
on the interval [0,1].
This is in particular the case of a continuum.
The choice of the solution depends upon the kind of facility, in particular
whether or not the facility is intented to meet basic needs (swimming
pool vs post office).
In some cases, the choice is difficult: where should a fire station
be located ?
157
Example: location of a noxious facility
Now, the distance to the facility measures the utility of agent i:
ui ( x)  x  ti
In the extreme case of a continuum, the egalitarian solution consists in
locating the facility anywhere because there is an individual in any
location.
The utilitarian solution now picks one of the extreme points.
The question is to compare the utilities at the end points.
158
Indeed, if f denotes the density function and  is the mean, we have:

1
0
1
x f ( x)dx   (1  x) f ( x)dx  2  1
0
Hence, location at 1 will be preferred if and only if  
1
.
2
f(x)
UT
0

1
x
159
Example: time sharing
The problem is to share a given length of time between m radio
programs to be broadcasted in a room where n individuals work.
Each individual is assumed to either like or dislike a program: utilities
are then either 0 or 1. Each program is supported by at least one
individual.
The problem is to allocate time in proportions t1,…,tm such that
tk  0 for all k and
t
k
1
160
Assume first that each individual likes one and only one program
and let nk denote the number of individuals who like program k:
0  nk  n for all k and
n
k
n
Utilitarism implies majority: it picks the program supported by the
largest group. In case where there is a tie, any combination is optimal.
Egalitarism does the opposite: each program is broadcasted equally i.e.
tk 
1
for all k
m
161
Assume now that individuals may be indifferent between radio
programs. Consider the following case where n = m = 5:
a
b
c
d
e
1
1
0
0
0
0
2
0
1
0
0
0
3
0
0
1
1
0
4
0
0
0
1
1
5
0
0
1
0
1
So as to equalize the portion of time each individual listen to a given
program, egalitarism suggests the following allocation:
2 2 1 1 1
x( , , , , )
7 7 7 7 7
162
a
b
c
d
e
1
1
0
0
0
0
2
0
1
0
0
0
3
0
0
1
1
0
4
0
0
0
1
1
5
0
0
1
0
1
Utilitarism instead suggest to forget about programs a and b, and to
concentrate on programs c, d and e, with an arbitrary allocation.
163
If one particular program is supported by a majority, for instance:
a
b
c
d
e
1
1
0
0
0
0
2
0
1
0
0
0
3
0
0
1
1
0
4
0
0
0
1
1
5
0
0
1
1
1
utilitarism would simply suggest to concentrate on that program,
without paying attention to those outside that majority.
164
2.5 Transferable utility games
165
TU-games
Given a collectivity N = {1,…,n}, a cooperative game with transferable
utility is defined by a "characteristic function" v that associates a real
number to any "coalition" S  N. Here v(S) is the worth of coalition S,
understood as the minimum it can secure for itself, independently of
what the players outside S do.
The set function v is assumed to be superadditive:
S  T    v(S )  v(T )  v(S  T )
a weaker requirement than convexity:
S , T  N  v(S )  v(T )  v(S  T )  v(S  T )
166
The problem is to share v(N) among the n players: find x = (x1,…,xn)
such that
x( N )  v( N )
The minimum requirements is individual rationality:
xi  v(i) for all i  N
This defines the set imputations:
I ( N , v)  {x 
n
| x( N )  v( N ), xi  v(i) for all i  N}
167
The core extends the rationality requirement from individuals to
coalitions:
x(S )  v(S ) for all S  N
The core is the set, possibly empty, of allocations satisfying these
conditions:
( N , v)  {x 
n
| x( N )  v( N ), x(S )  v(S ) for all S  N}
It is the set of allocations against which there can be no objections
from any coalition, including individuals. Hence
( N , v)  I ( N , v)
168
The core is not as such a solution. It is the set of "stable" allocations
and there may be no such allocations except for some classes of games
like for instance convex games.
There are two "rules" that defines "fair" allocations.
The Shapley value: it allocates v(N) on the basis of players marginal
contributions to all coalitions they belong to:
v( S )  v( S \ i )
It defines an imputation that may not belong to the core.
The nucleolus: it selects an allocation that is always defined and
belongs to the core when this one is nonempty.
169
Shapley value
To each permutation  = (i1,…,in)  N of the players is associated a
marginal contribution vector  () defined by:
i ( )  v(i1 )  v()  v(i1 )
1
i ( )  v(i1,..., ik )  v(i1,..., ik 1 )
k
(k  2,..., n)
The Shapley value is the average marginal contribution vector:
1
 ( N , v) 
 ( )

n!   N
170
Alternatively, the Shapley value can be written as:
i ( N , v) 
  (s) [v(S )  v(S \ i)]
SN
( S i )
n
( s  1)!(n  s )!
where the weights are given by  n ( s ) 
n!
The Shapley value is the unique allocation rule satisfying:
- symmetry:
contributions (substitute
treatments of equals)
players with identical marginal
players) get the same ( equal
-
null player: players never contributing (null players) get nothing
-
additivity: (N,v+w) = (N,v) + (N,w)
171
1 1
n  2 2  ( , )
2 2
2  (1,1)
1 1 1
n  3 3  ( , , )
3 6 3
3  (1, 2,1)
1 1 1 1
n  4 4  ( , , , )
4 12 12 4
4  (1,3,3,1)
1 1 1 1 1
n  5 5  ( , , , , )
5 20 30 20 5
5  (1, 4,6, 4,1)
1 1 1 1 1 1
n  6 6  ( , , , , , )
6 30 60 60 30 6
6  (1,5,10,10,5,1)
where n ( s)  Cns11 
(n  1)!
is the number of coalitions to which a given player belongs
(n  s)!( s  1)!
 n ( s) n ( s) 
1
for all s
n
172
Least core and nucleolus
The Shapley value is "fair" because it treats equal players equally and
does not remunerate non-contributing players. The nucleolus instead is
concerned with reducing the highest loss of the coalitions as measured
by the difference between wath a coalition is worth and what it gets:
e( x, S )  v(S )  x(S )
is the "excess" associated to imputation x and coalition S.
The least core is the set of imputations that minimize the largest excess:
Min xI ( N , v ) Max S  N e( x, S )
S , N
173
This is typically a set. The nucleolus goes further to eventually retain
a unique imputation:
to each imputation x is associated the vector  (x) of
dimension 2n – 2 obtained by placing the excesses e(x,S)
in a decreasing order
The nucleolus is then the unique imputations x that minimizes
lexicographically these vectors on the set of imputations I(N,v):
 (x )
L
 ( x) for all x  I ( N , v)
174
Example: "market" game
v(1) = v(2) = v(3) = v(23) = 0
v(12) = p2 ≤ p3
v(13) = v(123) = p3
The core is defined by:
( N , v)  { x  ( p,0, p3  p) p2  p  p3 }
In particular, if p3 = p2, then
( N , v)  { ( p3 ,0,0) }
175
1
2
3
123
0
200
100
132
0
0
300
213
200
0
100
231
300
0
0
312
300
0
0
321
300
0
0
1/6
1100
200
500
v(i) = 0
v(12) = p2 = 200
v(13) = p3 = 300
v(23) = 0
v(123) = p3 = 300
each row corresponds
to a permutation
chaque column corresponds
to a player
550 100 250
 ( N , v)  (
,
,
)  (183,33,83)
3
3
3
176
For any given coalition, the excess can be written as a fucntion of p:
e( p , S )   p
0
for S  {1}
for S  {2} and S  {13}
 p  p3 for S  {3} and S  {23}
 p2  p for S  {12}
For each p, order the excesses in a decreasing way:
(0, 0, p2  p, p  p3 , p  p3 ,  p) for p  [ p2 , p]
(0, 0, p  p3 , p  p3 , p2  p,  p) for p  [ p, p3 ]
p2  p3
where p 
2
177
p2
here the least core coincides with the
core and the nucleolus is its mid-point
p – p3
p  p3
p 2
2
0
p2
p
p3
(p2-p3)/2
p2-p3
p2 – p
- p3
–p
178
The nucleolus is the mid-point of the core:
p3  p2 
 p3  p2
 ( N , v)  
,0,

2
2


i.e (250, 0, 50) in the case where p3 = 300 and p2 = 200.
The nucleolus satisfies to two Shapley's axioms: symmetry and nul
player.
It does not satisfy additivity.
179
Example: crop game
Imagine a landlord and m (identical) workers, and a technology
described by a production function y = F(s) where s is the number
of workers:
v(S) = 0
if S does not include the landlord
v(S) = F(s – 1) if S includes the landlord
(he/she does not work)
In particular, v(i) = 0 for all i and v(N) = F(m).
We suppose that F is increasing with F(0) = 0, not more at this stage.
The associated game is superadditive. It is convex if returns to scale are
constant or increasing: linear or convex production function.
180
We first observe that the extreme allocation (F(m), 0, …,0) always
belongs to the core.
Let x be in the core. For all j  1, we have:
x( N \ j )  v( N \ j )  F (m  1)
where
x( N \ j)  x( N )  x j  F (m)  x j
Hence,
x j  F (m)  F (m  1)
 the most a worker can get within the core is
the marginal product [F(m) – F(m–1)]
181
Workers are substitutes: they get the same wage under the Shapley value.
We need only to compute what the value allocates to the landlord.
In a given permutation, only the position of the landlord counts.

if the landlord is in position k, he gets F(k-1)
and there are m + 1 positions possibles
 1 ( N , v) 
m 1
m
F (k  1) 
F (k )


m 1
m 1
1
k 1
1
k 1
182
F(m)
F(1) + F(2) + … + F(m)
F(k)
1 x F(k)
F(2)
F(1)
0
k k+1
m
183
decreasing
returns
F(m)
Workers
L>W
Landlord
m
  F (k )
k 0
0
m
184
constant
returns
F(m)
Workers
W=L
Landlord
0
m
185
increasing
returns
F(m)
T
Workers
L<W
Landlord
0
m
186
mixed
returns
F(m)
Workers
Landlord
0
m
187
The Talmud example
A man dies and his three wives have each a claim on his estate,
following past promises. The value of the estate falls short of the
total of the claims. Here is what a Mishnah suggests.
d1=100
d2=200
d3=300
E=100
33.3
33.3
33.3
E=200
50
75
75
E=300
50
100
150
EQUAL
?
UL
188
Aumann and Mashler (1985) have shown that the nucleolus actually
reproduces the Talmud figures for the following TU-game:
v(S )  Max  0, E  d ( N \ S ) 
Here v(S) represents the minimum coalition S can get:
it is the amount left once the outsiders have possibly
got their claims
In particular, v(N) = E.
The above game is known as "bankcruptcy game".
189
E = 200
d = (100,200,300)
v(S )  Max  0, E  d ( N \ S ) 
v(i) = 0 i = 1,2,3
v(12) = v(13) = 0
v(23) = 100
v(123) = 200
x1 , x2 , x3  0
x1  x2  x3  200  x1  100
x2  x3  100
Here players 2 and 3 are substitutes.
( N , v)  {( x1, x2 , x3 )  0 x1  a, x2  x3  200  a, 0  a  100 }
190
x1
(200,0,0)
200
0
I ( N , v)
x2
200
x3
200
x2 + x3 = 100
( N , v)
(0, 200,0)
x2 + x3 = 200
x1 = 0
(0,0, 200)
191
E = 200
d = (100,200,300)
v(1) = 0
v(2) = 0
v(3) = 0
v(12) = 0
v(13) = 0
v(23) = 100
v(123) = 200
v(S )  Max  0, E  d ( N \ S ) 
1
2
3
123
0
0
200
132
0
200
0
213
0
0
200
231
100
0
100
312
0
200
0
321
100
100
0
1/6
200
500
500
 200 500 500 
  ( N , v)  
,
,
  (33.7, 83.7, 83.7)
6
6 
 6
192
(200,0,0)
(100, 100, 0)
(100, 0, 100)
Equal
Nucleolus
Shapley
(0, 200,0)
(0,0, 200)
193
We observe that the four vertices of the core are precisely the four
marginal contribution vectors:
(0, 0, 200)
(0, 200, 0)
(100, 0, 100)
(100, 100, 0)
with multiplicity 2
with multiplicity 2
with multiplicity 1
with multiplicity 1
This is actually a characteristic of convex games. Actually:
the core of game is the convex hull of its marginal
contribution vectors if and only if it is a convex game
As a consequence, the Shapley value is in the core of convex games.
The bankcruptcy game is convex.
194
E = 200
d1 = 100 d2 = 200 d3 = 300
EQUAL
66.6
66.6
66.6
PROP
33.3
66.6
100
UG
66.6
66.6
66.6
UL
0
50
150
Nucleolus
50
75
75
Shapley
33.3
83.3
83.3
195
E = 100
d = (100,200,300)
v(S )  Max  0, E  d ( N \ S ) 
v(i) = 0 i = 1,2,3
v(12) = v(13) = v(23) = 0
v(123) = 200
x1 , x2 , x3  0
x1  x2  x3  200

( N , v)  I ( N , v)
The game is symmetric: all players are substitutes.
 i ( N , v )  i ( N , v ) 
200
3
196
E = 300
d = (100,200,300)
v(i) = 0 i = 1,2,3
v(12) = 0
v(13) = 100
v(23) = 200
v(123) = 300
v(S )  Max  0, E  d ( N \ S ) 
x1 , x2 , x3  0
x1  x2  x3  300
x1  x3  100
x2  x3  200
197
(300,0,0)
x1 + x3 = 100
x2 + x3 = 200
( N , v)
(0, 300,0)
(0,0, 300)
198
E = 300
d = (100,200,300)
v(i) = 0 i = 1,2,3
v(12) = 0
v(13) = 100
v(23) = 200
v(123) = 300
v(S )  Max  0, E  d ( N \ S ) 
1
2
3
123
0
0
300
132
0
200
100
213
0
0
300
231
100
0
200
312
100
200
0
321
100
200
0
1/6
300
600
900
 300 600 900 
  ( N , v)  
,
,
  (50, 100, 150)
6
6 
 6
199
(300,0,0)
x1 + x3 = 200
x1 + x2 = 150
x1 + x3 = 100
x2 + x3 = 200
Equal
Shapley = Nucleolus
(0, 300,0)
(0,0, 300)
200
(300,0,0)
(100, 0, 200)
(100, 200, 0)
(0, 300,0)
(0,200, 100)
(0,0, 300)
201
1
2
3
123
0
0
300
132
0
200
100
213
0
0
300
231
100
0
200
312
100
200
0
321
100
200
0
We observe again that the four vertices of the core are precisely
the four marginal contribution vectors:
(0, 0, 300)
(0, 200, 100)
(100, 0, 200)
(100, 200, 0)
with multiplicity 2
with multiplicity 1
with multiplicity 1
with multiplicity 2
confirming that the bankcruptcy game is convex.
202
Assignment games (Shapley and Shubik)
Consider a set N = {1,…,n} of agents and a set M = {1,…,m} (m  n)
of indivisible objects (say houses) to be allocated, one to each agent.
Each agent attaches a "utility" to each house. These data are
summarized in a utility matrix
[ui (h) | i  N , h  M ]
ui(h) is the reservation price of agent i for house h i.e. the maximum
price i is willing to pay for house h.
It is the value that agent i attach to house h expressed in monetary
terms.
203
Side payments being allowed, the associated TU-game is given by:
v(S )  Max f F

iS
ui ( f (i))
where F is the set of all functions f: N  M that associates a
house to each player.
Here v(S) is the cost of the houses that are optimally allocated to the
members of coalition S.
Consequently, (N,v) is a cost game. It is concave and thereby also
subadditive.
204
An optimal allocations of objects to players is associated to the
definition of C(N)
In the example below, it is (2,3,1): player 1 receives house 2, player 2
receives house 3, and player 3 receives house 1.
C(1) = 12
1
u1
u2
u3
3
9
9
C(2) = 9
C(3) = 9
C(12) = 21
2
12
6
6
3
9
6
3
C(13) = 21
C(23) = 15
C(123) = 27
205
An allocation (y1,…yn) of C(N) specifies for each player the
price he/she should pay for the object he/she has been assigned.
The associated prices are (12,6,9) and the Shapley value of the game is
given by
(N,C) = (12, 7.5, 7.5)
It implies the following side payments between players:
(0, 1.5, – 1.5)
i.e. player 1 stays put and player 2 pays 1.5 to player 3.
206
We observe that players 2 and 3 are substitute. The Shapley value is
obtained from the following table which associates marginal cost
vectors to players' permutations.
1
2
3
123
12
9
6
132
12
6
9
213
12
9
6
231
12
9
6
312
12
6
9
321
12
6
9
1/6
72
45
45
 (N,C) = (12, 7.5, 7.5)
207
The core is defined by the allocations satisfying the following
inequalities:
y1  y2  y3  27
y1  12
y1 = 12
y2  9
6  y2  9
y3  9
6  y3  9
y1  y2  21
y1  y3  21
optimal allocation before transfers
y2  y3  15
(12,6,9)
(12,9,6)
(12,7.5,7.5)
the Shapley value is located at the center of the core
208
(12,9,6)
(12,6,9)
set of
imputations
(27,0,0)
x2 = 6
x3 = 6
(9,9,9)
x2 = 9
x3 = 9
x1 = 12
(0,27,0)
(0,0,27)
209
3. Ordinal welfarism
210
A social choice procedure is a mapping F that associates alternatives
to preference profiles:
F : L( A)n  A
It associates to any profile p a subset of "winning" alternatives
F(p)  A. It is the collective choice set.
A social welfare function is a mapping F that associates "collective"
preferences to preference profiles:
F : L( A)n  L( A)
211
3.1 The case of two alternatives
212
Consider the case of 2 alternatives and n voters:
A = {0,1} and N = {1,…,n}
Assuming no indifference, a preference profile is a list of 0 and 1
of length n:
p = (p1,…,pn) where pi  L(A) = {0,1}
where
pi  1  1
i
0
pi  0  0
i
1
213
Example: n = 5 and p = (0,1,0,0,1)
3 in favour of 0
2 in favour of 1
There are 2n possible profiles.
The set of all possible preference profiles is {0,1}n.
A voting procedure is a mapping
F: {0,1}n  {0,1}
It associates to any profile p a subset F(p)  {0,1}.
F(p) is the "choice set".
214
There are 4 possible outcomes:
F(p) = {0}
F(p) = {1}
F(p) = {0,1}
F(p) = 
So ties are allowed.
The natural neutral mechanism to break a tie is the flipping of a coin.
215
Simple majority
F ( p )  {1}
F ( p )  {0}
n
n
if  pi 
2
i 1
n
n
if  pi 
2
i 1
F ( p )  {0,1} if
n
 pi 
i 1
n
2
 a tie is not a possible outcome of simple majority if n is odd
216
Unanimity
F ( p )  {1}
n
if
p
i 1
F ( p )  {0} if
n
p
i 1
F ( p)  
i
i
n
0
n
if 0   pi  n
i 1
217
A basic requirement to impose on a voting procedure is that it
produces a result:
Decisiveness A voting procedure is decisive if it never results
in the empty outcome:
F ( p)   for all p {0,1}n
Simple majority is always decisive. Unanimity is not.
218
What would be a fair voting procedure?
What are desirable properties a voting procedure should have
beyond decisiveness?
The result of a voting procedure should not depend on the identity of
the voters nor on the labelling of the alternatives:
voters and alternatives should be treated equally
219
Anonymity
A voting procedure F is anonymous if it symmetric in its n variables:
for any p  , permuting the voters leaves F(p) unchanged
For instance,
F (0,1,1,0,1)  F (1,0,1,0,1)  F (1,1,1,0,0)  ....
Anonymity clearly excludes dictatorship.
It is actually a stronger form of non-dictatorship.
220
Neutrality
A voting procedure F is neutral if permuting the choice of every voter
results in a permutation of the outcome:
for any p  P, F(1 – p) = 1 – F(p)
where 1 = (1,1,…,1).
For instance,
F (0,1,1,0,1)  {1}  F (1,0,0,1,0)  {0}
221
Proposition: A voting procedure is anonymous and neutral
if and only if it is the number of votes in favour
of an alternative which determines whether he/she
belongs to the choice set, i.e.
 n

F ( p)  G   pi 
 i 1 
for some increasing function G.
222
Alone, anonymity and neutrality allow for many different voting
procedures, including those based on stupid rules like:
F ( p)  {1}
1 1 n
4
if
  pi 
10 n i 1
10
F ( p)  {0} otherwise
If an alternative is elected and some voters change their minds in
favour of that candidate, it may be that he/she is not elected any more.
223
If, given the outcome F(p) corresponding to a preference profile p,
some voters change their mind in favour of a candidate who belongs
to the choice set F(p), we would expect that the resulting choice set
still includes that alternative.
Monotonicity A voting procedure is monotonic if
1 F ( p) and p  p  1 F ( p)
0  F ( p) and p  p  0  F ( p)
where p  p means pi  pi for all i.
224
An increased support for an alternative should never hurt.
An immediate consequence of monotonicity is strategyproofness:
a voter has no incentive to be insincere by
voting for the candidate he/she ranks second
Is it possible to characterize the procedures which satisfy these
3 axioms simultaneously ?
anonymity, neutrality and monotonicity
225
A quota procedure is defined by an integer
F ( p)  {1}
n
if
p
i 1
F ( p)  {0}
i
q,
n
 q  n, such that:
2
q
n
if n   pi  q 
i 1
n
p
i 1
i
 nq
F ( p)  {0,1} otherwise
Simple majority is defined by:
n 1
q
if n is odd
2
n
q   1 if n is even
2
226
Unanimity is also a quota procedure with q = n.
Proposition Quota procedures are the only voting procedures
which are anonymous, neutral and monotonic.
(i) quota procedures satisfy anonymity and neutrality by contruction:
their outcome depends on the sum of the pi's.
(ii) quota procedures satisfy monotonicity:
F(p) does not decrease when the sum of the pi's increases.
227
A stronger version of the monotonicity axiom is the following:
Strict monotonicity A voting procedure is strictly monotonic
(positive responsiveness)
if it monotonic and
F ( p)  {0,1} and p  p  F ( p)  {1}
F ( p)  {0,1} and p  p  F ( p)  {0}
where
p  p means pi  pi for all i and pj  p j for some j.
228
If some voters change their mind in favour of a candidate who
belongs to the initial choice set, then this alternative ends up
being the only winner.
In other words, either there was a tie and it disappears, or there
was a unique winner and he/she remains the unique winner.
Proposition Simple majority is the unique voting procedure which
(May, 1952)
is decisive, anonymous, neutral and strictly monotonic.
229
3.2 Social choice procedures
230
Borda method (1781)
- each of the m position is graded: m – 1 for the 1st,
m – 2 for the 2nd, …until 0 for the last
- looking at the preference ordering of each voter,
each alternative is graded accordingly
- adding the grades, each alternative receives a score
... the alternative(s) with the largest score wins.
231
n=7
a
b
c
d
e
Borda  b
a
d
b
e
c
a
d
b
e
c
a
14
c
b
d
e
a
b
17
c
d
b
a
e
c
16
b
c
d
a
e
e
c
d
b
a
d
16
e
7
232
Condorcet has criticized Borda's method.
Consider 3 alternatives and 30 voters,
19 with preferences a
b
c
11 with preferences b
c
a
For Condorcet, a should win while Borda assigns 41 to b against
38 to a.
Indeed a is preferred to b and c by 19 voters.
233
An alternative is Condorcet winner if...
... confronted to any other alternative, it comes before in more
than half of the orderings
1
a
2
a
3
a
4
c
5
c
6
b
7
e
b
b
d
b
d
c
c
c
d
b
d
b
d
d
d
e
e
e
a
a
b
e
c
c
a
e
e
a
(This does not define a decisive rule !)
234
Hare method (1861) "single transferable voting system"
- if an alternative comes on top of at least half of the orderings,
he/she wins
- if there is no such alternative, delete the alternative(s)
that are on top of the fewest ordering
- repeat the procedure with the remaining alternatives,...
235
a
b
c
d
e
a
d
b
e
c
a
d
b
e
c
c
b
d
e
a
c
d
b
a
e
b
c
d
a
e
e
c
d
b
a
delete d
236
a
b
c
e
a
b
e
c
a
b
e
c
c
b
e
a
c
b
a
e
b
c
a
e
e
c
b
a
delete b and e
237
a
c
a
c
a
c
c
a
c
a
c
a
c
a
delete a
Hare  c
238
Sequential pairwise voting (voting with an agenda)
The idea is that a sequence of alternatives is determined and followed.
For instance, d results from the sequence (a,b,c,d,e) but b that comes
out from the reverse sequence:
a
b
c
a
d
b
a
d
b
c
b
d
c
d
b
b
c
d
e
c
d
d
e
e
c
e
c
e
a
a
e
a
e
b
a
239
Pareto criteria
If all voters prefer x to y, then y cannot be in the social choice set.
a
a
a
c
c
b
e
b
d
d
b
a
c
c
c
b
b
a
b
a
a
d
e
e
e
d
d
b
e
c
c
d
e
e
d
240
Condorcet criteria
If there is a Condorcet winner, it must be in the social choice set.
1
a
2
a
3
a
4
c
5
c
6
b
7
e
b
b
d
b
d
c
c
c
d
b
d
b
d
d
d
e
e
e
a
a
b
e
c
c
a
e
e
a
241
Monotonicity criteria
Let the alternative x be in the social choice set for a given preference
profile p.
If the preference profile p is modified by moving up x in the ordering
of some voter,...
... x should remain in the social choice set.
242
Independence criteria (independence of irrelevant alternatives)
Assume that the social choice set includes x but not y.
If the preference profile P is modified, without altering the preferences
between x and y,...
... then the resulting choice set should still not include y.
243
Pareto
Condorcet
Monotonicity
Independance
Plurality
Yes
No
Yes
No
Borda
Yes
No
Yes
No
Hare
Yes
No
No
No
Agenda
No
Yes
Yes
No
Dictator
Yes
No
Yes
Yes
244
Plurality satisfies Pareto
If every voter prefers x to y, y cannot come on top of any ordering.
Borda satisfies Pareto
If x comes before y in all preference orderings, then x has more points
than y.
245
Hare satisfies Pareto
If every voter prefers x to y, y is not on top of any list.
Then, either some alternative is on top of more than
half of the orderings, it is the winner, not y,
or y (being absent from the the first row) is among
the alternatives to be deleted next.
Dictatorship satisfies Pareto:
if every voter prefers x to y, it is also the case of the dictator...
246
Sequential pairwise voting satisfies Condorcet
If an alternative is the Condorcet winner, it will by definition come out
of any sequence of pairwise votes.
Plurality satisfies monotonicity
If x be on top of the largest number of orderings, moving it up in some
ordering preserves this.
Borda satisfies monotonicity
Moving up an alternative in some of the orderings always increases the
number of points he/she gets...
247
Sequential pairwise voting satisfies monotonicity
Assume x is a social choice given a preference profile and
an agenda.
Moving x up in the preferences of some voter will certainly keep x in
the social choice set (with a larger margin).
Dictatorship satisfies monotonicity
If x is the social choice, it is on top of the dictator's ordering...
Dictatorship satisfies independence
If x is the social choice but not y, x is on top of the dictator's ordering
and will remain so...
248
Plurality does not satisfy Condorcet
1 to 4
5 to 7
8 and 9
a
b
c
b
c
b
c
a
a
a is plurality winner but b is Condorcet winner
249
Borda does not satisfy Condorcet
1, 2 and 3
a
4 and 5
b
b
c
c
a
b is Borda winner but a is Condorcet winner
250
Hare does not satisfy Condorcet
1 to 5
a
6 to 9
e
10 to 12
d
13 to 15
c
16 and 17
b
b
b
b
b
c
c
c
c
d
d
d
d
e
e
e
e
a
a
a
a
b is Condorcet winner but it will be deleted first
251
Dictatorship does not satisfy Condorcet
1
a
2
c
3
c
b
b
b
c
a
a
c is Condorcet winner while a is the "social"
choice if voter 1 is the dictator.
252
Hare does not satisfy monotonicity
1 to 7
a
8 to 12
c
13 to 16
b
17
b
b
a
c
a
c
b
a
c
a is the social choice according to Hare
253
If voter 17 moves a above b, ...
1 to 7
a
8 to 12
c
13 to 16
b
17
a
b
a
c
b
c
b
a
c
... c becomes the social choice
254
Plurality does not satisfy independence
a
a
b
c
b
b
c
b
c
c
a
a
a is the social choice and b is not
255
If voter 4 moves c between b and a, ...
a
a
b
b
b
b
c
c
c
c
a
a
... a and b are tied
256
Borda does not satisfy independence
1, 2 and 3
a
4 and 5
c
b
b
c
a
a is the social choice
257
If voters 4 and 5 move c between b and a, ...
1, 2 and 3
a
4 and 5
b
b
c
c
a
... b becomes the social choice
258
Hare does not satisfy independence
a
a
b
c
b
b
c
b
c
c
a
a
a is the social choice according to Hare
259
If voter 4 moves c between b and a, ...
a
a
b
b
b
b
c
c
c
c
a
a
... a and b are tied
260
Sequential pairwise voting does not satisfy Pareto
a
c
b
b
a
d
d
b
c
c
d
a
b dominates d in the sense of Pareto: all voters prefer b to d
but d results from the sequence (a,b,c,d) :
a defeats b, c defeats a but d defeats c.
261
Sequential pairwise voting does not satisfy Independance
c
a
b
b
c
a
a
b
c
The reverse sequence (c,b,a) produces a as social choice.
Interchanging c and b in the first ordering results in b as social choice
while no one has changed his/her mind about a and b.
262
An illustration: Bonn, Berlin or both ?
Bundestag, 20 June 1991
659 representatives, 3 alternatives:
a = government in Bonn and parliament in Berlin
b = government and parliament in Berlin
c = government and parliament in Bonn
A decision was eventually reached after a full day of debates.
263
Procedure adopted by the Council of Elders and the results:
Abstention
18/654
147/654
Bonn and Berlin
Yes
End
489/654 No
Abstention
29/657
Motion: NO to distinct locations
340/657
Yes
No
288/654
338/659
Abstention
1/659
Berlin
Bonn or Berlin
332/659
Bonn
264
Questions:
Which voting procedure should have been adopted ?
Does the actual voting procedure produce enough
information to enable a reconstruction of the preferences
of the 659 representatives ?
Would a different voting procedure have produced
a different outcome ?
265
Bonn-Berlin:
Leininger's results* based on a clever
reconstructed preference profile:
1. Majority would have been indecisive: 147/221/290.
2. Bonn would have been the plurality winner.
3. Berlin would have been the 2-step majority winner: 337/320.
4. Berlin is Condorcet winner:
B/A: 371/286
B/C: 337/320
A/C: 227/430
5. Bonn would have been the Borda winner:
B
C
A
A = 513
B = 708
C = 750
6. Berlin and Bonn would have probably won under approval voting.
*"The fatal vote: Bonn vs Berlin", Finanzarchiv, Neue Folge, Heft 1, 1993, 1-20
266
Scoring rules like Borda can be characterized. A scoring rule is defined
by a mapping that associates weights to alternatives (assuming strict
preferences) in terms of their positions in the preference lists.
Consistency
A social choice rule F is consistent if, for any two disjoint sets of
voters N and N', and preference profiles p and p' on a common a
set A of alternatives:
F ( p)
F ( p)    F ( p)
F ( p)  F ( p  p)
where p  p is the combined preference profile of N  N'.
267
Proposition (Young)
A voting procedure is anonymous, neutral and consistent
if and only if it is a scoring rule.
Remark: The Borda scoring rule has been axiomatized as well.
268
3.3 Impossibility theorems
269
Among the properties, the most desirable ones are certainly Pareto and
monotonicity. Condorcet comes next.
Independence appears as a strong requirement. It has indeed
be the object of much discussion in the literature.
We observe the following facts:
- only dictatorship satisfies the independence axiom
- only sequential pairwise voting satisfies the Condorcet axiom
270
Condorcet voting paradox
a
c
b
b
a
c
c
b
a
No Condorcet winner!
Whatever is the social choice, 2/3 of the voters are unhappy and
moreover, they agree on an other alternative !
271
There is a transitivity problem!
The collective preferences built by saying that
"x is preferred to y"
if and only if
"x is preferred to y by a majority of voters"
x
y
are not transitive:
a
b and b
c but c
a
... although individual preferences are.
272
One implicit assumption is made:
there is no retrictions on the preferences:
social choice function are defined for any
preference profile in L(A)n
The only requirement is that individual preferences are preorders.
273
Impossibility theorem 1 (Taylor)
There is no decisive social choice procedure satisfying
both the Condorcet and the independence criteria.
Proof:
- assume there exists such a procedure
- apply it to the preference profile underlying
the Condorcet paradox
- show that it produces no winner: none of the three
alternatives can be winning
274
Claim: a cannot be winning
(same arguments for b and c)
Consider the profile obtained from the Condorcet profile by moving
b down in the third list:
a
c
b
b
c
a
b
c
a
a
b
c
c
a
b
c
b
a
c is then Condorcet winner and must be in the choice set, not a.
Going back to the Condorcet profile by moving b up in the third list
should not affect the preferences between a and c.
So a should still be a non-winner.
275
Impossibility theorem 2 (Arrow)
Dictatorship is the only social welfare function
satisfying the Pareto and independence criteria
Impossibility theorem 3 (Gibbard)
Dictatorship is the only social choice procedure
satisfying the Pareto and monotonicity
276
These impossibility results remain true with a weaker Pareto
requirement:
If an alternative comes top in all preferences, then that
alternative must be the unique social choice.
This is indeed weaker than the original statement:
if all voters prefer x to y, then y cannot be
in the social choice set.
277
Strong monotonicity
A social choice procedure F is strongly monotone if for all preference
profile p and q in , and any alternative a in A:
if q is obtained from p by lifting a up in some preference list,
then either F(q) = F(p) or F(q) = a
Pushing up an alternative can only help that alternative.
Proposition (Muller and Satterthwaite)
Dictatorship is the only social choice procedure
satisfying the strong monotonicity
278
And what about the issue of manipulability:
do voters have an incentive to report thruthfully their
preferences i.e. to vote according to their preferences ?
This is the problem known as strategic voting.
It requires that voters have a fairly good idea of the preferences of
the others.
Non-manipulability could be one property that should be satisfied
by a "good" social procedure.
There too, there are impossibility results.
279
For a social procedure producing single outcomes, defining
manipulability is easy.
Let UA denote the set of utility functions representing complete
preorders on A.
Given a set A of alternatives, a social choice procedure F is nonmanipulable (or strategy-proof) if
ui ( F (u))  ui ( F (vi , ui ))
for all u U An , for all vi U A and for all i.
280
Example: Consider the Condorcet preference profile:
a
b
c
b c
c a
a b

a
b
c
b twice
c twice
a twice
If a two step procedure is followed, a being opposed to b first we get:
a vs b  a
a vs c  c
If voter 1 report the false preferences b
be b instead.
a
c the outcome would
281
Impossibility theorem 4 (Gibbard and Satterthwaite)
Assuming strict preferences, dictatorship is the only social
choice procedure that is onto* and strategy-proof.
The proof builds upon the impossibility theorem 3 according to which
Pareto and monotonicity implies dictatorship:
it is shown that a procedure satisfying strategy-proofness
also satisfies Pareto and monotonicity.
* i.e. surjective: each alternative can be winning for some preference profile.
282
Remark:
In the case of 2 alternatives, strategy-proofness is equivalent to
monotonicity and therefore, simple majority is strategy-proof.
Other questions:
Almost all social choice procedure are manipulable. The question
could then be: is it possible to measure the degree of manipulability ?
What about the possibility that a coalition of voters forms to jointly
agree on a voting strategy ?
283
3.4 Possibility theorems
284
One implicit assumption is that social choice procedures are defined
for any preference profile.
We shall see that restricting the possible preference profiles results
in possibility theorems.
We shall consider two kinds of restrictions, Sen coherence and single
peakedness, under which the collective preferences
defined by
x
y if and only if a majority of voters prefer x to y
are transitive.
285
Given a set A of m alternatives and a set N of voters, a preference
profile p  ( 1 ,..., i ,..., n ) is Sen coherent if, for all triplets (x,y,z)
in A, one of the following three situations arises:
x
i
y and x
i
z for all i  N
y
i
x and z
i
x for all i  N
y
i
x
i
z or z
i
x
i
y for all i  N
i.e. either x is preferred by all to y and z, or y and z are preferred to x
by all, or all place x between y and z.
286
Given a set A of m alternatives and a set N of voters, a preference profile
p  ( 1 ,..., i ,..., n ) is single peaked if there exists an ordering of the
alternatives such that each individual preference list has a peak.
For 3 alternatives {a,b,c} and an ordering, say (b,a,c), individual
preferences have a peak if one of the following situations arises:
a
b
b
a
c
c
a
b
c
a
c
b
a
b
a
b
c
c
a
b
a
c
c
b
287
Possibility theorem 1 (Sen)
Collective preferences derived from Sen coherent
preference profiles are transitive if n is odd.
Possibility theorem 2 (Black)
Collective preferences derived from single peaked
preference profiles are transitive if n is odd.
288
Possibility theorem 3 (Moulin)
If preferences are single-peaked and there is an odd number of
voters, there is a (unique) Condorcet winner, the "median peak".
Voters are ranked according to their peaks: a1  a2  ...  an .
The median peak is ak where k 
n 1
2
.
There is a strict majority (k ) "leftists" (ai  ak ).
There is also a strict majority (k ) of "rightists" (ai  ak ).
Leftists support ak when opposed to a greater outcome.
Rightists support ak when opposed to a smaller outcome.
289
Actually, the following proposition due to Moulin holds:
Restricted to preference profiles involving an odd number of
voters and for which a (unique) Condorcet winner (CW) exists,
any social choice procedure producing the Condorcet winner is
strategy-proof.
It is also coalitionally strategy-proof: no coalition of voters can
misreport its preferences and make its members better off.
290
Proof: Let D(A) be the set of individual preferences on A such that for
all profiles p in D(A)n, CW(p) exists.
Assume there exists a profile p in D(A)n, a coalition S in N and
preferences qS in D(A)s such that:
CW(p) = a and CW(qS,pN\S) = b  a
ui(a) < ui(b) for all i in S
set of voters preferring a to b
under profile p
 S  N(a,b| p) = 
The set N(a,b| p) is a strict majority and the set N(a,b| (qS,pN\S))
coincides with the set N(a,b| p). Hence b cannot be Condorcet
winner under the "false" profile (qS,pN\S).
291
Some interesting web sites:
social choice:
www.socialchoiceandbeyond.com
game theory (non-cooperative and cooperative):
www.citg.unige.it/siti_internet_web.html (in Italian)
www.econ.canterbury.ac.nz/personal_pages/paul_walker (historical)
arielrubinstein.tau.ac.il
www.economics.utoronto.ca/osborne/igt (the site of his book)
cooperative games:
www.econ.usu.edu/acaplan/tugames.htm (a nice piece of software for n = 3)
power indices:
powerslave.val.utu.fi
www.warwick.ac.uk/~ecaae (computation algorithms)
292