Social Choices

7/26/2019 Social Choices

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COLLECTIVE DECISION MAKING

Pierre Dehez

CORE

University of Louvain

[email protected]


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Outline

1. Preferences, utility and choices

2. Cardinal welfarism distributive justice

utilitarism vsegalitarism

Nash bargaining

social welfare orderings

transferable utility games

3. Ordinal welfarism the case of two alternatives

social choice procedures

impossibility theorems

possibility theorems


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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 theoryII: 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.


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1.Preferences, utility and choices


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Preferences

Preferences over a setAof alternativesare defined by a (binary)

relation overA:

bis not preferred to a

from which the strictpreference and indifferencerelations are deduced:

ais preferred to b

indifference between aand b

a b

a b

a b

[ and ]a b b a

[ and ]a b b a


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Preferences arerational if they verify the following properties:

- completeness:

- reflexivity:

- transitivity:

Completeness is by far the most demanding assumption!

A relation satisfying reflexivity and transitivity is apreorder.

We denote byL(A) the set of preorders on a setA.

[ or ] for all ,a b b a a b A

[ and ]a b b c a c

for alla a a A


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Ordinal utilities

A preference preorder carries no information on the intensity of

preferences:

if ais preferred to band cis preferred to d, we don't know

whether ais "more preferred" to bthan cis preferred to d

Under minimal assumptions, preferences can be represented by a

utility function which associates a real

number to each alternative, such that:

: : ( )u A a A u a

( ) ( )u a u b a b


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As such, this is an ordinalrepresentation of preferences:

only the sign of the difference provides

an information on the preferences between aand b

As a consequence, u and where is an arbitrary

increasing transformation, both represent the same preferences.

T(u) = u3is the simplest nonlinear transformation with range

( ) ( )u a u b

( ) ( ) 0

( ) ( ) 0

u a u b a b

u a u b a b

:T ( )v T u

.


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Choices

Given a set of alternativesAand preferences a choice is a best

element inA:

( ),L A

*

* for all

a A

a a a A

There may be several best elements. The set of solutions is called the

choice set.

or

* maximizes ( ) ona u a A


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Let denote the choice set associated to a setAof alternatives

and a preference relation Then:

( , )C A

There is indifference between the elements of a choice set.

In the multivalued case, a neutral mechanism is necessary to eventually

retain a uniquealternative.

For instance a random mechanism.

, ( , )a b C A a b

.


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Collectivity: preference profiles

Consider a setA of alternatives and nindividuals indexed by irunning

from 1 to n, each having a preference relation

Apreference profilePspecifies a preference relation for each member

of the group:

A utility profile can be associated to any alternative aA:

( ).i L A

1( ,..., ) ( )n

nP L A

1( ) ( ),..., ( ) n

nu a u a u a


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One of the questions addressed by social choice is the determination

of a collectivepreference ordering for comparing utility profiles.

There are several levels of independencethat 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


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Given a set of alternativeAand a preference profile onA

represented by utility functions u1,,un we define the attainable

utility set

The problem is then to pick up a point in this set, possibly given the

specification of a disagreement point din U(A).

1( ,..., ) ( )n

n L A

1( ) ( ),..., ( ) ,n

nU A u u u a u a a A


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1. Ordinal, non-comparable: full independence

This is the situation where each individual utility level is defined up toan arbitrary increasing transformation:

where the Ti's are arbitrary increasing transformation

1 1

1 1 1 1

( ,..., ) ( ,..., )

( ( ),..., ( )) ( ( ),..., ( ))

n n

n n n n

u u v v

T u T u T v T v

from into .


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2. Ordinal, comparable: independence of common utility space

This is the situation where individual utility levels are defined up to anarbitrary and commonincreasing transformation:

where Tis an arbitrary increasing transformation

1 1

1 1

( ,..., ) ( ,..., )

( ( ),..., ( )) ( ( ),..., ( ))

n n

n n

u u v v

T u T u T v T v

from into .


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4. Cardinal, partially comparable: independence of zero utilities

This is the situation where each individual utility level is defined up toan increasing and affine transformation:

Alternatively:

1 1

1 1 1 1

( ,..., ) ( ,..., )

( ,..., ) ( ,..., )

n n

n n n n

u u v v

u b u b v b v b

1for all , ..., .nb b

1 1 1 1( ,..., ) ( ,..., ) ( ,..., ) (0,..., 0)n n n nu u v v u v u v


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5. Cardinal, comparable: independence of utility scales and zero utilities

This is the situation where individual utility levels are defined up to anincreasing and affine commontransformation:

1 1

1 1

( ,..., ) ( ,..., )

( ,..., ) ( ,..., )

n n

n n

u u v v

a u b a u b a v b a v b

for all , , 0.a b a


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2.Cardinal welfarism


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2.1Distributive justice


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Liberalism: the social order emerges from the interaction of free wills.

Methodological individualismis at the root of liberalism.

Individuals are characterized by values, rightsand 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.


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Principle 1: ex ante equality

There arebasic rightslike 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 unequalexogenous rights which justify unequal

treatment.


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Principle 3: reward or penalize

justifies unequal sharesyi's of resources to compensate

for voluntary differencesin 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).


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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.


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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


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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 awaste of time auctioning is better if

individuals are comparable, because otherwise

it favors the rich


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Assumption: equal exogenous rights

the allocation depends only on the

distribution of claims or liabilities

An allocation methodis a rule that associates an allocation

to any given allocation problem such that

1 1( ,..., ) ( , ,..., )n nx x E d d

1( , ,..., )nE d d ( ) .x N E


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x2


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0

2

x1

d2

d1

PROP

11

1 2

22

1 2

dx E

d dd

x Ed d

22 1

1

dx xd

1 2x x E


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x2


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0

2

x1

d2

d1

ES

SURPLUS: d1+d2 < E and d1 > d2

d1d2

2 1 2 1( )x x d d

2 1x x

1 21

2 12

2

2

E d d

x

E d dx

x2


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0

2

x1

d2

d1

ES

SURPLUS: d1+d2 < E and d1< d2

d2d1

2 1 2 1( )x x d d 2 1x x

1 21

2 12

2

2

E d d

x

E d dx


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x2


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0x1

d2

d1

UG

2 1x x


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Uniform gain rule (in case of a deficit)


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Uniform gain rule(in case of a deficit)

satisfies

1

( , )

where satisfies ( , )

i i

n

ii

x Min z d

z E Min z d

for all .i ix d i

Herezcan be interpreted as the common gain.

This rule is also called "constrained" egalitarian.

y = f(z) 2y z y d z


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0

y f( )

zd2 d1

d1+d2

y2y d z

1 2( ) ( , ) ( , )f z Min z d Min z d

1 2d d

2d2

d2


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y = f(z) DEFICIT: d1+d2 > E


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0zd2 d1

d1+d2

1y d z 1 2 2y d d z

1 2( ) ( , 0) ( , 0)f z Max d z Max d z

1 2d d

d1d2

d1

DEFICIT: d1+d2 E

x2DEFICIT: d1+d2 > E


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0x1

d2

d1d1d2

2 1 2 1( )x x d d

DEFICIT: d1 d2 E

1 21

2 12

2

2

E d dx

E d dx

2 1x x

UL


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Proportional (surplus/deficit)( )

i i

Ex d

d N


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Proportional (surplus/deficit)

Equal surplus (surplus)

Uniform gain (surplus)

Uniform gain (deficit)

Uniform loss (deficit)( ,0)

where is such that

i i

i

x Max d z

z x E

( )i i

d N

1( ( ))i ix d E d Nn

( , )

where is such that

i i

i

x Min z d

z x E

( , )

where is such that

i i

i

x Max z d

z x E


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Surplus: n= 5,E= 80 and d= (20, 16, 10, 8, 6)


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1 216 20 and 165

Ex x

3 4 5

3614.7 14.7

3

Ex x x

( ) 60d N E


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E= di= 20 16 10 8 6

20 PRO 6 7 5 3 3 3 2 7 2

60id


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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

40 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

50 PRO 16.7 13.3 8.3 6.7 5

UG 13 13 10 8 6UL 18 14 8 6 4

80 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

120 PRO 40 32 20 16 12

UG 24 24 24 24 24

ES 32 28 22 20 18

deficit

su

rplus


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1. Invariance to transfers


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if iandj"merge" into a single individual, is the resulting

share equalto the sum of the individuals' shares ?

Only theproportional ruleis invariant to transfers.

The uniform gain ruleis not: merging leads to a smalleror equal share.

The uniform loss ruleis not: merging leads to a higher

or equal share.

2. Truncation property


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In case of a deficit, a solution satisfies the truncation property

if truncating the claims toE

does not affect the resulting allocation.

The uniform gainrulesatisfies the truncation property.

The uniform loss ruleand theproportional rule do not.

,i i id d Min d E


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1 2y y


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y1

y2

0 t1

t2

FlatHead

Levelling

y1y2

2 1 1 2( )t t y y 2 1t t

22 1

1

yt t

y

1 2y y

1 2y y


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y1

y2

0 t1

t2

FlatHead

Levelling

y1y2

2 1 1 2( )t t y y 2 1t t

22 1

1

yt t

y

1 2y y

Principles


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1. Fair ranking

A higher income justifies both a higher tax burden and a higher after-

tax income :

Under this principle, equal incomes are taxed equally.

i ji j

i i j j

t ty yy t y t


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2. Progressive tax


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A higher income justifies a higher the tax rate:

3. Regressive tax

A higher income justifies a lower the tax rate:

orji i i

i j

i j j j

tt t yy y

y y t y

jii j

i j

tty y

y y


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progressive region: levelling tax


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0

progressive region: levelling tax

is the most progressive method

y1

y2

t1

t2

y1y2 T

T

2 2

1 1

t y

t y

The exponential methodis defined by:


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where is choosen such that

It is progressive forp> 1and regressive forp< 1.

It is the flat tax forp= 1.

It is the head tax forp= 0.

, for some 0pi i it Min y y p

.it T

t2


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y1

y2

0 t1y1y2

Equal sacrifice


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"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 incon-

venience from his share of the payment than

every other person experiences from his."

John Stuart Mill,Principle of Economics, 1848

Let ui(y) be the utility associated to incomeyby individuali.


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Equal sacrifice means choosing taxes in such a way that differences of

utilities are equalized:

To avoid interpersonal utility comparisons, we postulate a common

utility function u (a kind of social norm):

Mill proposed to use the Bernoulli utility function logy.

( ) ( ) for alli i i i iu y u y t z i

( ) ( ) for alli i iu y u y t z i

yields theproportional tax:

j jy ty t

( ) logu y y


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Equal relativesacrifice means choosing taxes in such a way that ratios

of utilities are equalized:

It is merely equivalentto equal absolute sacrifice: the log of the ratio

equals the difference of the log.

j ji i

i i j j i j

y ty t

y t y t y y

( )

for all( )i i

i

u y t

z iu y


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The solution can then be alternatively written as:

1

E Mi E d Mi E d


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The contested garment rule satisfies both the truncation property and

the concession property. Actually, it is the only 2-person rule satisfyingthese two properties. They define it.

An allocation rule has the contested garment propertyif, when

applied to a 2-person problem, it coincides with the contested garmentsolution.

1 1 2

2 2 1

, ,2

1, ,

2

x E Min E d Min E d

x E Min E d Min E d


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2.2 Egalitarism vsutilitarism

Egalitarian vsutilitarian solutions

egalitarian solution (compensation):


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egalitarian solution(compensation):

utilitarian solution(sum-fitness):

1find ( ,..., ) such that ( ) ( ) for all , and ( )n i i j jx x u x u x i j x N E

1find ( ,..., ) that maximizes ( ) subject to ( )n i ix x u x x N E

The egalitarian solution solution may not be defined. The proper

formulation should instead be the following:


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1find ( , ..., ) 0 such that ( )and

0 ( ) ( )

n

i i i j j j

x x x N E

x u x Min u x

Whether or not the utility function are concave (decreasing marginalutility) 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.

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


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the revised definition of the egalitarian solution and the first order

conditionassociated to the utilitarian solution:

0 ( ) ( )i i i j j jx u x Min u x

0 ( ) ( )i i i j j jx u x Max u x

Hence, the utilitarian solution with utility functions uicorresponds to

the egalitarian solution with utility functionsui'.

If concavity is quite natural in a context of income distribution,

convexity may be adequate in other context e.g. medical rationing.


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The egalitarian solutionreads:

0 ( ) ( )i i i j N j jx u x Min u x


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becauseuis increasing. Hence

We observe that this solution is equivalentto the uniform gain solutionapplied to the problem of dividing the amount

with claims

0 ( ) ( )

( )

i i i j N j j

i i j N j j

x u x Min u x

x Min x

' ( )E E N

for all , .i i j jx x i j

.i id

The utilitarian solutionis the solution of the following maximization

problem:


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Using the 1storder conditions, we have:

because u'is decreasing. Hence

0 ( ) ( )

( )

i i i j N j j

i i j N j j

x u x Max u x

x Max x

( ) subject to:

0 1,...,

j j j

i

Max u x x E

x i n

for all , .i i j jx x i j

If now uis a strictly convexfunction, 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


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of the choice of the base utility functionthat only needs to be

increasing.

The utilitarian solution instead allocates all the resource to the

richest individual!

Indeed ifpositiveamountsxiandxjare allocated to the iandj

such that strict convexity implies:

i.e. transferringxjto i increases the sum of the utilities.

i j

( ) ( ) ( ) ( )i i j j i i j ju x u x u x x u

Another example

Assume the utility functions are of the form


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Assume the utility functions are of the form

where the i's are positive "productivities" and uis some base

and strictly concaveutility function such that u(0) = 0.

Here the two principles give opposite recommendations.

( ) ( )i i i iu x u x

The egalitarian solutionsimply equalizes utilities:

( ) ( ) for all ,i i j ju x u x i j


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The utility function ubeing strictly increasing, shares and

productivities are negatively correlated:

The utilitarian solutionis defined by the 1storder conditions

By strict concavity, shares and productivities are nowpositively

correlated:

( ) ( ) for all ,i i j ju x u x i j

i j i jx x

i j i jx x


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2.3Nash bargaining

Consider a game in strategic (normal) form (S1, S2, u1, u2) involving two

players.

We de te b A the et f e e e ll i f el ted


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We denote byAthe set of consequences, allowing for correlated

strategies and we work directly on the expectedutility set

Players may agree on a choice of strategies, knowing that incase 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 diis the security level of player i.

2 1 2( ), ( ) ,U u u u a u a a A

A bargaining problem is defined by pair (U,d) where

Uis a subset of that is closed, convex and bounded above2


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dis a point in Usuch that there exists some uU, u>> d

Example: the battle of sexes

strategic form correlated strategiesa2 b2

a1 2,1 0,0

b1 0,0 1,2

a2 b2

a1 p1 p2

b1 p3 p4

0 1 and 1i ip p

(1 2)

u2battle of sexes


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(1,2)

(2,1)

(0,0) u1

U

a2 b2

a1 2,1 0,0

b1 0,0 1,2

In general, Uis the convex hull of the utility pairs corresponding

topure strategies:

u2


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u1

C7

C8

U

U

Bargaining problem need not result from a game situation. This is the

case of allocation problems like the bankcruptcy problem.

u2


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u1

L

L

C2

C1

U

C1+ C2>L

A solution to a bargaining problem (U,d) is a point u* in Usatisfying the

following minimalproperties:

collective rationality: such that * and *u U u u u u


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collective rationality:

individual rationality:

We look for a ruleassociating a solution to any bargaining problem

(U,d).

A bargaining problem (U,d) is symmetricif d1= d2and inter-changing

the players results in the same set Ui.e. the 45line

is a symmetry axis of U.

such that and u U u u u u

*u d

u2

individual + collective rationality


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u1d1

d2

U

d

symmetric bargaining problem


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45

d


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These three axioms determine the solution of symmetric

bargaining problems:


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(U,d)

d

(1,2)

u2 battle of sexes


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( , )

(2,1)

(0,0) u1

U

3 3( , )2 2

Because utilities are expected utilities we need the following

further axiom.

Independence with respect to preference representation (covariance)


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p p p p ( )

(U,d) (V,c) wherevi= ai+ biui (bi> 0)

ci= ai+ bidi

i(V,c) = a

i+ b

i

i(U,d) (i = 1,2)

Indeed we are in a cardinalframework with non comparable utilities

(independence of utility scales).

b 5

i ii

i i

u dv

b d


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d1=2 b1=71

1

b2-d2=2

b1-d1=5

(4.5, 4)

(2.5, 1)

(0.5, 0.5)

d2=3

b2=5i i


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The following axiom extends the solution to all bargaining problems.

Independence with respect to irrelevant alternatives:


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If (U,d) and (V,d) are such that

UV and (V,d) U

then (U,d) = (V,d).

(V,d) U


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(U,d) = (V,d)

UV

u2

Here the bargaining problem (V,d) is defined by the line tangent to Uat its mid point:


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u1d

U

(U,d)

this is theNash solution

u2contour curves are

rectangular hyperbolas


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u1d

(u1d1)(u2d2) = constantu*

U

Indeed the line segment tangent to a rectangular hyperbola and restricted

to the axis is divided in its middle at the tangency point.


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Hence, the Nash solution is nothing but the solution of the maximizationof theproduct of the gains on U:

MaxuU(u1d1)(u2d2)

u2

L Ci>L/2


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0 u1

L

C2

C1

u1* = u2* =L/2u*

L/2

L/2

u2

L

C2


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0 u1

L

C2

C1

u1* =LC2u2* = C2

u*L/2

L/2

u2

Problem with the Nash solution: truncating the Uset leavesthe solution unchanged !


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0 u1

u*

Relative utilitarism (Kalai et Smorodinsky)

Each individual has an aspiration levelbi defined as the maximum utility

level compatible with individual rationality


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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.

u2

bu*


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u1d1

d2

U

d

"idal" point

u2


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0 u1

u2

L

bi= Ci

ui* =Ci L


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0 u1L

C2

C1

u*

C1+ C2


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2.4 Social welfare orderings

Welfarism postulates that the welfareof individuals is the only

ingredient to be used to compare states of the world.

Cardinal welfarismassumes that


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- 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 doesnot enter into account.

The task of the benevolent dictator is to compare utility profiles

and to identify the best profile.1( ,..., )nu u

Efficiency-fitnessis one of the basic concept of welfarism. It underlies

utilitarism.

LetAdenote the set of feasible states. A statex in APareto-dominates


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a statex'inAif

i.e. there is unanimityto move from statexto statex'.

A feasible statexis Pareto optimalif it is notdominated by any

feasible state.

The other principle is compensation. It underlies egalitarism.

with strict inequality( ) ( ') for all for at least on, ei iu x u x i i

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:

.


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- utilitarian:

- Nash:

- egalitarian (leximin):

1 1( ,..., ) ( ,..., )n n i iu u u u u u

1 1 1 1( ,..., ) ( ,..., ) ( ,..., ) ( ,..., )n n n L nu u u u v v v v

where is the applied to a reordering

of the utility profile

lexicographi

s in an incr

c

e

orderin

asing .

g

way

L

1 1( ,..., ) ( ,..., )n n i iu u u u u u

Lifeboat Consider the case of 5 individuals and the following feasible

arrangments:

{{1,2},{1,3},{1,4},{2,3,5}{3,4,5},{2,4,5}}A


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Assume first that all individuals value equally being (10) and not being

onboard (1).

Utilitarismrecommends choosing one of the 3-person arrangments.

Egalitarismrecommends the same solution:

(1,1, ,10,10) (1,1, ,10,10)10 1L

Assume now that utilities differ:

1 2 3 4 5


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Utilitarismnow recommends choosing either {1,2} or {1,3}. Theranking is given by:

in 10 6 6 5 3

out 0 1 1 1 0

{1,2} {1,3} {1,4} {2,3,5} {2,4,5} {3,4,5}

1 2 3 4 5

in 10 6 6 5 3

out 0 1 1 1 0


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Egalitarismrecommends {2,3,5}.

Indeed the corresponding ranking is:

obtained from:

(0,1, 3, 6, 6) (0,1, 3, 5, 6) (0,1,1, 6,10) (0,1,1, 5,10)L L L

{2,3,5} {2,4,5} {3,4,5} {1,2} {1,3} {1,4}

Collective utility function

Most social welfare orderings can be represented by a collective utility

function W(u1,,un).


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A collective utility function Wis additiveif there exists some

increasing functionfsuch that:

for all (u1,,un).

1( ,..., ) ( )n iW u u f u

Additive collective utility functions

Social welfare orderings are assumed to be complete and transitive.

i ddi i l i


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Five additional assumptions

1. Monotonicity:

2.Symmetry:

1 1for all and ( ,..., ) ( ,..., )i i j j n nu u i j u u u u u u

1 1

1 1

if ( ,..., ) is obtained from ( ,..., ) by permuting

individuals, then ( ,..., ) ( ,..., )

n n

n n

u u u u

u u u u

Monotonicity is compatible with Pareto optimality:

1 1

1 1

if ( ,..., ) Pareto-dominates ( ,..., )

then ( ,..., ) ( ,..., )

n nu u u u

u u u u


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Hence, maximal elementson the set of feasible statesAof a monotonic

social welfare ordering are Pareto-optimal.

Symmetry is equivalent to "equal treatment of equals": only differences

in utilities may justify discrimination.

1 1n n

3. Ignoring unconcerned individuals:

( , ) ( , ) ( , ) ( , ) for all ,i i i iu a u a u b u b a b

where ( | ).i ju u j i


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Hence social welfare orderings depends only on the welfare of the

individuals who are affected.

Proposition Any social welfare ordering represented by anadditivecollective utility function satisfies the above property.

Under continuity, the converse is true: ignoringunconcerned

individual implies additivity.

4. Pigou-Dalton transfer principle: aversion for inequality

If the utility profiles are such that:1 1( , ..., ) and ( , ..., )n nu u u u

1 2


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then

i.e. operating a transfer that reduces the inequality between any two

individuals does not lead to a less preferred utility profile.

1 2

1 1 2 2

for all 1,2

and

i i

u uu u i

u u a u u a

1 1( ,..., ) ( ,..., ).n nu u u u

5. Independence of common scale

A common rescalingof every individual utility function leaves

the social welfare ordering unaffected:


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Applied to an additive collective function, this property reads:

Restricting to increasing and continuous functions f leads to

1 1 1 1( ,..., ) ( ,..., ) ( ,..., ) ( ,..., )n n n nu u u u u u u u

whenever 0 and 0 for all .i iu u i

( ) ( ) 0 ( ) ( ) 0i i i if u f u f u f u

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:


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1

1

1

( ,..., ) for some 0 ( )

( ,..., ) log ( ) log

1( ,..., ) for some 0 ( )

p p

n i

n i

p

n p

i

W u u u p f u u

W u u u f u u

W u u p f u uu

Maximizing is equivalent to maximizing Indeed,

log is an increasing function and we have:

log logi i

u u

.iulog iu


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Hence, is called theNash collective utility function. It is the

limit of the other two families of utility function forp 0.

The classical utilitarian utilityfunction

is obtained by settingp= 1 in the first family.

i i

log iu

1( ,..., )n iW u u u

Proposition An additive utility function

meets the Pigou-Dalton transfer principle if and only if the functionf

1( ,..., ) ( )n iW u u f u


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is concave.

For instance, the quadratic utility function

promotes inequality. Indeed, because transferring

utility to one individual is always preferable.

21( ,..., )n iW u u u

22

i iu u

Conclusion: if we impose the five requirements

- monotonicity and symmetry

- ignoring unconcerned individuals

- aversion for inequality

- independance of common scale


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- independance of common scale

we are left with the following family of utility functions:

including their limits forp0.

It is a one dimensional familydefined by a single parameter

1

1

( ,..., ) for some , 0 1

( ,..., ) for some 0

p

n i

pn i

W u u u p p

W u u u p

.p

Leximinegalitarian social welfare ordering

Equalization of utilities may not be possible because the ranges of

the utility functions differ.


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Equalization of utilities may be incompatible with Pareto efficiency.

The leximin social welfare orderingselects the most egalitarian among

the Pareto optimal allocations.

The leximin welfare ordering cannotbe represented by a collective

utility function.

However, it belongs to the family of additive concave collective utility

f i i li i


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functions in a limit sense.

Proposition The social welfare ordering represented by the

collective utility function

converges to the leximin welfare ordering

1( ,..., ) p

n iW u u u

for .p

u2u1= u2

UT

no equalityefficiency trade-off


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u1

u1+ u2= constant

EG = LEXU(A)

u2u1= u2

u1+ u2= constant

no equalityefficiency trade-off


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u1

UT

EG = LEXU(A)

u2u1= u2

UT

LEX

equalityefficiency trade-off


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u1

u1+ u2= constant

U(A)

u2u1= u2

NASH


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u1

u1u2= constant

U(A)

Independence of the common utility space

The leximin ordering is invariant with respect to a common

transformationof the utilities:


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Proposition Leximin is the onlysocial welfare ordering satisfying the

Pigou-Dalton transfer principle and the independence of the common

utility space.

1 1

1 1

( , ..., ) ( ,..., )

( ( ),..., ( )) ( ( ),..., ( ))

n L n

n L n

u u v v

T u T u T v T v

Independence of zero utilities

The utilitarian social welfare ordering is invariant of zero utilities:

1 1( ,..., ) ( ,..., )n nu u v v


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1 1 1 1

( ,..., ) ( ,..., )

( ,..., ) ( ,..., )n n n n

u u v v

u w u w v w v w

1for all ( ,..., ), ornw w

1 1 1 1( ,..., ) ( ,..., ) ( ,..., ) (0,..., 0)n n n nu u v v u v u v

Proposition The utilitarian social welfare ordering is the only social

welfare ordering satisfying independence of zero utilities.

Independence of utility scales

The Nash social welfare ordering is independent of utility scales:

1 1( ,..., ) ( ,..., )n nu u v v


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1 1

1 1 1 1 1 1

( , , ) ( , , )

( ,..., ) ( ,..., )

n n

n n n n n na u b a u b a v b a v b

Proposition The Nash social welfare ordering is the only social welfare

ordering satisfying independence of utility scales.

for all 0 and .i ia b

Example: location of a facility

Consider the "linear" city represented by the interval [0,1] along which

individuals are located:


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Ifxdenotes the location of the facility, the disutilityof agent iis

measured by its distance to thex:

( )i iu x x t

individual is located at [0,1]ii t

If there are agents located at 0 or 1, the egalitarian solutionconsists in

placing the facility in the middle: The corresponding ordered

utility vector is of the form (1 /2,1/2,.).

It differs from the utilitarian solution which picks the median

1/ 2.x


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d e s o e u so u o w c p c s e ed

defined by:x

1 1 { | } and { | }

2 2i ii t x i t x

This is indeed the point where total disutility is minimum: moving

awayin any directionincreases the disutility of at least 1/2 of

the individuals.

Both solutions coincide when the individuals are uniformlydistributed

on the interval [0,1].

This is in particular the case of a continuum.


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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 vspost office).

In some cases, the choice is difficult: where should a fire station

be located ?

Example: location of a noxiousfacility

Now, the distance to the facility measures the utilityof agent i:

( )i iu x x t


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In the extreme case of a continuum, the egalitarian solution consists in

locating the facility anywherebecause 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.

Indeed, iff denotes the density function and is the mean, we have:

Hence, location at 1will be preferred if and only if

1 1

0 0( ) (1 ) ( ) 2 1x f x dx x f x dx

1 .2


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2

f(x)

x0 1

UT

Example: time sharing

The problem is to share a given length of time between mradio

programsto be broadcasted in a room where nindividuals work.


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Each individual is assumed to either like or dislikea 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,,tmsuch that

0 for all and 1k kt k t

Assume first that each individual likes one and only oneprogram

and let nkdenote the number of individuals who like program k:

0 for all andk kn n k n n


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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 equallyi.e.

1for allkt k

m

Assume now that individuals may be indifferentbetween radio

programs. Consider the following case where n= m= 5:

a b c d e

1 1 0 0 0 02 0 1 0 0 0


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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, egalitarismsuggests the following allocation:

2 2 1 1 1( , , , , )7 7 7 7 7

x

a b c d e

1 1 0 0 0 0

2 0 1 0 0 0

3 0 0 1 1 0


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4 0 0 0 1 1

5 0 0 1 0 1

Utilitarism instead suggest to forget about programs aand b, and to

concentrate on programs c, dande, with an arbitrary allocation.

a b c d e

1 1 0 0 0 0

2 0 1 0 0 0

3 0 0 1 1 0

If one particular program is supported by a majority, for instance:


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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.


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2.5 Transferable utility games

TU-games

Given a collectivityN= {1,,n}, a cooperative game with transferable

utility is defined by a "characteristic function" vthat associates a real

number to any "coalition" SN. Here v(S) is the worthof coalition S,understood as the minimum it can secure for itself, independently of


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understood as the minimumit can secure for itself, independently of

what the players outside Sdo.

The set function vis assumed to be superadditive:

a weaker requirement than convexity:

( ) ( ) ( )S T v S v T v S T

, ( ) ( ) ( ) ( )S T N v S v T v S T v S T

The problem is to share v(N) among the nplayers: findx= (x1,,xn)such that

The minimum requirements is individual rationality:

( ) ( )x N v N


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This defines the set imputations:

( ) for allix v i i N

( , ) { | ( ) ( ), ( ) for all }n iI N v x x N v N x v i i N

The coreextends the rationality requirement from individuals tocoalitions:

The core is the set, possibly empty, of allocations satisfying theseconditions:

( ) ( ) for allx S v S S N


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It is the set of allocations against which there can be no objections

from any coalition, including individuals. Hence

( , ) { | ( ) ( ), ( ) ( ) for all }nN v x x N v N x S v S S N

( , ) ( , )N v I N v

The core is not as such a solution. It is the set of "stable" allocationsand 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.


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The Shapley value: it allocates v(N) on the basis of players marginal

contributions to all coalitions they belong to:

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.

( ) ( \ )v S v S i

Shapley value

To each permutation = (i1,,in) Nof the players is associated a

marginal contribution vector () defined by:

1 1 1( ) ( ) ( ) ( )i v i v v i


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The Shapley value is the averagemarginal contribution vector:

1( , ) ( )

!N

N vn

1

1 1 1( ) ( ,..., ) ( ,..., ) ( 2,..., )ki k kv i i v i i k n

Alternatively, the Shapley value can be written as:

where the weights are given by

( )

( , ) ( ) [ ( ) ( \ )]i nS N

S i

N v s v S v S i

( 1)!( )!( )!

n

s n ss


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The Shapley value is the unique allocation rule satisfying:

- symmetry: players with identical marginalcontributions (substitute players) get the same (equal

treatments of equals)

- null player: players never contributing (null players) get nothing

- additivity: (N,v+w) = (N,v) + (N,w)

!n

2 2

3 3

4 4

1 12 ( , ) (1,1)

2 2

1 1 13 ( , , ) (1,2,1)

3 6 3

1 1 1 14 ( , , , ) (1,3,3,1)

4 12 12 4

1 1 1 1 1

n

n

n


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5 5

6 6

1 1 1 1 15 ( , , , , ) (1,4,6,4,1)

5 20 30 20 5

1 1 1 1 1 16 ( , , , , , ) (1,5,10,10,5,1)

6 30 60 60 30 6

n

n

1

1

1

1

( )!where ( ) is the number of coalitions to which a given player belongs

( )!( )!

1( ) ( ) for all

s

n n

n n

ns C

n s s

s s sn

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:


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is the "excess" associated to imputationxand coalition S.

The least core is the set of imputations that minimize the largest excess:

( , ) ( ) ( )e x S v S x S

( , ),

Min Max ( , )x I N v S NS N

e x S

This is typically a set. The nucleolus goes further to eventually retaina uniqueimputation:

to each imputationxis associated the vector (x) of

dimension 2n2 obtained by placing the excesses e(x,S)

in a decreasing order


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The nucleolus is then the unique imputations that minimizes

lexicographicallythese vectors on the set of imputationsI(N,v):

( ) ( ) for all ( , )Lx x x I N v

x

Example: "market" game

v(1) = v(2) = v(3) = v(23) = 0

v(12) =p2 p3

v(13) = v(123) =p3


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The core is defined by:

In particular, ifp3=p2, then

3 2 3( , ) { ( ,0, ) }N v x p p p p p p

3( , ) { ( ,0,0) }N v p

1 2 3

123 0 200 100

132 0 0 300

213 200 0 100

231 300 0 0

v(i) = 0

v(12) =p2 = 200

v(13) =p3 = 300

v(23) = 0

v(123) =p3 = 300


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231 300 0 0

312 300 0 0

321 300 0 0

1/6 1100 200 500

550 100 250( , ) ( , , ) (183,33,83)

3 3 3N v

each row corresponds

to a permutation

chaque column correspondsto a player

176

For any given coalition, the excess can be written as a fucntion ofp:

3

2

( , ) for {1}

0 for {2} and {13}

for {3} and {23}

for {12}

e p S p S

S S

p p S S

p p S


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For eachp, order the excesses in a decreasing way:

2 3 3 2

3 3 2 3

(0, 0, , , , ) for [ , ]

(0, 0, , , , ) for [ , ]

p p p p p p p p p p

p p p p p p p p p p

2 3where2

p pp

pp3p20

pp3

p2

2 3

2

p pp

here the least core coincides with thecore and the nucleolus is its mid-point


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- p3

(p2-p3)/2

p2

-p3

p

p2p

The nucleolus is the mid-point of the core:

i e (250 0 50) in the case where p3 = 300 and p2 = 200

3 2 3 2( , ) ,0,2 2

p p p pN v


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i.e (250, 0, 50) in the case wherep3 300 andp2 200.

The nucleolus satisfies to two Shapley's axioms: symmetry and nulplayer.

It does notsatisfy additivity.

Example: crop game

Imagine a landlord and m(identical) workers, and a technologydescribed by a production functiony=F(s) wheresis the numberof workers:

v(S) = 0 if Sdoes not include the landlord


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v(S) =F(s1) if Sincludes the landlord

(he/she does not work)

In particular, v(i) = 0 for all i and v(N) =F(m).

We suppose thatFis increasingwithF(0) = 0, not more at this stage.

The associated game is superadditive. It is convexif returns to scale are

constant or increasing: linear or convex production function.

We first observe that the extreme allocation (F(m), 0, ,0) always

belongs to the core.

Letxbe in the core. For allj1, we have:

where

( \ ) ( \ ) ( 1)x N j v N j F m


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where

Hence,

the most a worker can get within the core is

the marginal product [F(m)F(m1)]

( \ ) ( ) ( )j jx N j x N x F m x

( ) ( 1)jx F m F m

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 thepositionof the landlord counts.

if the landlord is in position k, he gets F(k-1)


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if the landlord is in position k, he getsF(k1)

and there are m + 1 positions possibles

1

1

1 1

1 11

1 1( , ) ( ) ( )

m m

k k

N v F k F km m

F(m)

F(k) 1xF(k)F(1) +F(2) + +F(m)


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m0

F(1)

F(2)

k k+1

F(m)

L > W

decreasing

returns

Workers


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m0

L > W

0

( )

m

k

F k

Landlord

F(m)

W = L

constant

returns

Workers


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m0

W L

Landlord

F(m)

T

L < W

increasing

returns

Workers


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m0

L < W

Landlord

F(m) mixed

returns

Workers


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m0

Landlord

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.

d 100 d 200 d 300 EQUAL


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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

?

Aumann and Mashler (1985) have shown that the nucleolus actuallyreproduces the Talmud figures for the following TU-game:

Here v(S) represents the minimumcoalition Scan get:

( ) 0, ( \ )v S Max E d N S


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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".

E= 200d= (100,200,300)

v(i) = 0 i = 1,2,3

v(12) = v(13) = 0v(23) = 100

v(123) = 200

( ) 0, ( \ )v S Max E d N S

1 2 3

1 2 3 1

2 3

, , 0

200 100

100

x x x

x x x x

x x


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v(123) 200

Here players 2 and 3 are substitutes.

3

1 2 3 1 2 3( , ) {( , , ) 0 , 200 , 0 100}N v x x x x a x x a a

(200,0,0)

x1

x3x2

0

200

200

200

( , )I N v


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(0, 200,0)

x2+ x3=100

(0,0, 200)x2+ x3=200 x1=0

( , )N v

E= 200d= (100,200,300)

v(1) = 0

v(2) = 0v(3) = 0

( ) 0, ( \ )v S Max E d N S

1 2 3

123 0 0 200

132 0 200 0

213 0 0 200

231 100 0 100


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v(12) = 0

v(13) = 0

v(23) = 100

v(123) = 200

231 100 0 100

312 0 200 0

321 100 100 01/6 200 500 500

200 500 500( , ) , , (33.7, 83.7, 83.7)

6 6 6N v

(200,0,0)

(100 100 0)


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(0, 200,0) (0,0, 200)

Nucleolus

Shapley

Equal

(100, 100, 0) (100, 0, 100)

We observe that the four vertices of the core are precisely the fourmarginal contribution vectors:

(0, 0, 200) with multiplicity 2


(100, 0, 100) with multiplicity 1(100, 100, 0) with multiplicity 1


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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.

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


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195

UG 66.6 66.6 66.6

UL 0 50 150

Nucleolus 50 75 75

Shapley 33.3 83.3 83.3

E= 100d= (100,200,300)

v(i) = 0 i = 1,2,3

v(12) = v(13) = v(23) = 0

v(123) = 200

( ) 0, ( \ )v S Max E d N S


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The game is symmetric: all players are substitutes.

1 2 3

1 2 3

, , 0

200

x x x

x x x

( , ) ( , )N v I N v

200( , ) ( , )

3i iN v N v

E= 300d= (100,200,300)

v(i) = 0 i= 1,2,3

v(12) = 0

v(13) = 100

v(23) = 200

( ) 0, ( \ )v S Max E d N S

1 2 3

1 2 3

1 3

, , 0

300100

x x x

x x xx x


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( )

v(123) = 300 2 3 200x x

(300,0,0)


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(0, 300,0) (0,0, 300)

x2

+ x3

=200

x1+ x3=100

( , )N v

E= 300d= (100,200,300)

v(i) = 0 i= 1,2,3

v(12) = 0

v(13) = 100

v(23) = 200

( ) 0, ( \ )v S Max E d N S

1 2 3

123 0 0 300

132 0 200 100213 0 0 300

231 100 0 200


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v(123) = 300231 100 0 200

312 100 200 0

321 100 200 0

1/6 300 600 900

300 600 900( , ) , , (50, 100, 150)

6 6 6N v

(300,0,0)

+ 100

x1+ x3=200

x1+ x2=150


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200(0, 300,0) (0,0, 300)

Shapley = Nucleolus

Equal

x2

+ x3

=200

x1+ x3=100

(300,0,0)


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201(0, 300,0) (0,0, 300)(0,200, 100)

(100, 200, 0) (100, 0, 200)

We observe again that the four vertices of the core are precisely

the four marginal contribution vectors:

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


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confirming that the bankcruptcy game is convex.

Assignment games (Shapley and Shubik)

Consider a setN = {1,,n} of agents and a setM = {1,,m} (mn)

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


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ui(h) is the reservation priceof agent i for house hi.e. the maximum

price i is willing to pay for house h.

It is the valuethat agent iattach to house hexpressed in monetary

terms.

[ ( ) | , ]iu h i N h M

203

Side payments being allowed, the associated TU-game is given by:

whereFis the set of all functionsf:NMthat associates ahouse to each player.

( ) ( ( ))f F ii Sv S Max u f i


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Here v(S) is the cost of the houses that are optimally allocated to the

members of coalition S.

Consequently, (N,v)isa cost game. It is concaveand thereby also

subadditive.

204

An optimal allocations of objects to players is associated to thedefinition of C(N)

In the example below, it is (2,3,1): player 1 receives house 2, player 2

receiveshouse3, and player 3 receives house 1.

u1 u2 u3C(1) = 12

C(2) 9


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1 2 3

1 3 9 9

2 12 6 6

3 9 6 3

C(2) = 9

C(3) = 9

C(12) = 21

C(13) = 21

C(23) = 15

C(123) = 27

205


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We observe that players 2 and 3 are substitute. The Shapley value isobtained 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


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213 12 9 6

231 12 9 6

312 12 6 9321 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 followinginequalities:

1 2 3

1

2

3

1 2

27

12

99

21

y y y

y

yy

y y

y1= 12

6 y29

6 y39


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1 2

1 3

2 3

21

21

15

y y

y y

y y

(12,6,9)(12,9,6)

(12,7.5,7.5)

optimal allocation before transfers

the Shapley value is located at the center of the core

208

(27,0,0)

x3= 9

x2= 6x3= 6

x2= 9

set of

imputations

(9,9,9)

(12,6,9)(12,9,6)


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(0,27,0) (0,0,27)

x1= 12

209

3. Ordinal welfarism


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A social choice procedureis a mappingFthat associates alternativesto preference profiles:

: ( )nF L A A

It associates to any profilepa subset of "winning" alternatives

F(p) A.It is the collective choice set.


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: ( ) ( )nF L A L A

A social welfare functionis a mappingFthat associates "collective"preferences to preference profiles:

3.1 The case of two alternatives


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Consider the case of 2 alternativesand nvoters:

A= {0,1} andN= {1,,n}

Assuming no indifference, apreference profileis a list of 0 and 1

of length n:

p= (p1,,pn) wherepiL(A) = {0,1}


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where

1 1 0

0 0 1

i

i

i

i

p

p

Example: n= 5 andp= (0,1,0,0,1) 3 in favour of 02 in favour of 1

There are 2npossible profiles.

The set of all possible preference profiles is {0,1}n.

A voting procedureis a mapping


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F:{0,1}n {0,1}

It associates to any profilepa subsetF(p) {0,1}.

F(p) is the "choice set".

There are 4 possible outcomes:

F(p) = {0}

F(p) = {1}

F(p) = {0,1}

F( )


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F(p) =

So tiesare allowed.

The natural neutral mechanism to break a tie is the flipping of a coin.

Simple majority

1

1

( ) {1} if 2

( ) {0} if 2

n

i

i

n

i

i

nF p p

nF p p


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1

( ) {0,1} if2

n

ii

nF p p

a tie is not a possible outcome of simple majority if nis odd

Unanimity

1

1

( ) {1} if

( ) {0} if 0

n

i

i

n

i

i

F p p n

F p p


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1

( ) if 0n

ii

F p p n

A basic requirement to impose on a voting procedure is that itproduces a result:

Decisiveness A voting procedure is decisiveif it never results

in the empty outcome:

( ) for all {0,1}nF p p


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Simple majority is always decisive. Unanimity is not.

What would be a fairvoting 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:


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voters and alternatives should be treated equally

Anonymity

A voting procedureFis anonymousif it symmetric in its nvariables:

for anyp , permuting the voters leavesF(p) unchanged

For instance,


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(0,1,1,0,1) (1,0,1,0,1) (1,1,1,0,0) ....F F F

Anonymity clearly excludes dictatorship.

It is actually a stronger form of non-dictatorship.

Neutrality

A voting procedureFis neutralif permuting the choice of every voter

results in a permutation of the outcome:

for anyp P, F(1p) = 1F(p)

where 1= (1,1,,1).


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For instance,

(0,1,1,0,1) {1} (1,0,0,1,0) {0}F F

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.

1

( )n

i

i

F p G p


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for some increasing function G.

Alone, anonymity and neutrality allow for many different votingprocedures, including those based on stupid rules like:

1

1 1 4( ) {1} if

10 10( ) {0} otherwise

n

i

i

F p p

nF p


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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.

If, given the outcomeF(p) corresponding to a preference profilep,some voters change their mind in favourof a candidate who belongs

to the choice setF(p), we would expect that the resulting choice set

still includes that alternative.

Monotonicity A voting procedure is monotonicif


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1 ( ) and 1 ( )

0 ( ) and 0 ( )

F p p p F p

F p p p F p

where means for all .i ip p p p i

An increased support for analternativeshould 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


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Is it possible to characterize the procedures which satisfy these

3 axioms simultaneously ?

anonymity, neutrality and monotonicity

A quota procedureis defined by an integer , , such that:2nq q n

1

1 1

( ) {1} if

( ) {0} if

( ) {0,1} otherwise

n

i

i

n n

i i

i i

F p p q

F p n p q p n q

F p


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Simple majority is defined by:

1 if is odd2

1 if is even2

nq n

nq n


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( ) {0,1} and ( ) {1}

( ) {0,1} and ( ) {0}

F p p p F p

F p p p F p

A stronger version of the monotonicity axiom is the following:

Strict monotonicity A voting procedure isstrictly monotonic

(positive responsiveness) if it monotonic and


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means for all and for some .i i j jp p p p i p p j

where

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 uniquevoting procedure which


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p p j y q g p

(May, 1952) is decisive, anonymous, neutral and strictlymonotonic.

3.2 Social choice procedures


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Borda method (1781)

- each of the mposition is graded: m1 for the 1st,

m2 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


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adding the grades, each alternative receives a score

... the alternative(s) with the largest score wins.

a a a c c b e

b d d b d c c

c b b d b d d

d e e e a a b

e c c a e e a

n= 7


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Bordaba b c d e

14 17 16 16 7

Condorcet has criticized Borda's method.

Consider 3 alternatives and 30 voters,

19 with preferences

11 with preferences

For Condorcet a should win while Borda assigns 41 to b against

a b c

b c a


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For Condorcet, ashould win while Borda assigns 41 to bagainst

38 to a.

Indeed ais preferred to b and cby 19 voters.

An alternative isCondorcet winner if...

... confronted to any other alternative, it comes before in more

than half of the orderings

1 2 3 4 5 6 7a a a c c b e

b b d b d c c

d b d b d d


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(This does not define a decisive rule !)

c d b d b d d

d e e e a a b

e c c a e e a

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,...


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repeat the procedure with the remaining alternatives,...

a a a c c b e

b d d b d c c

c b b d b d d

d e e e a a b

e c c a e e a


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delete d

a a a c c b e

b b b b b c c

c e e e a a b

e c c a e e a


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delete b ande

a a a c c c c

c c c a a a a

delete a


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Harec

delete a

Sequential pairwise voting (voting with an agenda)

The idea is that a sequence of alternatives is determined and followed.

For instance, dresults from the sequence (a,b,c,d,e) but bthat comes

out from the reverse sequence:

a a a c c b e


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b d d b d c c

c b b d b d d

d e e e a a b

e c c a e e a

Pareto criteria

If all voters preferxtoy, thenycannot 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


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d e e e d d b

e c c d e e d

Condorcet criteria

If there is a Condorcet winner, it must be in the social choice set.

1 2 3 4 5 6 7

a a a c c b e

b b d b d c c


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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

Monotonicity criteria

Let the alternativexbe in the social choice set for a given preference

profilep.

If the preference profilepis modified by moving upxin the orderingof some voter,...

...xshould remain in the social choice set.


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Independence criteria(independence of irrelevant alternatives)

Assume that the social choice set includesxbut noty.

If the preference profilePis modified, without altering the preferences

betweenxandy,...

... then the resulting choice set should still not includey.


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Pareto Condorcet Monotonicity Independance

Plurality Yes No Yes No

Borda Yes No Yes No

Hare Yes No No No

Agenda N Y Y N


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Agenda No Yes Yes No

Dictator Yes No Yes Yes

Plurality satisfies Pareto

If every voter prefersxtoy,ycannot come on top of any ordering.

Borda satisfies Pareto

Ifxcomes beforeyin all preference orderings, thenxhas more points

thany.


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Hare satisfies Pareto

If every voter prefersxtoy,yis not on top of any list.

Then, either some alternative is on top of more than

half of the orderings, it is the winner, noty,

ory(being absent from the the first row) is among

the alternatives to be deleted next.


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Dictatorship satisfies Pareto:

if every voter prefersxtoy, it is also the case of the dictator...


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Sequential pairwise voting satisfies monotonicity

Assumexis a social choice given a preference profile and

an agenda.

Movingxup in the preferences of some voter will certainly keepxin

the social choice set (with a larger margin).

Dictatorship satisfies monotonicity

Ifxis the social choice, it is on top of the dictator's ordering...


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x s e soc a c o ce, s o op o e d c a o s o de g...

Dictatorship satisfies independence

Ifxis the social choice but noty,xis on top of the dictator's ordering

and will remain so...

Plurality does not satisfy Condorcet

1 to 4 5 to 7 8 and 9

a b c

b c b

c a a


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ais plurality winner but bis Condorcet winner

Borda does not satisfy Condorcet

1, 2 and 3 4 and 5

a b

b c

c a


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bis Borda winner but ais Condorcet winner

Hare does not satisfy Condorcet

1 to 5 6 to 9 10 to 12 13 to 15 16 and 17

a e d c b

b b b b cc c c d d

d d e e e

e a a a a


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bis Condorcet winner but it will be deleted first

e a a a a

Dictatorship does not satisfy Condorcet

1 2 3

a c c

b b b

c a a


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cis Condorcet winner while ais the "social"

choice if voter 1 is the dictator.

Hare does not satisfy monotonicity

1 to 7 8 to 12 13 to 16 17

a c b b

b a c a

c b a c


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ais the social choice according to Hare

If voter 17 moves aabove b, ...

1 to 7 8 to 12 13 to 16 17

a c b a

b a c b

c b a c


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...cbecomes the social choice

Plurality does not satisfy independence

a a b c

b b c b

c c a a


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ais the social choice and bis not

If voter 4 moves cbetween band a, ...

a a b b

b b c c

c c a a


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... aand bare tied

Borda does not satisfy independence

1, 2 and 3 4 and 5

a c

b b

c a


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ais the social choice

If voters 4 and 5 move cbetween band a, ...

1, 2 and 3 4 and 5

a b

b c

c a


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... bbecomes the social choice

Hare does not satisfy independence

a a b c

b b c b

c c a a


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ais the social choice according to Hare

If voter 4 moves cbetween band a, ...

a a b b

b b c c

c c a a


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... aand bare tied

Sequential pairwise voting does not satisfy Pareto

a c b

b a d

d b cc d a


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bdominates din the sense of Pareto: all voters prefer bto dbut dresults from the sequence (a,b,c,d) :

adefeats b, cdefeats abut ddefeats c.

Sequential pairwise voting does not satisfy Independance

c a b

b c aa b c

The reverse sequence (c,b,a) produces aas social choice.


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Interchanging cand bin the first ordering results in bas social choice

while no one has changed his/her mind about aand b.

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


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A decision was eventually reached after a full day of debates.

Bonn and Berlin

Motion: NO to distinct locations

No

YesEnd

340/657

489/654

147/654

Procedure adopted by the Council of Elders and the results:

Yes No 288/654

Abstention

Abstention

18/654

29/657


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Bonn or Berlin

Berlin338/659

Bonn332/659

Abstention

1/659

Questions:

Which voting procedure should have been adopted ?

Does the actual voting procedure produce enoughinformation to enable a reconstruction of the preferences

of the 659 representatives ?


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Would a different voting procedure have produceda different outcome ?

Bonn-Berlin: Leininger's results* based on a clever

reconstructedpreference 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: 227/430A = 513

B C A


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5. Bonn would have been the Borda winner:

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

B = 708C= 750

Scoring rules like Borda can be characterized. A scoring rule is defined

by a mapping that associates weights to alternatives (assuming strictpreferences) in terms of their positions in the preference lists.

Consistency

A social choice ruleFis consistent if, for any two disjoint sets of

votersNandN',and preference profilespandp'on a common a

setA of alternatives:


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where is the combined preference profile ofNN'.

( ) ( ) ( ) ( ) ( )F p F p F p F p F p p

p p

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.


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3.3 Impossibility theorems


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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


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- only sequential pairwise voting satisfies the Condorcet axiom

Condorcet voting paradox

h i h i l h i / f h h d

a c b

b a c

c b a

No Condorcet winner!


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Whatever is the social choice, 2/3 of the voters are unhappy andmoreover, they agree on an other alternative !

There is a transitivityproblem!

The collective preferences built by saying that

"xis preferred toy"

if and only if"xis preferred toyby a majority of voters"

are not transitive:

x y

d bb b


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... although individual preferences are.

and buta b b c c a

One implicit assumption is made:

there is no retrictions on the preferences:

social choice function are defined for any

preference profile inL(A)n

The only requirement is that individual preferences are preorders.


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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

h th t it d i f th th


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- show that it produces no winner: none of the threealternatives can be winning

Claim: acannot be winning

Consider the profile obtained from the Condorcet profile by moving

bdown in the third list:

a c b

b a c

c b a

a c c

b a b

c b a

cis then Condorcet winner and must be in the choice set, not a.

(same arguments forband c)


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Going back to the Condorcet profile by moving bup in the third list

should not affect the preferences between aand c.

So ashould still be a non-winner.

Impossibility theorem 2 (Arrow)

Dictatorship is the only social welfare functionsatisfying the Pareto and independence criteria

Impossibility theorem 3 (Gibbard)

Dictatorship is the only social choice procedureti f i th P t d t i it


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satisfying the Pareto and monotonicity


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Strong monotonicity

A social choice procedureFis strongly monotone if for all preference

profilep andqin , and any alternative ainA:

if qis obtained frompby lifting aup in some preference list,

then eitherF(q) =F(p) or F(q) = a

Pushing up an alternative can only help that alternative.

Proposition (Muller and Satterthwaite)

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