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EC9A4EC9A4 Social Choice and Social Choice and

VotingVoting Lecture 2Lecture 2

Prof. Francesco SquintaniProf. Francesco Squintani

f.squintani@warwick.ac.uk f.squintani@warwick.ac.uk

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Summary from previous Summary from previous lecturelecture

We have defined the general set up of the We have defined the general set up of the socialsocial

choice problem.choice problem.

We have shown that majority voting is We have shown that majority voting is particularlyparticularly

valuable to choose between two alternatives.valuable to choose between two alternatives.

We have proved Arrow’s theorem: the onlyWe have proved Arrow’s theorem: the only

transitive complete social rule satisfying weaktransitive complete social rule satisfying weak

Pareto, IIA and unrestricted domain is a Pareto, IIA and unrestricted domain is a

dictatorial rule.dictatorial rule.

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Preview of the lecturePreview of the lecture

We will extend Arrow’ theorem to social We will extend Arrow’ theorem to social choicechoice

functions.functions.

We will introduce the possibility of We will introduce the possibility of interpersonalinterpersonal

comparisons of utility.comparisons of utility.

We describe different concept of social We describe different concept of social welfare: welfare:

the utilitarian Arrowian representation the utilitarian Arrowian representation

and the maximin Rawlsian representation.and the maximin Rawlsian representation.

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Social Choice FunctionsSocial Choice FunctionsThe Arrow theorem concerns the The Arrow theorem concerns the

impossibility to impossibility to find a social preference order that find a social preference order that

satisfies minimalsatisfies minimalrequirements.requirements.

But the object of social choice may be But the object of social choice may be restricted torestricted to

find simply a socially optimal find simply a socially optimal alternative ratheralternative rather

than to rank all possible alternatives.than to rank all possible alternatives.

We will show that a similar We will show that a similar impossibility resultimpossibility result

holds when considering social choice holds when considering social choice functions.functions.

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Let Let RR be a subset of the set of all be a subset of the set of all profile of preferences profile of preferences RR = R(1),… = R(1),…R(N).R(N).

A social choice function f defined on A social choice function f defined on RR assigns a assigns a

single element to every profile single element to every profile RR in in RR..

The function f is The function f is monotonicmonotonic if, for any if, for any R R and and R’ R’ s.t.s.t.

the alternative x = f (the alternative x = f (RR) maintains its ) maintains its position position

from from R R to to R’ R’ we have that x = f (we have that x = f (R’R’).).

The alternative x The alternative x maintains its maintains its positionposition from from R R to to R’R’,,

if x R(i) y implies x R’(i) y for all i and if x R(i) y implies x R’(i) y for all i and y. y.

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The social choice function f is The social choice function f is weakly weakly ParetianParetian if if

y = f (y = f (RR) whenever there is x s.t. x R(i) ) whenever there is x s.t. x R(i) y for all i.y for all i.

An individual n is An individual n is dictatorialdictatorial for f if, for f if, for every for every RR

f (f (RR) ) {x : x R(n) y for all y in X}.{x : x R(n) y for all y in X}.

Theorem.Theorem. Suppose that there are at Suppose that there are at least three least three

distinct alternatives. Then any weakly distinct alternatives. Then any weakly Paretian Paretian

and monotonic social function defined and monotonic social function defined on theon the

whole domain of preferences is whole domain of preferences is dictatorial.dictatorial.

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Proof.Proof. The proof shows the result as a The proof shows the result as a corollary of corollary of

Arrow impossibility theorem. We Arrow impossibility theorem. We derive a social derive a social

rule F(rule F(RR) from f, using the properties ) from f, using the properties of f on all of f on all RR. .

We show that F satisfies all Arrow We show that F satisfies all Arrow axioms but ND.axioms but ND.

Step 0. Given a subset X’ of X and Step 0. Given a subset X’ of X and RR, , we say that we say that

R’R’ takes X’ to the top from takes X’ to the top from RR, if for , if for every i,every i,

x P’(i) y for all x in X’ and y not in X’;x P’(i) y for all x in X’ and y not in X’; x R(i) y if and only if x R’(i) y for all x, x R(i) y if and only if x R’(i) y for all x,

y in X’.y in X’.

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Step 1. If both Step 1. If both R’R’ and and R’’R’’ take X’ to the take X’ to the top from top from

RR, then f(, then f(R’R’) = f() = f(R’’R’’).).

In fact, by WP, f(In fact, by WP, f(R’R’) is in X’. Because ) is in X’. Because f(f(R’R’) maintains its position from ) maintains its position from R’R’ to to R’’R’’, f(, f(R’R’) = f() = f(R’’R’’) by monotonicity.) by monotonicity.

Step 2. For every Step 2. For every RR, we let `x F(, we let `x F(RR) y’ if ) y’ if x = f(x = f(R’R’) )

when when R’ R’ is any profile that takes {x,y} is any profile that takes {x,y} to the top of to the top of

profile profile R.R.

By Step 1, F is well defined. By Step 1, F is well defined.

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Step 3. Because f is weakly Paretian, if Step 3. Because f is weakly Paretian, if R’R’ takes {x,y} takes {x,y}

to the top from to the top from RR, then f(, then f(RR) in {x,y}. By ) in {x,y}. By step 1 and step 1 and

2, either x F(2, either x F(RR) y or y F() y or y F(RR) x (but not ) x (but not both).both).

Hence F is complete.Hence F is complete.

Step 4. Suppose that x F(Step 4. Suppose that x F(RR) y and y F() y and y F(RR) ) z. z.

If If R’’R’’ takes {x,y,z} to the top of takes {x,y,z} to the top of RR, then , then F(F(R’’R’’) in ) in

{x,y,z}, because f is weakly Paretian.{x,y,z}, because f is weakly Paretian.

Say by contradiction that y = f(Say by contradiction that y = f(R’’R’’). ).

Consider Consider R’R’ that takes {x,y} to the top that takes {x,y} to the top from from R’’R’’..

By monotonicity, f(By monotonicity, f(R’R’) = y.) = y.

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But But R’R’ also takes {x,y} to the top from also takes {x,y} to the top from RR..

This contradicts x F(This contradicts x F(RR)) y. In sum, y = f y. In sum, y = f ((R’’R’’).).

Similarly z = f (Similarly z = f (R’’R’’), by contradiction ), by contradiction with y F(with y F(RR) z.) z.

Thus x = f (Thus x = f (R’’R’’).).

Let Let R’R’ take {x,z} to the top from take {x,z} to the top from R’’R’’..

By monotonicity, x = f (By monotonicity, x = f (R’R’). ).

But But R’R’ also takes {x,z} to the top from also takes {x,z} to the top from RR,,

and hence x F(and hence x F(RR)) z.z.

Hence F(Hence F(RR)) is transitive.is transitive.

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Step 5. The social rule F(.) satisfies Step 5. The social rule F(.) satisfies Weak Pareto.Weak Pareto.

In fact, if x R(i) y for all i, then x = f In fact, if x R(i) y for all i, then x = f ((R’R’) whenever ) whenever R’R’ takes {x,y} to the takes {x,y} to the top from top from RR..

Thus x F(Thus x F(RR) y, by step 1.) y, by step 1.

Step 6. The social rule F(.) satisfies IIA.Step 6. The social rule F(.) satisfies IIA.

In fact, if In fact, if RR and and R’R’ have the same have the same ordering for x,y, and ordering for x,y, and R’’R’’ takes {x,y} to takes {x,y} to the top of the top of RR, then it also takes {x,y} to , then it also takes {x,y} to the top of the top of R’R’..

Hence, x = f (Hence, x = f (R’’R’’) implies: x F() implies: x F(RR) y and ) y and x F(x F(R’R’) y.) y.

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Step 7. The social choice function f is Step 7. The social choice function f is dictatorial.dictatorial.

By Arrow theorem, there is an agent i By Arrow theorem, there is an agent i such thatsuch that

for every profile for every profile RR, we have x F(, we have x F(RR) y ) y when x R(i)y.when x R(i)y.

Thus if x R(i) y for all y, then x = f(Thus if x R(i) y for all y, then x = f(RR).).

This final step concludes the proof.This final step concludes the proof.

We have shown how to construct the We have shown how to construct the social rule social rule

F(F(RR) starting from the social choice ) starting from the social choice function f.function f.

Applying Arrow’s theorem, we have Applying Arrow’s theorem, we have shown that shown that

there is a dictator for F(there is a dictator for F(RR), and so f is ), and so f is dictatorial.dictatorial.

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Interpersonal Interpersonal comparisonscomparisons

We will assume that all social choice We will assume that all social choice functions ffunctions f

satisfy welfarism, i.e. U, P and IIA, satisfy welfarism, i.e. U, P and IIA, and continuity.and continuity.

Hence there is a continuous function Hence there is a continuous function W such that:W such that:

V(x) V(x) >> V(y) if and only if W( V(y) if and only if W(uu(x)) (x)) >> W(W(uu(y)).(y)).

The welfare function depends only on The welfare function depends only on the utility the utility

ranking, not on how the ranking ranking, not on how the ranking comes about.comes about.

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Under Arrow axiom, utilities are Under Arrow axiom, utilities are measured along measured along

an an ordinal scale, ordinal scale, and are and are non-comparablenon-comparable across across

individuals. individuals.

Specifically, the function W aggregates Specifically, the function W aggregates thethe

preferences (upreferences (uii))i=1,…,ni=1,…,n if and only if W if and only if W aggregates aggregates

the preferences (v(uthe preferences (v(uii))))i=1,…,ni=1,…,n, for all , for all increasing increasing

transformation vtransformation vi i (u(uii), for any i, ), for any i, independently across i.independently across i.

We modify the framework to allow for We modify the framework to allow for cardinal cardinal

comparisons of utility, and comparability comparisons of utility, and comparability acrossacross

individuals.individuals.

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Suppose that preferences are fully Suppose that preferences are fully comparable but comparable but

measured on the ordinal scale.measured on the ordinal scale.The social ranking V must be invariant to The social ranking V must be invariant to

arbitrary,arbitrary,but common, increasing transformations but common, increasing transformations

vvii applied applied

to every individual’s utility function uto every individual’s utility function uii..

Specifically, the function W aggregates Specifically, the function W aggregates thethe

preferences (upreferences (uii))i=1,…,ni=1,…,n if and only if W if and only if W

aggregates the preferences (v(uaggregates the preferences (v(uii))))i=1,…,ni=1,…,n, , for all for all

increasing transformation vincreasing transformation vi i (.), such that (.), such that v’v’i i is is

constant across i.constant across i.

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Suppose that preferences are fully Suppose that preferences are fully comparable andcomparable and

measured on the cardinal scale.measured on the cardinal scale.The social ranking V must be invariant to The social ranking V must be invariant to

increasing,increasing,

linear transformations v(ulinear transformations v(uii)= a)= aii+bu+buii, , where b is where b is

common to every individual.common to every individual.

Specifically, the function W aggregates Specifically, the function W aggregates thethe

preferences (upreferences (uii))i=1,…,ni=1,…,n if and only if W if and only if W aggregates theaggregates the

preferences (v(upreferences (v(uii))))i=1,…,ni=1,…,n, for all , for all transformation vtransformation vii(.),(.),

such that vsuch that vii(u(uii)= a)= aii+bu+buii, with b>0., with b>0.

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HE: Let HE: Let uu and and u’u’ be two distinct utility be two distinct utility vectors. vectors.

Suppose that uSuppose that uk k = u’= u’k k for all k other than i for all k other than i and j.and j.

If uIf ui i > u’> u’i i > u’> u’j j > u> uj j , then W(, then W(u’u’) > W() > W(uu).).

Condition HE states that the society has a Condition HE states that the society has a preferencepreference

towards decreasing the dispersion of towards decreasing the dispersion of utilities acrossutilities across

individuals.individuals.

Rawlsian FormRawlsian Form

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AN: The social rule W is AN: The social rule W is anonymousanonymous if for every if for every

permutation p, W(upermutation p, W(u11, …, u, …, uNN)=F(u)=F(up(1)p(1),…u,…up(N)p(N)).).

Theorem: Suppose that preferences are fullyTheorem: Suppose that preferences are fully

comparable and measured on the ordinal scale. comparable and measured on the ordinal scale.

The social welfare function W satisfies Weak The social welfare function W satisfies Weak

Pareto, Anonymity and Hammond equality Pareto, Anonymity and Hammond equality

if and only if it takes the Rawlsian form if and only if it takes the Rawlsian form

W(u)=min{uW(u)=min{u11, …, u, …, uNN}.}.

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ProofProof. It is easy to see that the function. It is easy to see that the function

W(u)=min{uW(u)=min{u11, …, u, …, uNN} satisfies Weak } satisfies Weak

Pareto, Anonymity and Hammond Pareto, Anonymity and Hammond equality.equality.

To show the converse, we will see only To show the converse, we will see only the case the case

for N=2.for N=2.

Consider aConsider autility index utility index uu,,

with uwith u1 1 > u> u22

u1

u2

u

u*I

II

III

IV

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By anonymity, the utility profile By anonymity, the utility profile u* must be ranked

in the same way as u: hence W(u) = W(u*). By Weak Pareto, all u’ such that u’ > u or u’ > u* must be such that W(u’) > W(u).

Hence the whole areain blue is s.t.W(u’) > W(u).

u1

u2

u

u*

I

II

III

IV

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By Weak Pareto, all u’ such that u’ < u or

u’ < u* must be such that W(u’) < W(u).

Hence the whole area in green is such that

W(u’) < W(u).

u1

u2

u

u*I

II

III

IV

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Pick a point u’ in zone III. To be in III, it must be

that u2 < u’2 < u’1 < u1

Every linear transform v such that vi(ui)=ui

yields: u2 < v2(u’2) < v1 (u’1) < u1

This concludesthat all pointsin III are rankedthe same waywrt to u. u1

u2

u

u*I

IIIII

IV

u’

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To be in III, it must be that u2 < u’2 < u’1 < u1

Hammond equality implies that W(u’) > W(u).

u1

u2

u

u*I

IIIII

IV

u’

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By anonymity, the ranking of each u’ in III relative

to u must be the same as the ranking of any utility

vector u’’ in II: W(u’’)>W(u).

u1

u2

u

u*I

IIIII

IV

u’

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Any linear transform v such that v1(u’1) = u1,

v2(u’2) = u2, yields:

W (v(u)) < W (v1 (u’1), v2(u’2)) = W (u).

Hence all the utility vectors u’’ in IV are ranked

opposite to all utility vectors u’ in III, relative to u. u1

u2

u

u*I

IIIII

IV

u’

u’’

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Hence W(u) > W(u’’) for all u’’ in IV, and, by

anonymity, W(u) > W(u’’) for all u’’ in I.

u1

u2

u

I

IIIII

IV u’’

u’’

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We conclude that V(II) and V(III) > W(u) > V(I) and

V(IV).We are left to consider the boundaries

of these sets.

u1

u2

u

u*I

IIIII

IV

u’

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Because W is continuous, the boundaries opposite

to each other, relative to u must be indifferent to u’.

Thus, the boundaries between II and III and the

blue set must be better than u in W terms.The boundariesbetween I and IVand the green setmust be indifferent to u. u1

u2

u

u*I

IIIII

IV

u’

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We have obtained the Rawlsian indifference

curves, where W (u) = min {u1 , u2}.

u1

u2

u

u*

IIIII

IV

u

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Utilitarian FormUtilitarian Form

Theorem: Suppose that preferences are Theorem: Suppose that preferences are

fully comparable and measured on the fully comparable and measured on the

cardinal scale. cardinal scale. The social welfare function W satisfies The social welfare function W satisfies

Weak Weak Pareto and Anonymity if and only if it Pareto and Anonymity if and only if it

takes the takes the utilitarian form:utilitarian form:

W(u)= uW(u)= u11 +… + u +… + uNN..

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ProofProof. It is easy to see that the . It is easy to see that the functionfunction

W(u)= uW(u)= u11+ …+ u+ …+ uNN, satisfies Weak , satisfies Weak Pareto and Pareto and

Anonymity.Anonymity.To show the converse, we will see only To show the converse, we will see only

the case the case for N=2.for N=2.

Consider aConsider autility index utility index uu,,

with uwith u1 1 = u= u22

u1

u2

u’

u*

u

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Define the constant k = uDefine the constant k = u11 + u + u22..

Consider the locus k(u) = {(uConsider the locus k(u) = {(u11, u, u22) : k ) : k = u= u11 + u + u22}. }.

For any vector u’ on k(u), the vector For any vector u’ on k(u), the vector u* such that u* such that

u* = (u’u* = (u’2 2 , u’, u’11) is also on k(u).) is also on k(u).

By Anonymity,By Anonymity,

W(u’)=W(u*).W(u’)=W(u*).u1

u2

u’

u*

u

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Suppose now that W(u) > W(u’). Suppose now that W(u) > W(u’).

Under CS/IC, this ranking must be Under CS/IC, this ranking must be invariant toinvariant to

transformation vtransformation vii(u(uii) = a) = ai i + bu+ buii

u1

u2

u’

u*

u

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Let vLet vii(u(uii) = (u) = (uii - u’ - u’ii)+ u)+ ui i for i=1,2.for i=1,2.

Hence, (vHence, (v11(u’(u’11), v), vii(u’(u’ii)) = u and (v)) = u and (v11(u(u11), ), vvii(u(uii)) = u*.)) = u*.

If W(u) > W(u’), then W(u*) > W(u), If W(u) > W(u’), then W(u*) > W(u), which which

contradicts W(u’) = W(u*). contradicts W(u’) = W(u*).

u1

u2

u’

u*

u

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If W(u) < W(u’), then W(u*) < W(u), If W(u) < W(u’), then W(u*) < W(u), which which

contradicts W(u’) = W(u*). contradicts W(u’) = W(u*).

Hence we conclude that W(u)=W(u’) Hence we conclude that W(u)=W(u’) for all for all

vectors u’ on k(u).vectors u’ on k(u).

u1

u2

u’

u*

u

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By Weak Pareto, each vector u’’ to the By Weak Pareto, each vector u’’ to the north-east north-east

of a vector u’ on k(u) is strictly of a vector u’ on k(u) is strictly preferred to u.preferred to u.

Thus W(u’’) > W(u) for u’’ such that Thus W(u’’) > W(u) for u’’ such that

u’’u’’11+ u’’+ u’’2 2 > u> u11+ u+ u2 .2 .

u1

u2

u

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Similarly, W(u’’) < W(u) for u’’ such Similarly, W(u’’) < W(u) for u’’ such that that

u’’u’’11+ u’’+ u’’2 2 < u< u11+ u+ u2 2

u1

u2

u

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We concluded that the indifference We concluded that the indifference curve of anycurve of any

vector u is k(u) = {(uvector u is k(u) = {(u11, u, u22) : k = u) : k = u11 + + uu22}. }.

HenceHence

W(u) = uW(u) = u11+ u+ u2 2

IndifferenceIndifference

curves arecurves are

straight lines straight lines

of slope -1.of slope -1.

u1

u2

uu

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If we drop the axiom of Anonymity, the If we drop the axiom of Anonymity, the full range full range

of generalized utilitarian orderings is of generalized utilitarian orderings is allowed. allowed.

These are the linear social welfare These are the linear social welfare functionsfunctions

W(u)= aW(u)= a1 1 uu1 1 +…+ a+…+ aN N uuNN, with a, with ai i >> 0 for 0 for all i, all i,

and aand aii > 0 > 0

for some i.for some i.

IndifferenceIndifference

curves are linescurves are lines

of negativeof negative

slope.slope.

u1

u2

u

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The maximin Rawlsian form and the utilitarian The maximin Rawlsian form and the utilitarian form both belong to constant elasticity classform both belong to constant elasticity classwith the formula:with the formula:

W = ( uW = ( u11 +… + u +… + uNN

) )

where 0 = where 0 = and and is the constantis the constantelasticity of social substitution between any elasticity of social substitution between any

pair pair of individuals.of individuals.

As As approachesapproachesW approaches the W approaches the utilitarian formutilitarian form

As As approaches –infinity,approaches –infinity,W approaches the W approaches the Rawlsian form.Rawlsian form.

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A Theory of JusticeA Theory of Justice

Behind a ``veil of ignorance’’, an individual Behind a ``veil of ignorance’’, an individual doesdoes

not know which position she will take in a not know which position she will take in a society.society.

Will she be rich or poor, successful or Will she be rich or poor, successful or

unsuccessful?unsuccessful?

If she assigns equal probability to any of theIf she assigns equal probability to any of the

possible economic and social identities that possible economic and social identities that

exist in the society, a rational evaluation wouldexist in the society, a rational evaluation would

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evaluate welfare according to the expected utilityevaluate welfare according to the expected utility

[u[u11(x) + … + u(x) + … + uNN(x) ]/N(x) ]/N

This is equivalent to adopt the utilitarian This is equivalent to adopt the utilitarian criterion:criterion:

W(u(x)) = uW(u(x)) = u11(x) + … + u(x) + … + uNN(x).(x).

But the approach is also consistent with every But the approach is also consistent with every CES CES

form, embodying different degrees of risk form, embodying different degrees of risk aversion.aversion.

Consider the positive transformation Consider the positive transformation

vvii(x) = - u(x) = - uii(x)(x)-a -a with a>0. with a>0.

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Suppose that uSuppose that uii(x) represents utility over social (x) represents utility over social

states “with certainty,” whereas vstates “with certainty,” whereas vii(x) (x) represents represents

utility over social states “with uncertainty.”utility over social states “with uncertainty.”

In the form vIn the form vii(x) = - u(x) = - uii(x) (x) -a -a , a>0 represents the , a>0 represents the

degree of risk aversion.degree of risk aversion.

Suppose that the social welfare function is Suppose that the social welfare function is given by given by

the expected utility:the expected utility:

W = [vW = [v11(x) +…+ v(x) +…+ vNN(x)]/N (x)]/N

= [-u= [-u11(x)(x)-a -a - …-u- …-uNN(x)(x)-a-a]/N]/N

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Because the monotonic transformation Because the monotonic transformation

W = ( -uW = ( -u11(x)(x)-a -a - …-u- …-uNN(x)(x)-a -a ))-1/a-1/a

equivalently represents welfare, we obtain equivalently represents welfare, we obtain that anythat any

CES form is compatible with the expectedCES form is compatible with the expectedutility formulation behind a veil of ignorance.utility formulation behind a veil of ignorance.

The extreme risk aversion case of CES is The extreme risk aversion case of CES is

W = min {uW = min {u11(x), …, u(x), …, uNN(x)}, the Rawlsian form(x)}, the Rawlsian formthat describes a social concern for the agent that describes a social concern for the agent

with with the lowest utility.the lowest utility.

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ConclusionConclusion

We have extended Arrow’ theorem to social We have extended Arrow’ theorem to social

choice functions.choice functions.

We have introduced the possibility ofWe have introduced the possibility of

interpersonal comparisons of utility.interpersonal comparisons of utility.

We have described different concept of We have described different concept of social social

welfare: the utilitarian Arrowian welfare: the utilitarian Arrowian representation andrepresentation and

the maximin Rawlsian representation.the maximin Rawlsian representation.

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Preview of the next Preview of the next lecturelecture

We will introduce single-peaked We will introduce single-peaked utilities.utilities.

We will prove Black’s theorem: We will prove Black’s theorem: Majority voting isMajority voting is

socially fair when utilities are single-socially fair when utilities are single-peaked.peaked.

We will prove Median Voter We will prove Median Voter Convergence in theConvergence in the

Downsian model of elections.Downsian model of elections.

We will introduce the probabilistic We will introduce the probabilistic voting model.voting model.