Welfarist escape out of
Arrow’s theorem
What are the individual preferences
standing for ?
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What does it mean to say that Bob prefers social state
x to social state y ?
Economic theory is not very precise in its
interpretation of preferences
A preference is usually considered to be an ordering
of social states that reflects the individual’s
« objective » or « interest » and which rationalizes
individual’s choice
More precise definition: preferences reflects the
individual’s « well-being » (happiness, joy,
satisfaction, welfare, etc.)
What happens if one views the problem of defining
general interest as a function of individual well-being
rather than individual preferences ?
Philosophical tradition: Utilitarianism (Beccaria,
Hume, Bentham): The best social objective is to
achieve the maximal « aggregate happiness ».
What is happiness ?
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Objective approach: happiness is an objective
mental state
Subjective approach: happiness is the extent to
which desires are satisfied
See James Griffin « Well being: Its meaning,
measurement and moral importance », London,
Clarendon 1988
Can happiness be measured ?
Can happiness be compared accross individuals ?
If the answers given to these two questions are
positive, how should we aggregate individuals’
happinesses ?
Can we measure happiness ? (1)
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
Suppose Ri is an ordering of social states
according to i’s well-being.
Can we get a « measure » of this happiness ?
In a weak ordinal sense, the answer is yes
(provided that the set X is finite or, if X is some
closed and convex subset of +nl , if Ri is
continuous (Debreu (1954))
Let Ui: X   be a numerical representation of Ri

Ui is such that, for every x and y in X,
Ui(x)  Ui(y)  x Ri y

Ordinal measure of happiness
Can we measure happiness ? (2)
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Ordinal measure of happiness: defined up to an increasing
transform.
Definition: g: A  (where A  ) is an increasing
function if, for all a, b  A, a > b  g(a) > g(b)
If Ui is a numerical representation of Ri, and if g:   is
an increasing function, then the function h: X   defined
by: h(x) = g(U(x)) is also a numerical representation of Ri
Example : if Ri is the ordering on +2 defined by:
(x1,x2) Ri (y1,y2)  lnx1 + lnx2  lny1 + lny2 , then the
functions defined, for every (z1,z2), by:
U(z1,z2) = lnz1 + lnz2
G(z1,z2) = e U(z1,z2) = elnz1elnz2 = z1z2
H(z1,z2) = -1/G(z1,z2) = -1/(z1z2) all represent numerically
R
Can we measure happiness ? (3)
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
The three functions of the previous example are
ordinally equivalent.
Definition: Function U is said to be ordinally
equivalent to function G (both functions having X
as domain) if, for some increasing function g: 
, one has U(x) = g(U(x)) for every x  X
Remark: ordinal equivalence is a symmetric
relation, because if g :   is increasing, then
its inverse is also increasing.
Ordinal measurement of well-being is weak
because all ordinally equivalent functions provide
the same information about this well-being.
Can we measure happiness ? (4)
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
Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being
true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true
for every increasing function g:  .
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being
true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true
for every increasing function g:  . For example,
having 3 > (4+1)/2 does not imply having 33 > (43+13)/2
Can we measure happiness ? (5)
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Stronger measurement of well-being: cardinal.
Suppose U: X  and G: X  are two measures of wellbeing. We say that they are cardinally equivalent if and
only if there exists a real number a and a strictly positive
real number b such that, for every x  X, U(x) = a + bG(x).
We say that a cardinal measure of well-being is unique up
to an increasing affine transform (g:   is affine if, for
every c  , it writes g(c) = a + bc for some real numbers a
and b
Statements about welfare changes make sense with
cardinal measurement
If U(x)-U(y) > U(w)-U(z), then (a+bU(x)-(a+bU(y)) =
b[U(x)-U(y)] > b[U(w)-U(z)] (if b > 0)
= (a + bU(w)-(a+bU(z))
Can we measure happiness ? (6)
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
Example of cardinal measurement in sciences:
temperature. Various measures of temperature
(Kelvin, Celsius, Farenheit)
Suppose U(x) is the temperature of x in Celcius.
Then G(x) = 32 + 9U(x)/5 is the temperature of x
in Farenheit and H(x) = -273 + U(x) is the
temperature of x in Kelvin
With cardinal measurement, units and zero are
meaningless but a difference in values is
meaningful.
Can we measure happiness ? (7)
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Measurement can even more precise than
cardinal. An example is age, which is what we call
ratio-scale measurable.
If U(x) is the age of x in years, then G(x) = 12U(x)
is the age of x in months and H(x) = U(x)/100 is
the age of x in centuries. Zero matters for age. A
ratio scale measure keeps constant the ratio.
Statements like « my happiness today is one third
of what it was yesterday » are meaningful if
happiness is measured by a ratio-scale
Functions U: X  and G: X  are said to be
ratio-scale equivalent if and only if there exists a
strictly positive real number b such that, for every
x  X, U(x) = bG(x).
Can we measure happiness ? (8)
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
Notice that the precision of a measurement is defined
by the « size » of the class of functions that are
considered equivalent.
Ordinal measurement is not precise because the
class of functions that provide the same information
on well-being is large. It contains indeed all functions
that can be obtained from another by mean of an
increasing transform.
Cardinal measurement is more precise because the
class of functions that convey the same information
than a given function is restricted to those functions
that can be obtained by applying an affine increasing
transform
Ratio-scale measurement is even more precise
because equivalent measures are restricted to those
that are related by a increasing linear function.
Can we measure happiness ? (9)

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
What kind of measurement of happiness is available ?
Ordinal measurement is « easy »: you need to observe
the individual choosing in various circumstances and
to assume that her choices are driven by the pursuit
of happiness. If choices are consistent (satisfy
revealed preferences axioms), you can obtain from
choices an ordering of all objects of choice, which can
be represented by a utility function
Cardinal measurement seems plausible by
introspection. But we haven’t find yet a device (rod)
for measuring differences in well-being (like the
difference between the position of a mercury column
when water boils and its position when water freezes).
Ratio-scale is even more demanding: it assumes the
existence of a zero level of happiness (above you are
happy, below you are sad). Not implausible, but
difficult to find. Level at which an individual is
indifferent between dying and living ?
Can we define general interest as a
function of individuals’ well-being ?
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As before, we assume that there are n individuals
Ui: X   a (utility) function that measures individual
i’s well-being in the various social states
(U1 ,…, Un): a profile of individual utility functions
the set of all logically conceivable real valued
functions on X
DU  n the domain of « plausible » profiles of utility
functions
A social welfare functional is a mapping W: DU  
that associates to every profile (U1 ,…, Un) of
individual utility functions a binary relation R =
W(U1,…,Un))
Problem: how to find a « good » social welfare
functional ?
Examples of social welfare functionals

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
Utilitarianism: x R y iUi(x)  iUi(y) where R =
W(U1,…,Un)
x is no worse than y iff the sum of happiness is no
smaller in x than in y
Venerable ethical theory: Beccaria, Bentham,
Hume, Stuart Mills.
Max-min (Rawls): x R y  min (U1(x),…, Un(x)) 
min (U1(y),…, Un(y)) where R = W(U1,…,Un)
x is no worse than y if the least happy person in x
is at least as well-off as the least happy person in
y
Contrasting utilitarianism and max-min
u2
utility possibility set
u1 = u2
u1
Contrasting utilitarianism and max-min
u2
u’
-1
Utilitarian optimum
u1 = u2
u
u
u’
u1
Contrasting utilitarianism and max-min
u2
u’
-1
u1 = u2
Rawlsian
optimum
u
u
u’
u1
Contrasting utilitarianism and max-min
u2
Utilitarian optimum
u1 = u2
Rawlsian
optimum
Best feasible
egalitarian
outcome
u1
Contrasting utilitarianism and Max-min
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
Max-min and utilitarianism satisfy the weak Pareto
principle (if everybody (including the least happy) is
better off, then things are improving).
Max-min is the most egalitarian ranking that satisfies
the weak Pareto principle
Max-min does not satisfy the strong Pareto
principle (Max min does not consider to be good a
change that does not hurt anyone and that benefits
everybody except the least happy person)
Utilitarianism does not exhibit any aversion to
happiness-inequality. It is only concerned with the sum,
no matter how the sum is distributed
Examples of social welfare functionals
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Utilitarianism and Max-min are particular
(extreme) cases of a more general family of social
welfare functionals
Mean of order r family (for a real number r  1)
x R y [iUi(x)r]1/r  [iUi(y)r]1/r if r  0 and
x R y ilnUi(x)  ilnUi(y) otherwise (where R
= W(U1,…,Un))
If r =1, Utilitarianism
As r  -, the functional approaches Max-min
r  1 if and only if the functional is weakly averse
to happiness inequality.
Mean-of-order r functional
u2
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
r=+
u1
Mean-of-order r functional
u2
u1 = u2
r=+
Max-max indifference
curve
u1
Extension of Max-min
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Max-min functional does not respect the strong
Pareto principle
There is an extension of this functional that does:
Lexi-min (due to Kolm (1972)
Lexi-min: x R y  There exists some j  N such
that U(j)(x)  U(j)(y) and U(j’)(x) = U(j’)(y) for all j’
< j where, for every z  X, (U(1)(z),…,U(n)(z)) is
the (ordered) permutation of (U1(z)…Un(z)) such
that U(j+1)(z)  U(j)(z) for every j = 1,…,n-1 (R =
W(U1,…,Un))
Information used by a social
welfare functional
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When defining a social welfare functional, it
is important to specify the information on the
individuals’ utility functions used by the
functional
Is individual utility ordinally measurable,
cardinally measurable, ratio-scale
measurable ?
Are individuals’ utilities interpersonally
comparable ?
Information used by a social
welfare functional (ordinal)
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A social welfare functional W: DU  uses ordinal
and non-comparable (ONC) information on
individual well-being iff for all (U1,…Un) and
(G1,…,Gn)  DU such that Ui = gi(Gi) for some
increasing functions gi:   (for i = 1,…n), one
has W (U1,…Un) = W(G1,…,Gn)
A social welfare functional W: DU  uses ordinal
and perfectly comparable (OC) information on
individual well-being iff for all (U1,…Un) and
(G1,…,Gn)  DU such that Ui = g(Gi) for some
increasing function g:   (for i = 1,…n), one has
W (U1,…Un) = W(G1,…,Gn)
Information used by a social
welfare functional (cardinal)



A social welfare functional W: DU  uses cardinal and
non-comparable (CNC) information on individual wellbeing iff for all (U1,…Un) and (G1,…,Gn)  DU such that Ui =
aiGi+bi for some strictly positive real number ai and real
number bi (for i = 1,…n), one has W (U1,…Un) =
W(G1,…,Gn)
A social welfare functional W: DU  uses cardinal and
unit-comparable (CUC) information on individual wellbeing iff for all (U1,…Un) and (G1,…,Gn)  DU such that Ui =
aGi+bi for some strictly positive real number a and real
number bi (for i = 1,…n), one has W (U1,…Un) =
W(G1,…,Gn)
A social welfare functional W: DU  uses cardinal and
fully comparable (CFC) information on individual wellbeing iff for all (U1,…Un) and (G1,…,Gn)  DU such that Ui =
aGi+b for some strictly positive real number a and real
number b (for i = 1,…n), one has W (U1,…Un) =
W(G1,…,Gn)
Information used by a social
welfare functional (ratio-scale)
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
A social welfare functional W: DU  uses ratio-scale
and non-comparable (RSNC) information on individual
well-being iff for all (U1,…Un) and (G1,…,Gn)  DU such
that Ui = aiGi for some strictly positive real number ai (for i
= 1,…n), one has W (U1,…Un) = W(G1,…,Gn)
A social welfare functional W: DU  uses ratio-scale
and comparable (RSC) information on individual wellbeing iff for all (U1,…Un) and (G1,…,Gn)  DU such that Ui
= aGi for some strictly positive real number a (for i =
1,…n), one has W (U1,…Un) = W(G1,…,Gn)
Information used by a social
welfare functional
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
There are some connections between these various
informational invariance requirements
Specifically, ONC  CNC  CUC  CFC  RSFC and,
similarly, OFC  CFC and CUC  CFC. On the other
hand, it is important to notice that CUC does not imply nor
is implied by OFC.
What information on individual’s well-being are the
examples of welfare functional given above using ?
Information used by a social
welfare functional

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
Max-min, Max-max, lexi-min, lexi-max are all using
OFC information.
Utilitarianism: uses CUC information
Mean of order r: uses RSC information.
Under various informational assumptions, can we obtain
sensible welfare functionals ?
Desirable properties on the Social
Welfare functional
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
1) Non-dictatorship. There exists no individual h in N
such that, for all social states x and y, for all profiles
(U1,…,Un)  DU, Uh(x) > Uh(y) implies x P y (where R =
W(U1,…,Un))
2) Collective rationality. The social ranking should always
be an ordering (that is, the image of W should be )
3) Unrestricted domain. DU = n (all logically conceivable
combination of utility functions are a priori possible)
Desirable properties on the
Social Welfare Functional



4a) Strong Pareto. For all social states x and y, for all
profiles (Ui,…,Un)  DU , Ui(x)  Ui(y) for all i  N and Uh(x) >
Uh(y) for some h should imply x P y (where R =
W(U1,…,Un))
4b) Pareto Indifference. For all social states x and y, for
all profiles (Ui,…,Un)  DU , Ui(x) = Ui(y) for all i  N implies
x I y (where R = W(U1,…,Un))
5) Binary independance from irrelevant alternatives.
For every two profiles (U1,…,Un) and (U’1,…,U’n)  DU and
every two social states x and y such that Ui(x) = U’i(x) and
Ui(y) = U’i(y) for all i, one must have x R y  x R’ y where R
= W(U1,…,Un)) and R’ = W(U’1,…,U’n))
Welfarist lemma: If a social welfare
functional W satisfies 2, 3, 4b and 5, there
exists an ordering R* on n such that, for
all profiles (U1,…,Un)  DU,
x R y  (U1(x),…,Un(x)) R* (U1(y),…,Un(y))
(where R = W(U1,…,Un))
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4b and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof.
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 1: {x,y}  {x’,y’} = .
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 1: {x,y}  {x’,y’} = . By unrestricted domain,
one can find a profile of utility functions (U’’1,…,U’’n)  DU
such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) =
Ui(y) for all i  N.
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 1: {x,y}  {x’,y’} = . By unrestricted domain,
one can find a profile of utility functions (U’’1,…,U’’n)  DU
such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) =
Ui(y) for all i  N. By the independence axiom, x R y  x
R’’ y.
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 1: {x,y}  {x’,y’} = . By unrestricted domain,
one can find a profile of utility functions (U’’1,…,U’’n)  DU
such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) =
Ui(y) for all i  N. By the independence axiom, x R y  x
R’’ y. By Pareto indifference, x’ R’’ y’ and by,
independence again, x’ R’ y’.
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x).
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y).
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
 z R’’’ y (independence)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
 z R’’’ y (independence)
 z R’’’ x (Pareto-indifference and transitivity)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
 z R’’’ y (independence)
 z R’’’ x (Pareto-indifference and transitivity)
 z R’’ x (independence)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
 z R’’’ y (independence)
 z R’’’ x (Pareto-indifference and transitivity)
 z R’’ x (independence)
 y R’’ x (Pareto indifference and transitivity)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 2: (x’,y’) = (y,x). By unrestricted domain,
and since X contains at least 3 distinct elements, there is
a z distinct from x and y and profiles of utility functions
(U’’1,…,U’’n), (U’’’1,…,U’’’n) and (U’’’’1,…,U’’’’n) such that
Ui(x) = U’’i(y) = U’’i(z) = U’’’i(z) =U’’’’i(x) = U’’’i(z) and Ui(y)
= U’’i(x) = U’’’i(x) = U’’’’i(y) = U’’’i(y). Now:
x R y  x R’’’’ y (independence)
 z R’’’’ y (Pareto-indifference and transitivity)
 z R’’’ y (independence)
 z R’’’ x (Pareto-indifference and transitivity)
 z R’’ x (independence)
 y R’’ x (Pareto indifference and transitivity)
 y R’ x (Independence)
Proof of the lemma


Sublemma (neutrality): If W is a social welfare
functional satisfying 2, 3, 4a and 5, then, for all social
states x, y, x’ and y’  X and all profiles (U1,…,Un) and
(U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i 
N implies that x R y  x’ R’ y’ where R = W(U1,…,Un))
and R’ = W(U’1,…,U’n))
Proof. Case 3 and others: combine the two previous
cases
Proof of the welfarist lemma




The binary relation R* on n defined by a R* b   x, y 
X and a profile (U1,…,Un)  DU for which Ui(x) = ai and
Ui(y) = bi for all i  N such that x R y (for R = W
(U1,…,Un)) is uniquely defined by the preceding
sublemma.
The only thing that remains to be shown is that this
binary relation is reflexive, complete and transitive.
Reflexivity is clear. Completeness is also clear if R is
complete
For transitivity, let a, b and c  n be such that a R* b and
b R* c. By unrestricted domain, one can find a profile
(U1,…,Un)  DU and social states x, y and z  X such that
Ui(x) = ai, Ui(y) = bi and Ui(z) = ci. By the preceding
sublemma, we have x R y and y R z and, since R is
transitive, x R z, which implies therefore a R* c.
Welfarist lemma




Quite powerful: The only information that matters for
comparing social states is the utility levels achieved in
those states
Ranking of social states can be represented by a ranking
of utilities vectors achieved in those states.
This lemma can be used to see whether Arrow’s
impossibility result is robust to the replacement of
information on preference by information on happiness
As can be guessed, this robustness check will depend
upon the precision of the information that is available on
individual’s happiness.
Arrow’s theorem remains if happiness
is not interpersonnaly comparable


Theorem: If a social welfare functional W: DU  
satisfies conditions 2-5 and uses CNC or ONC
information on individuals well-being, then W is
dictatorial.
Proof: Diagrammatic (using the welfarist theorem, and
illustrating for two individuals)
Illustration
u2
u
u1
Illustration
u2
A
u
u
u1
Illustration
u2
A
u
u
B
u1
Illustration
u2
A
C
u
u
B
u1
Illustration
u2
A
C
u
u
B
D
u1
Illustration
u2
A
C
Better than
u by Pareto
u
u
B
D
u1
Illustration
u2
A
C
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
By NC, all points
in C are ranked
in the
same way
vis-à-vis u
A
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
By NC, all points
in C are ranked
in the
same way
vis-à-vis u
A
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration


The social ranking of a =(a1,a2) and
u=(u1,u2) must be the same than the
social ranking of (1a1+1, 2a2+2) and
(1u1+1, 2u2+2) for every numbers i >
0 and i (i = 1, 2).
Using i = (ui-bi)/(ui-ai) > 0 and
i = ui(bi-ai)/(ui-ai), this implies that the
social ranking of b = (1a1+1, 2a2+2)
and u = (1u1+1, 2u2+2) must be the
same than the social ranking of a and u
Illustration
u2
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
all points here
are also ranked
in the same way
vis-à-vis u
u1
Illustration
by Pareto, a and b
can not be
u2 indifferent to u
(and to themselves)
by transitivity)
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
all points here
are also ranked
in the same way
vis-à-vis u
u1
Illustration
u2
A
C
u
u
B
by NC, the
(strict) ranking
of region C
vis-à-vis u must
be the opposite
of the (strict)
ranking of D
vis-à-vis u
D
u1
Illustration
u2
A
C
u
u
B
D
u1
Illustration
u2
A
C c
uu
B
D
d
u1
Illustration



The social ranking of c =(c1,c2) and
u =(u1,u2) must be the same than the
social ranking of (1c1+1, 2c2+2) and
(1u1+1, 2u2+2) for every numbers i >
0 and i (i = 1, 2).
Using i = (di-ui)/(ui-ci) > 0 and
i = (u2i-dici)/(ui-ci), this implies that the
social ranking of u = (1c1+1, 2c2+2)
and d = (1u1+1, 2u2+2) must be the
same than the social ranking of c and u
If c is above u, d is below u and if c is
below u, d is above u
Illustration
u2
A
C
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
A
Worse
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
Better
u1
Illustration
u2
A
Worse
u
u
B
Better
u1
Illustration
u2
A
Worse
Individual 1
is the dictator
u
u
B
Better
u1
Illustration
u2
A
C
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
A
Better
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
Worse
u1
Illustration
u2
A
Better
u
u
B
Worse
u1
Illustration
u2
A
Better
Individual 2
Is the dictator
u
u
B
Worse
u1
Moral of this story




Arrow’s theorem is robust to the replacement of
preferences by well-being if well-being can not be
compared interpersonally (notice that cardinal
measurability does not help if no interpersonal comparison
is possible)
What if well-being is ratio-scale measurable and
interpersonnally non-comparable ?
Welfarist theorem gives nice geometric intuition on what’s
going on, see Blackorby, Donaldson and Weymark (1984),
International Economic Review
Generalization to n individuals is easy
Allowing ordinal comparability






A strengthening of non-dictatorship: Anonymity
A social welfare functional W is anonymous if for every
two profiles (U1,…,Un) and (U’1,…,U’n)  DU such that
(U1,…,Un) is a permutation of (U’1,…,U’n), one has R =
R’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n))
Dictatorship of individual h is clearly not anonymous.
Hence, by virtue of the previous theorem, there are no
anonymous social welfare functionals that use ON or
CN information on individual’s well-being and that
satisfy axioms 2)-5).
We will now show that this impossibility vanishes if we
allow for ordinal comparisons of well-being accross
individuals.
Specifically, we are going to show that if we allow the
social welfare functional to uses OC information on
individual well-being, then the only anonymous social
welfare functionals are positional dictatorships
Positional dictatorship




A social welfare functional W is a positional dictatorship if
there exists a rank r  {1,…,n} such that, for every two
social states x and y, and every profile (U1,…,Un) of utility
functions U(r)(x) > U(r)(y)  x P y where R = W(U1,…,Un))
and, for every z  X, (U(1)(z),…,U(n)(z)) is the ordered
permutation of (U1(z)…,Un(z)) satisfying U(i)(z)  U(i+1)(z)
for every i = 1,…,n-1
Max-min and Lexi-min are positional dictatorships (for r =
1). So is Max-max (r = n). Another one would be the
dictatorship of the smallest integer greater than or equal
to n/2 (median)
Positional dictatorship rules only specify the social
ranking that prevails when the positional dictator has a
strict preference. They don’t impose anything on the
social ranking when the positional dictator is indifferent.
Hence, positional dictatorship does not enable a
distinction between lexi-min and max-min.
A new theorem:


Theorem: A social welfare functional W: DU  
is anonymous, satisfies conditions 2-5 and uses
OC information on individuals well-being if and
only if W is a positional dictatorship.
Proof: Diagrammatic (using the welfarist
theorem, and illustrating for two individuals)
Illustration
u2
IV
III
u’(.)
u’1
= u’(2)
u2 = u1
II
IX
V
I
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
III
u’(.)
u’1
= u’(2)
II
IX
V
I
VIII
u’ and u’(.) are
indifferent by
anonymity
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
III
u’(.)
u’1
= u’(2)
II
IX
V
better than u’
(and than u’(.)
by Pareto
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
u’(.)
u’1
= u’(2)
better than u’ (and
than u’(.) by Pareto
III
IX
V
better than u’
(and than u’(.)
by Pareto
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
u’(.)
u’1
= u’(2)
better than u’ (and
than u’(.) by Pareto
better than u’ (and
than u’(.) by Pareto
IX
V
better than u’
(and than u’(.)
by Pareto
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
u’(.)
u’1
= u’(2)
IX
V
better
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
u’(.)
u’1
= u’(2)
IX
V
better
VIII
u’
u’2
VI
u’2 = u’(1)
X
VII
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
better
VIII
u’
u’2
worse
u’2 = u’(1)
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
all points
in this zone
are ranked
in the same
way vis-à-vis
u’ (and u’(.)
by transitivity)
u’2
worse
u’2 = u’(1)
better
u’
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
u’
u’2
worse
u’2 = u’(1)
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
u’
u’2
worse
u’2 = u’(1)
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
u’
u’2
worse
u’2 = u’(1)
X
b1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
u’
u’2
worse
u’2 = u’(1)
X
b1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
b2
u’
u’2
worse
u’2 = u’(1)
X
b1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
b2
u’
u’2
worse
u’2 = u’(1)
X
b1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
b2
u’
u’2
worse
u’2 = u’(1)
X
b1
a1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
better
b
b2
u’
u’2
worse
u’2 = u’(1)
X
b1
a1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
IX
a
a2
better
b
b2
u’
u’2
worse
u’2 = u’(1)
X
b1
a1
u’1
u1


Since W uses ordinally comparable
information on individuals’ well-beings, the
social ranking of a = (a1,a2) and u’ =(u’1,u’2)
must be the same than the social ranking
of (g(a1),g(a2) and (g(u1), g(u2)) for every
increasing real valued function g
If we consider the function g whose graph
is the following, then this implies than the
social ranking of b vis-à-vis u’ must be the
same than that of a vis-à-vis u.
u2
u2 = u1
u’1
IX
a
a2
b
b2
u’2
u’2
b1
a1
u’1
u1
u2
u2 = u1
u’1
IX
a
b1 = a2
b
b2
u’2
u’2
b1 = a2
a1
u’1
u1
u2
u2 = u1
u’1
g(x)
Remark: we use the fact
that we are free to use any
increasing function
IX
a
b1 = a2
b
b2
u’2
u’2
b1 = a2
a1
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
all points
in this zone
are ranked
in the same
way vis-à-vis
u’ (and u’(.)
by transitivity)
u’2
worse
u’2 = u’(1)
better
u’
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
by anonymity
same is true
for this zone
all points
in this zone
are ranked
in the same
way vis-à-vis
u’ (and u’(.)
by transitivity)
u’2
worse
u’2 = u’(1)
better
u’
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
by Pareto,
these points
can not be
indifferent to
u’ (and by
transitivity, to
themselves)
better
u’
u’2
worse
u’2 = u’(1)
X
u’1
u1
u2
u2 = u1
IV
u’1
= u’(2)
u’(.)
by Pareto,
these points
can not be
indifferent to
u’ (and by
transitivity, to
themselves)
better
u’
u’2
worse
u’2 = u’(1)
ranking of those points vis-à vis u’
is the opposite than those
in the yellow zone
u’1
u1
u2
Same is true for those points
IV
u’1
= u’(2)
u2 = u1
u’(.)
by Pareto,
these points
can not be
indifferent to
u’ (and by
transitivity, to
themselves)
better
u’
u’2
worse
u’2 = u’(1)
ranking of those points vis-à vis u’
is the opposite than those
in the yellow zone
u’1
u1
u2
u’1
= u’(2)
u2 = u1
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
u’1
u1
u2
u’1
= u’(2)
Hence we are left with two possibilities
u2 = u1
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
u’1
u1
u2
u’1
= u’(2)
Hence we are left with two possibilities
u2 = u1
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
u’1
u1
u2
u’1
= u’(2)
Max is the dictator
u2 = u1
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
u’1
u1
u2
u’1
= u’(2)
Min is the dictator
u2 = u1
u’(.)
better
u’
u’2
worse
u’2 = u’(1)
u’1
u1
Remarks on this theorem





If we drop anonymity, we get other kinds of dictatorships
(including non-anonymous ones)
Generalizations to more than two individuals is
cumbersome (see Gevers, Econometrica (1979) and
Roberts R. Eco. Stud. (1980).
Max dictatorship is not very appealing. Can we eliminate it
?
Yes if we impose an axiom of « minimal equity », due to
Hammond (Econometrica, 1976)
A social welfare functional W satisfies Hammond’s
minimal equity principle if for every profile (U1,…,Un) and
every two social states x and y for which there are
individuals i and j such that Uh(x) = Uh(z) for all h  i, j, and
Uj(y) > Uj(x) > Ui(x) > Ui(y), one has x P y where R =
W(U1,…,Un))
The lexi-min theorem:


Theorem: A social welfare functional
W: DU   is anonymous, satisfies conditions 25, uses OC information on individuals well-being
and satisfies Hammond’s equity principle if and
only if it is the Lexi-min .
Proof: See Blackorby, Donaldson & Weymark
(1984) theorem 6.1 for a diagrammatic twoindividuals proof or Hammond (1976) for a
complete proof of this.
Further remarks on lexi-min



It is not a continuous ranking of alternatives
Maxi-min by contrast is continuous (even
thought it violates the strong Pareto principle)
Suppose we replace in the previous theorem
strong Pareto by weak Pareto, and that we
add continuity, can we get Maxi-min ?
Continuity ?
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
We go continuously
from the better…
u’
worse
u’2
u’2 = u’(1)
u’1
better
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse to the
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse to the
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse Without encountering
indifference
u’2 = u’(1)
u’1
u1
Continuity


A social welfare functional W satisfying 2,3,
4a and 5 is continuous if for every profile
(U1,…,Un), the welfarist ordering R* of n that
corresponds to R by the welfarist theorem is
continuous where R = W(U1,…,Un))
An ordering R* of n is continuous if, for
every u  n, the sets NWR*(u) = {u’ n: u’ R*
u} and NBR*(u) = {u’  n: u R* u’} are both
closed in n
Bad news ?



Theorem 1: There are no anonymous and continuous
social welfare functionals W: DU   that use OC
information on individuals’ well-being and that satisfy
collective rationality, weak Pareto, Pareto-indifference,
unrestricted domain, binary independance and
Hammond’s equity if n > 2
Theorem 2: If n = 2, an anonymous and continuous
social welfare functional W: DU   using OC
information on individuals’ well-being satisfies
collective rationality, weak Pareto, Pareto-indifference,
unrestricted domain, binary independance and
Hammond’s equity if and only if it is the max-min
Hence, no characterization of max-min in this setting.
Cardinal measurability and unit
comparability


Theorem: An anonymous social welfare
functional W: DU   satisfies conditions 2-5
and uses CUC information on individuals wellbeing if and only if it is utilitarian.
Proof: Diagrammatic again
u2
u2 = u1
b
b2
IX
c
-1
b
a
a2
b1
c
a1
2c
u1
u2
u2 = u1
b
b2
Indifferent by
anonymity
IX
c
-1
b
a
a2
b1
c
a1
2c
u1
u2
u2 = u1
b
b2
a must be ranked
vis-à-vis c the same
way than c is vis-à-vis b
IX
c
-1
b
a
a2
b1
c
a1
2c
u1
u2
u2 = u1
b
b2
a must be ranked
vis-à-vis c the same
way than c is vis-à-vis b
(because of CUC)
IX
c
-1
b
a
a2
b1
c
a1
2c
u1
u2
u2 = u1
b
b2
a must be ranked
vis-à-vis c the same
way than c is vis-à-vis b
(because of CUC)
IX
c = a2
+c–a2
-1
b
a
a2
b1
c = a1 + c-a1
a1
2c
u1
u2
b2= 2c
–a2
u2 = u1
b
a must be ranked
vis-à-vis c the same
way than c is vis-à-vis b
(because of CUC)
IX
c = a2
+c–a2
-1
b
a
a2
b1 = 2c-a1
c = a1 + c-a1
a1
2c
u1
u2
b2= 2c
–a2
u2 = u1
b
IX
c = a2
+c–a2
-1
The only possibility
compatible with indifference
between a and b is
indifference between c and
a, or b.
b
a
a2
b1 = 2c-a1
c = a1 + c-a1
a1
2c
u1
u2
b2= 2c
–a2
u2 = u1
b
Hence all utility vectors
on the isoutility line are
socially indifferent
IX
c = a2
+c–a2
-1
b
a
a2
b1 = 2c-a1
c = a1 + c-a1
a1
2c
u1
u2
b2= 2c
–a2
u2 = u1
b
IX
c = a2
+c–a2
-1
By Pareto, all utility
vectors on the north
east of the isoutility line
are better than any point
on the line
b
a
a2
b1 = 2c-a1
c = a1 + c-a1
a1
2c
u1
Remarks on this utilitarian theorem



No need of continuity
If anonymity is dropped, then asymmetric
utilitarianism emerges (social ranking R is
defined by: x R y  iNiUi(x)  iNiUi(y) for
some non-negative real numbers i (i = 1,…,n)
(numbers are strictly positive if strong Pareto
is satisfied).
Notice that if weak Pareto only is required
(some i can be zero), this family of social
orderings contains standard dictatorship
(which is not surprising)
Other axiomatic justifications of
utilitarianism




Maskin (1978). Uses CFC along with continuity
and a separability condition (independence
with respect to unconcerned individuals)
Harsanyi (1953) impartial observer theorem.
Society is looked at from behind a « veil of
ignorance ». We must choose a social state
without knowing in which shoes we are going
to be, but by assuming an equal chance of
being in anybody’s shoes
If the « social planner » who looks at society
from behind this veil of ignorance has VonNeuman Morgenstern preferences, he should
order social state on the basis of the expected
utility of being anyone
This argument is flawed
Generalized utilitarianism





Utilitarianism is insensitive to utility inequality
A social ranking that is more general than
utilitarianism is, as we have seen, the mean of order
r
But one could also consider a more general family
of social rankings: symmetric generalized
utilitarianism
x R y ig(Ui(x))  ig(Ui(y)) where R =
W(U1,…,Un) for some increasing function g: n 
Mean of order r is a special case of this where g is
defined by g(u) = u1/r if r > 0, g(u) = ln(u) if r = 0 and
g(u) = -u1/r if r < 0
Generalized utilitarianism


Theorem: An anonymous social welfare
functional W: +n   satisfies Paretoindifference, strong Pareto, continuity,
independence with respect to unconcerned
individuals and binary independence of
irrelevant alternative if and only if it is a
generalized utilitarian ranking
Proof: See Blackorby, Bossert and Donaldson,
Population Issues in Social Choice Theory,
Welfare economics and Ethics, Cambrige
University Press, 2005, theorem 4.7
Remarks on this theorem (1)






Does not ride on measurability assumption on wellbeing
Does not restrict the g function.
A way to restrict the g function is to impose utility
inequality aversion property on the social ranking
An example of inequality aversion: Hammond’s weak
equity principle
Another example (weaker than Hammond’s): PigouDalton principle of equity
A social welfare functional W satisfies the PigouDalton equity principle if for every profile (U1,…,Un)
and every two social states x and y for which there are
individuals i and j and a number  > 0 such that Uh(x) =
Uh(z) for all h  i, j, and Uj(x) = Uj(y) -   Ui(x) = Ui(y) +
, one has x P y where R = W(U1,…,Un))
Remarks on this theorem (2)






Both equity principles incorporate implicitly
interpersonnal comparability and measurability
assumptions on well-being
Utility levels must be compared accross individuals to
make sense of Hammond’s equity principles.
Utility differences of  between two individuals must also
be meaningful in order for the Pigou-Dalton equity
principle of transfer to make sense
Hammond’s equity implies Pigou-Dalton equity but not
vice-versa
Pigou-Dalton equity leads to a significant restriction of
the g function: concavity
g is concave if, for all numbers u and v and every
number   [0,1], one has g(u+(1-)v)  g(u)+(1-)g(v)
Concavity?
g(x)
g(u)
g(u+(1-)v)
IX
g(u) +
a
(1-)g(v)
b
g(v)
v
u+(1-)v
u
Equity respectful Generalized utilitarianism

Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 

Proof: The only thing that needs to be proved is
the concavity of g.
Equity respectful Generalized utilitarianism

Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 

Proof: The only thing that needs to be proved is
the concavity of g. It can be checked that if g is
concave, then generalized utilitarianism satisfies
the Pigou Dalton equity principle
Equity respectful Generalized utilitarianism

Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 

Proof: The only thing that needs to be proved is
the concavity of g. Conversely, assume that g is
not concave. That, is assume that there are some
u, v and  such that g(u+(1-)v) < g(u)+(1-)g(v)
Equity respectful Generalized utilitarianism

Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 

Proof: The only thing that needs to be proved is
the concavity of g. Without loss of generality, one
can assume that u > v and  =1/2
Equity respectful Generalized utilitarianism

Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 

Proof: The only thing that needs to be proved is
the concavity of g. By unrestricted domain,
consider a profile U1,…,Un of utility functions and
two alternatives x and y such that:
Equity respectful Generalized utilitarianism


Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 
Proof:. Uh(x) = Uh(z) for all h  i, j, and Uj(y) = v <
Uj(x) = (v +u)/2 = Uj(x) < u = Uj(y).
Equity respectful Generalized utilitarianism


Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 
Proof:. Now: x R y  ig(Ui(x))  ig(Ui(y)) 
g(Ui(x))+ g(Uj(x))  g(Ui(y))+ g(Uj(y)) 
2g((v +u)/2)  g(v)+ g(u)
Equity respectful Generalized utilitarianism


Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 
Proof:. Now: x R y  ig(Ui(x))  ig(Ui(y)) 
g(Ui(x))+ g(Uj(x))  g(Ui(y))+ g(Uj(y)) 
2g((v +u)/2)  g(v)+ g(u). This inequality is violated
by assumption.
Equity respectful Generalized utilitarianism


Theorem: An anonymous social welfare functional
W: +n   satisfies Pareto-indifference, strong
Pareto, continuity, independence with respect to
unconcerned individuals, binary independence of
irrelevant alternative and Pigou-Dalton equity
principle if and only if x R y ig(Ui(x)) 
ig(Ui(y)) where R = W(U1,…,Un) for some
increasing and concave function g: n 
Proof:. Now: x R y  ig(Ui(x))  ig(Ui(y)) 
g(Ui(x))+ g(Uj(x))  g(Ui(y))+ g(Uj(y)) 
2g((v +u)/2)  g(v)+ g(u). This inequality is violated
by assumption. Hence Pigou-Dalton is violated.
Ratio-scale comparability





Requires a meaning to be given to zero levels of
happiness
A negative happiness is not the same thing then a positive
one.
Suppose that we restrict the domain DU of admissible
profiles of utility functions to n+ where + is the set of all
functions U: X  +
A social welfare functional satisfies independence with
respect to unconcerned individuals if, for all profiles
(U1,…,Un) of utility functions and all social states w, x, y and
z  X, the existence of a group G of individuals such that
Ug(w) = Ug(x) and Ug(y) = Ug(z) for all g  G and Uh(w) =
Uh(y) and Uh(x) = Uh(z) for all h  N\G implies that w R x  y
R z where R = W(U1,…,Un))
Says that the social ranking of two states should not depend
upon the utility level of the individuals who are indifferent
between them (these individuals are « unconcerned »)
Ratio scale comparability


Theorem: An anonymous social welfare
functional W: +n   satisfies Paretoindifference, strong Pareto, continuity,
independence with respect to unconcerned
individuals, binary independence of irrelevant
alternative and RSFC if and only if it is the
mean of order r ranking
Proof: See Blackorby and Donaldson,
International Economic Review (1982),
theorem 2
Some issues with variable population





We have been so far assuming that the
number of individuals is fixed.
Yet there are many normative issues that
require the comparison of societies with
different numbers of members
Is it good to add new people to actual
societies (demographic policies) ?
With varying number of individuals, defining
general interest as a function of individuals
interest becomes tricky
How can someone compares her well-being
with the situation in which she does not
come to existence ?
Some issues with variable population





Problem (under welfarism and anonymity):
Comparing utility vectors of different dimensions.
(u1,…,um) a vector of utilities in a society with m
persons; (v1,…,vn) a vector of utilities in a society
with n persons (remember that individual’s name
does not matter under anonymity)
X = nn
for all u  X, n(u) is the dimension of u (number of
people)
How should we compare these vectors ?
Some issues with variable population


Classical utilitarianism
u RCU v  n(u)i=1 ui  n(v)i=1 vi
Critical level utilitarianism
u RCLU v  n(u)i=1 (ui –c(u))  n(v)i=1 (vi –c(v))
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where c is a « critical utility level » which in
general depends upon the distribution of
utilities)
Average Utilitarianism
u RAU v  (n(u)i=1 ui)/n(u)  (n(v)i=1 vi)/n(v)
Note: AU (c(u) =(n-1/n)(u)) and CU (c=0)
are particular cases of CL
Some issues with variable population
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Classical utilitarianism : Generates the
repugnant conclusion (Parfitt, reason and
persons, 1984). For any positive level of wellbeing , however small, it is always possible to
improve upon the current state by packing the
earth with people even if these people only
enjoy  level of utility
Average Utilitarianism: Avoids the repugnant
conclusion, because adding people is good only
if their well-being is above the average.
See Blackorby, Bossert, Donaldson: Population
issues in Social Choice Theory, Wefare
Economics, and ethics, Cambridge U.Press,
2005.