Arrow’s theorem and the
problem of social choice
Can we define the general interest on
the basis of individuals interests ?
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





X, a set of mutually exclusive social states
(complete descriptions of all relevant aspects of a
society)
N a set of individuals N = {1,..,n} indexed by i
Ri a preference ordering of individual i on X (with
asymmetric and symmetric factors Pi and Ii)
x Ri y means « individual i weakly prefers state x
to state y »
Pi = « strict preference », Ii = « indifference »
An ordering is a reflexive, complete and transitive
binary relation
The interpretation given to preferences is unclear
in Arrow’s work (influenced by economic theory)
Can we define the general interest on
the basis of individuals interests ?
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




<Ri > = (R1 ,…, Rn) a profile of individual
preferences
 the set of all binary relations on X
  , the set of all orderings on X
D  n, the set of all admissible profiles
Arrow’s problem: to find a « collective decision
rule » C: D   that associates to every profile
<Ri > of individual preferences a binary relation R
= C(<Ri >)
x R y means that the general interest is (weakly)
better served with x than with y when individual
preferences are (<Ri >)
Examples of collective decision rules
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
1: Dictatorship of individual h: x R y if and only x Rh y
(not very attractive)
2: ranking of social states according to an exogenous
code (say the Charia). Assume that the exogenous code
ranks any pair of social alternatives as per the ordering 
(x  y means that x (women can not drive a car) is weakly
preferable to y (women drive a car). Then C(<Ri >) =  for
all <Ri >  D is a collective decision rule. Notice that even
if everybody in the society thinks that y is strictly preferred
to x, the social ranking states that x is better than y.
Examples of collective decision rules
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
3: Unanimity rule (Pareto criterion): x R y if
and only if x Ri y for all individual i. Interesting but
deeply incomplete (does not rank alternatives for
which individuals preferences conflict)
4: Majority rule. x R y if and only if
#{i  N:x Ri y}  #{i  N :y Ri x}. Widely used, but
does not always lead to a transitive ranking of
social states (Condorcet paradox).
the Condorcet paradox
the Condorcet paradox
Individual 1
Individual 2
Individual 3
the Condorcet paradox
Individual 1
Ségolène
Nicolas
François
Individual 2
Individual 3
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
A majority (1 and 3) prefers Ségolène to Nicolas
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
A majority (1 and 3) prefers Ségolène to Nicolas
A majority (1 and 2) prefers Nicolas to François
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
A majority (1 and 3) prefers Ségolène to Nicolas
A majority (1 and 2) prefers Nicolas to François
Transitivity would require that Ségolène be
socially preferred to François
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
A majority (1 and 3) prefers Ségolène to Nicolas
A majority (1 and 2) prefers Nicolas to François
Transitivity would require that Ségolène be
socially preferred to François but………….
the Condorcet paradox
Individual 1
Individual 2
Ségolène
Nicolas
François
Nicolas
François
Ségolène
Individual 3
François
Ségolène
Nicolas
A majority (1 and 3) prefers Ségolène to Nicolas
A majority (1 and 2) prefers Nicolas to François
Transitivity would require that Ségolène be
socially preferred to François but………….
A majority (2 and 3) prefers strictly François to Ségolène
Example 5: Positional Borda
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

Works if X is finite.
For every individual i and social state x,
define the « Borda score » of x for i as the
number of social states that i considers
(weakly) worse than x. Borda rule ranks
social states on the basis of the sum, over
all individuals, of their Borda scores
Let us illustrate this rule through an
example
Borda rule
Individual 1
Individual 2
Individual 3
Ségolène
Nicolas
Jean-Marie
François
Nicolas
François
Jean-Marie
Ségolène
François
Ségolène
Nicolas
Jean-Marie
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François
1
Nicolas
François
Jean-Marie
Ségolène
Sum of scores Ségolène = 8
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François
1
Nicolas
François
Jean-Marie
Ségolène
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François
1
Nicolas
François
Jean-Marie
Ségolène
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Nicolas is the best alternative, followed closely by Ségolène
and François. Jean-Marie is the last
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Problem: The social ranking of François, Nicolas and Ségolène
depends upon the position of Jean-Marie
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Raising Jean-Marie above Nicolas for 1 and Jean-Marie below
Ségolène for 2 changes the social ranking of Ségolène and Nicolas
Borda rule
Individual 1
Individual 2
Ségolène 4
Nicolas
3
Jean-Marie 2
François 1
Nicolas
François
Jean-Marie
Ségolène
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Raising Jean-Marie above Nicolas for 1 and Jean-Marie below
Ségolène for 2 changes the social ranking of Ségolène and Nicolas
Borda rule
Individual 1
Individual 2
Ségolène 4
Jean-Marie 3
Nicolas
2
François 1
Nicolas
François
Ségolène
Jean-Marie
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Raising Jean-Marie above Nicolas for 1 and Jean-Marie below
Ségolène for 2 changes the social ranking of Ségolène and Nicolas
Borda rule
Individual 1
Individual 2
Ségolène 4
Jean-Marie 3
Nicolas
2
François 1
Nicolas
François
Ségolène
Jean-Marie
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 9
Sum of scores Nicolas = 8
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Raising Jean-Marie above Nicolas for 1 and Jean-Marie below
Ségolène for 2 changes the social ranking of Ségolène and Nicolas
Borda rule
Individual 1
Individual 2
Ségolène 4
Jean-Marie 3
Nicolas
2
François 1
Nicolas
François
Ségolène
Jean-Marie
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 9
Sum of scores Nicolas = 8
Sum of scores François = 8
Sum of scores Jean-Marie = 5
Raising Jean-Marie above Nicolas for 1 and Jean-Marie below
Ségolène for 2 changes the social ranking of Ségolène and Nicolas
Borda rule
Individual 1
Individual 2
Ségolène 4
Jean-Marie 3
Nicolas
2
François 1
Nicolas
François
Ségolène
Jean-Marie
Individual 3
4
3
2
1
François
Ségolène
Nicolas
Jean-Marie
4
3
2
1
Sum of scores Ségolène = 9
Sum of scores Nicolas = 8
Sum of scores François = 8
Sum of scores Jean-Marie = 5
The social ranking of Ségolène and Nicolas depends on the individual
ranking of Nicolas vs Jean-Marie or Ségolène vs Jean-Marie
Are there other collective
decision rules ?
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Arrow (1951) proposes an axiomatic
approach to this problem
He proposes five axioms that, he thought,
should be satisfied by any collective decison
rule
He shows that there is no rule satisfying all
these properties
Famous impossibility theorem, that throw a
lot of pessimism on the prospect of obtaining
a good definition of general interest as a
function of the individual interest
Five desirable properties on
the collective decision rule
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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 <Ri>, x
Ph y implies x P y (where R = C(<Ri>)
2) Collective rationality. The social ranking should always
be an ordering (that is, the image of C should be )
(violated by the unanimity and the majority rule; violated
also by the Lorenz domination criterion if X is the set of all
income distributions)
3) Unrestricted domain. D = n (all logically conceivable
preferences are a priori possible)
Five desirable properties on
the collective decision rule
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
4) Weak Pareto principle. For all social states x and y, for
all profiles <Ri>  D , x Pi y for all i  N should imply x P y
(where R = C(<Ri>) (violated by the collective decision rule
coming from an exogenous tradition code)
5) Binary independance from irrelevant alternatives.
For every two profiles <Ri> and <R’i>  D and every two
social states x and y such that x Ri y  x R’i y for all i, one
must have x R y  x R’ y where R = C(<Ri>) and R’ =
C(<R’i>). The social ranking of x and y should only depend
upon the individual rankings of x and y.
Arrow’s theorem: There does not
exist any collective decision
function C: D   that satisfies
axioms 1-5
Proof of Arrow’s theorem (Sen 1970)
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
Given two social states x and y and a collective
decision function C, say that group of individuals
G  N is semi-decisive on x and y if and only if
x Pi y for all i  G and y Ph x for all h  N\G implies
x P y (where R = C(<Ri>))
Analogously, say that group G is decisive on x
and y if x Pi y for all i  G implies x P y
Clearly, decisiveness on x and y implies semidecisiveness on that same pair of states.
Strategy of the proof: every collective decision
function satisfying axioms 2-5 implies the
existence of a decisive individual (a dictator).
Proof of Arrow’s theorem (Sen 1970)




We are going to prove two lemmas.
In the first (field expansion) lemma, we will show that
if the collective decision function admits a group G that
is semi-decisive on a pair of social states, then this
group will in fact be decisive on every pair of states
In the second (group contraction) lemma, we are
going to use the first lemma to show that if a group G
containing at least two individuals is decisive on a pair
of social states, then it contains a proper subgroup of
individuals that is decisive on that pair.
These two lemmas prove the theorem because, by the
Pareto principle, we know that the whole society is
decisive on every pair of social states.
Field expansion lemma

Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} .
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G. Since G
is semi-decisive on x and y, x P y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G. Since G
is semi-decisive on x and y, x P y. Since C satisfies the
Pareto principle, y P b and, since the social ranking is
transitive, x P b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G. Since G
is semi-decisive on x and y, x P y. Since C satisfies the
Pareto principle, y P b and, since the social ranking is
transitive, x P b. Now, by binary independence of
irrelevant alternatives, the social ranking of x (= a) and b
only depends upon the individual preferences over a and
b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G. Since G
is semi-decisive on x and y, x P y. Since C satisfies the
Pareto principle, y P b and, since the social ranking is
transitive, x P b. Now, by binary independence of
irrelevant alternatives, the social ranking of x (= a) and b
only depends upon the individual preferences over a and
b. Yet only the preferences over a and b of the members
of G have been specified here.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
1: a = x and b {x,y} . Since C is defined for every
preference profile, consider a profile such that x Pi y Pi b
for all i in G and y Ph x and y Ph b for all h  N\G. Since G
is semi-decisive on x and y, x P y. Since C satisfies the
Pareto principle, y P b and, since the social ranking is
transitive, x P b. Now, by binary independence of
irrelevant alternatives, the social ranking of x (= a) and b
only depends upon the individual preferences over a and
b. Yet only the preferences over a and b of the members
of G have been specified here. Hence G is decisive on a
and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} .
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G. Since G is semidecisive on x and y, one has x P y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G. Since G is semidecisive on x and y, one has x P y. Since C satisfies the
Pareto principle, a P x and, since the social ranking is
transitive, a P y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G. Since G is semidecisive on x and y, one has x P y. Since C satisfies the
Pareto principle, a P x and, since the social ranking is
transitive, a P y. Now, by binary independence of
irrelevant alternatives, the social ranking of a and b (= y)
only depends upon the individual preferences over a and
b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G. Since G is semidecisive on x and y, one has x P y. Since C satisfies the
Pareto principle, a P x and, since the social ranking is
transitive, a P y. Now, by binary independence of
irrelevant alternatives, the social ranking of a and b (= y)
only depends upon the individual preferences over a and
b. Yet only the preferences over a and b of the members
of G have been specified here.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
2: b = y and a {x,y} . As before, using unrestricted
domain, consider a profile such that a Pi x Pi y for all i in
G and y Ph x and a Ph x for all h  N\G. Since G is semidecisive on x and y, one has x P y. Since C satisfies the
Pareto principle, a P x and, since the social ranking is
transitive, a P y. Now, by binary independence of
irrelevant alternatives, the social ranking of a and b (= y)
only depends upon the individual preferences over a and
b. Yet only the preferences over a and b of the members
of G have been specified here. Hence G is decisive on a
and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
3: a = x, b = y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
3: a = x, b = y. Consider any z  X with z {x,y}.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
3: a = x, b = y. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
3: a = x, b = y. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z. Of course if G is decisive on x and z, it is
quasi-decisive on x and z.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
3: a = x, b = y. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z. Of course if G is decisive on x and z, it is
quasi-decisive on x and z. Applying case 1 again, we are
led to the conclusion that G is decisive on x and y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
4: a = y, b = x.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
4: a = y, b = x. Consider any z  X with z {x,y}.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
4: a = y, b = x. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
4: a = y, b = x. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z. Of course if G is decisive on x and z, it is
quasi-decisive on x and z.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
4: a = y, b = x. Consider any z  X with z {x,y}. By case
1), if G is semi-decisive on x and y, it must be decisive
on x and z. Of course if G is decisive on x and z, it is
quasi-decisive on x and z. Applying case 2), we are led
to the conclusion that G is decisive on x and y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
5: a {x,y}, b = x.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
5: a {x,y}, b = x. Since G is semi-decisive on x and y, it
is decisive on x and a by case 1) and, therefore quasidecisive on x (= b) and a.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
5: a {x,y}, b = x. Since G is semi-decisive on x and y, it
is decisive on x and a by case 1) and, therefore quasidecisive on x (= b) and a. By case 4), if a group is quasidecisive on b and a, it is decisive on a and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
6: b {x,y}, a = y.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
6: b {x,y}, a = y. Since G is semi-decisive on x and y, it
is decisive on b and y by case 2) and, therefore quasidecisive on b and a (=y).
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
6: b {x,y}, a = y. Since G is semi-decisive on x and y, it
is decisive on b and y by case 2) and, therefore quasidecisive on b and a (=y). By case 4), if a group is quasidecisive on b and a, it is decisive on a and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
7: {a,b}  {x,y} = .
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
7: {a,b}  {x,y} = . By case 1), if G is semi-decisive on x
and y, it is decisive on x and b and, and therefore, quasidecisive on x and b.
Field expansion lemma


Lemma: If C is a collective decision function satisfying
axioms 2-5, then, if a group G is semi-decisive on two
social states x and y, G is in fact decisive on all social
states a and b.
Proof: We consider several cases.
7: {a,b}  {x,y} = . By case 1), if G is semi-decisive on x
and y, it is decisive on x and b and, and therefore, quasidecisive on x and b. But by case 2), this implies that G
is decisive on a and b.
Group Contraction lemma
Group Contraction lemma

Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof:
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z. Since the social ranking R is complete, either x
P z or z R x.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z. Since the social ranking R is complete, either x
P z or z R x. By binary independance of irrelevant alternatives,
the social ranking of any pair of social states only depends
upon the individual rankings of that pair.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z. Since the social ranking R is complete, either x
P z or z R x. By binary independance of irrelevant alternatives,
the social ranking of any pair of social states only depends
upon the individual rankings of that pair. If x P z, we are led to
the conclusion that G1 is quasi-decisive on x and z.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z. Since the social ranking R is complete, either x
P z or z R x. By binary independance of irrelevant alternatives,
the social ranking of any pair of social states only depends
upon the individual rankings of that pair. If x P z, we are led to
the conclusion that G1 is quasi-decisive on x and z. If z R x, we
obtain from the transitivity of R that y P x and, therefore, that
G2 is quasi-decisive on y and x.
Group Contraction lemma


Lemma: If C is a collective decision function satisfying axioms
2-5, then, if a group G containing at least two individuals is
decisive on all social states x and y, then it contains a proper
subset of individuals that are also decisive on x and y.
Proof: Partition G into two non-empty and disjoint subsets G1
and G2 and let x, y and z be 3 social states. By unrestricted
domain, consider a profile where x Ph y Ph z for all h in G1, y Pi
z Pi x for all i in G2 and z Pj x Pj y for all j in N\G. Since G is
decisive, y P z. Since the social ranking R is complete, either x
P z or z R x. By binary independance of irrelevant alternatives,
the social ranking of any pair of social states only depends
upon the individual rankings of that pair. If x P z, we are led to
the conclusion that G1 is quasi-decisive on x and z. If z R x, we
obtain from the transitivity of R that y P x and, therefore, that
G2 is quasi-decisive on y and x. By the Field expansion
lemma, it follows that either G1 or G2 is decisive on every pair
of social states.
All Arrow’s axioms are independent





Dictatorship of individual h satisfies Pareto, collective rationality,
binary independence of irrelevant alternatives and unrestricted
domain but violates non-dictatorship
The Tradition ordering satisfies non-dictatorship, collective
rationality, binary independance of irrelevant alternative and
unrestricted domain, but violates Pareto
The majority rule satisfies non-dictatorship, Pareto, binary
independence of irrelevant alternative and unrestricted domain but
violates collective rationality (as does the unanimity rule)
The Borda rule satisfies non-dictatorship, Pareto, unrestricted
domain and collective rationality, but violates binary independence
of irrelevant alternatives
We’ll see later that there are collective decisions functions that
violate unrestricted domain but that satisfies all other axioms
Escape out of Arrow’s theorem






Natural strategy: relaxing the axioms
It is difficult to quarel with non-dictatorship
We can relax the assumption that the social
ranking of social states is an ordering (in
particular we may accept that it be
« incomplete »)
We can relax unrestricted domain
We can relax binary independance of irrelevant
alternatives
Should we relax Pareto ?
Should we relax the Pareto principle ? (1)






Most economists, who use the Pareto principle as the main
criterion for efficiency, would say no!
Many economists abuse of the Pareto principle
Given a set A in X, say that state a is efficient in A if there
are no other state in A that everybody weakly prefers to a
and at least somebody strictly prefers to a.
Common abuse: if a is efficient in A and b is not efficient in
A, then a is socially better than b
Other abuse (potential Pareto) a is socially better than b if
it is possible, being at a, to compensate the loosers in the
move from b to a while keeping the gainers gainers!
Only one use is admissible: if everybody believes that x is
weakly better than y, then x is socially weakly better than y.
Illustration: An Edgeworth Box
xA2
B
xB1
y
2
z
x
xA1
A
1
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
x is efficient
z is not efficient
y
x
z
xA1
A
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
y
x
z
x is efficient
z is not efficient
x is not socially
better than z as
per the Pareto
principle
xA1
A
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
y is better than
z as per the
Pareto principle
y
x
z
xA1
A
xB 2
Should we relax the Pareto principle ? (2)





Three variants of the Pareto principle
Weak Pareto: if x Pi y for all i  N, then x P y
Pareto indifference: if x Ii y for all i  N, then x I y
Strong Pareto: if x Ri y for all i for all i  N and x Ph y
for at least one individual h, then x P y
A famous critique of the Pareto-principle: When
combined with unrestricted domain, it may hurt
widely accepted liberal values (Sen (1970) liberal
paradox).
Sen (1970) liberal paradox (1)



Minimal liberalism: respect for an individual
personal sphere (John Stuart Mills)
For example, x is a social state in which Mary
sleeps on her belly and y is a social state that
is identical to x in every respect other than the
fact that, in y, Mary sleeps on her back
Minimal liberalism would impose, it seems, that
Mary be decisive (dictator) on the ranking of x
and y.
Sen (1970) liberal paradox (2)


Minimal liberalism: There exists two
individuals h and i  N, and four social states w,
x,, y and z such that h is decisive over x and y
and i is decisive over w and z
Sen impossibility theorem: There does not
exist any collective decision function C: D
satisfying unrestricted domain, weak pareto
and minimal liberalism.
Proof of Sen’s impossibility result







One novel: Lady Chatterley’s lover
2 individuals (Prude and Libertin)
4 social states: Everybody reads the book (w), nobody reads
the book (x), Prude only reads it (y), Libertin only reads it (z),
By liberalism, Prude is decisive on x and y (and on w and z) and
Libertin is decisive on x and z (and on w and y)
By unrestricted domain, the profile where Prude prefers x to y
and y to z and where Libertin prefers y to z and z to x is
possible
By minimal liberalism (decisiveness of Prude on x and y), x is
socially better than y and, by Pareto, y is socially better than z.
It follows by transitivity that x is socially better than z even
thought the liberal respect of the decisiveness of Libertin over z
and x would have required z to be socially better than x
Sen liberal paradox




Shows a problem between liberalism and respect of
preferences when the domain is unrestricted
When people are allowed to have any preference
(even for things that are « not of their business »), it is
impossible to respect these preferences (in the Pareto
sense) and the individual’s sovereignty over their
personal sphere
Sen Liberal paradox: attacks the combination of the
Pareto principle and unrestricted domain
Suggests that unrestricted domain may be a strong
assumption.
Relaxing unrestricted domain
for Arrow’s theorem (1)





One possibility: imposing additional structural
assumptions on the set X
For example X could be the set of all allocations of l
goods (l > 1) accross the n individuals (that is X = nl)
In this framework, it would be natural to impose
additional assumptions on individual preferences.
For instance, individuals could be selfish (they care
only about what they get). They could also have
preferences that are convex, continuous, and
monotonic (more of each good is better)
Unfortunately, most domain restrictions of this kind
(economic domains) do not provide escape out of the
nihilism of Arrow’s theorem.
Relaxing unrestricted domain
for Arrow’s theorem (2)






A classical restriction: single peakedness
Suppose there is a universally recognized ordering  of the set X
of alternatives (e.g. the position of policies on a left-right
spectrum)
An individual preference ordering Ri is single-peaked for  if, for
all three states x, y and z such that x  y  z , x Pi z  y Pi z and
z Pi x  y Pi x
A profile <Ri> is single peaked if there exists an ordering  for
which all individual preferences are single-peaked.
Dsp  n the set of all single-single peaked profiles
Theorem (Black 1947) If the number of individuals is odd, and D
= Dsp then there exists a non-dictatorial collective decision
function C: D  satisfying Pareto and binary independence of
irrelevant alternatives. The majority rule is one such collective
decision function.
Single peaked preference ?
Single-peaked
left
Ségolène
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Ségolène
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Ségolène
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Ségolène
François
Nicolas
right
Single peaked preference ?
Not Single-peaked
left
Ségolène
François
Nicolas
right
Single peaked preference ?
Not Single-peaked
left
Ségolène
François
Nicolas
right
Comments on Black theorem



Widely used in public economics
In any set of social states where each
individual has a most preferred state, the
social state that beats any other by a majority
of vote (Condorcet winner) is the most
preferred alternative of the individual whose
peak is the median of all individuals peaks
(median voter theorem)
Notice the odd restriction on the number of
individuals
Even with single-peaked preferences, the
majority rule is not transitive if the number
of individuals is even
Individual 1
Ségolène
François
Nicolas
Individual 2
Individual 3
Individual 4
François
Ségolène
Nicolas
Nicolas
François
Ségolène
Nicolas
François
Ségolène
Preferences are single peaked (on the left-right axe)
Nicolas is weakly preferred, socially, to François
François is strictly preferred to Ségolène
but Nicolas is not strictly preferred to Ségolène
Domain restrictions that garantees
transitivity of majority voting




Sen and Pattanaik (1969) Extremal Restriction
condition
A profile of preferences <Ri> satisfies the Extremal
Restriction condition if and only if, for all social states x,
y and z, the existence of an individual i for which x Pi y
Pi z must imply, for all individuals h for which z Ph x,
that z Ph y Ph x.
Theorem (Sen and Pattanaik (1969). A profile of
preferences <Ri> satisfies the extremal restriction
condition if and only if the majority rule defined on this
profile is transitive.
See W. Gaertner « Domain Conditions in Social Choice
Theory », Cambridge University Press, 2001.
Relaxing « Binary independence
of irrelevant alternatives »




Justification of this axiom: information
parcimoniousness
De Borda rule violates it
In economic domains, there are various
social orderings who violates this axiom
but satisfy all the other Arrow’s axioms
An example: Aggregate consumer’s
surplus
Aggregate consumer’s surplus ?





X = +nl (set of all allocations of consumption
bundles)
xi  +l individual i’s bundle in x
Ri, a continuous, convex, monotonic and
selfish ordering on +nl
Selfishness means that for all i  N, w, x, y
and z  in +nl such that wi = xi and yi = zi,
x Ri y  w Ri z
Selfishness means that we can view individual
preferences as being only defined on +l
Aggregate consumer’s surplus ?








Individuals live in a perfectly competitive environment
Individual i faces prices p =(p1,….,pl) and wealth wi.
B(p,wi)={x  +l p.x  wi } (Budget set)
Individual ordering Ri on +l induces the dual (indirect)
ordering RDi of all prices/wealth configurations (p,w) 
+l+1 as follows: (p,w) RDi (p’,w’)  for all x’B(p’,w’),
there exists x  B(p,wi) for which x Ri x’.
Ui: +l , a numerical representation of Ri (Ui(x) 
Ui(y)  x Ri y) (such a numerical representation exists
by Debreu (1954) theorem; it is unique up to a
monotonic transform)
Vi: +l+1  a numerical representation of RDi
Vi(p,wi) = « the maximal utility achieved by i when
facing prices p  +l and having a wealth wi »
Problem of applied cost-benefit analysis: ranking
various prices and wealth configurations
Aggregate consumer’s surplus ?


A money-metric representation of individual
preferences
For every prices configuration p  +l and utility
level u, define E(p,u) by:
l
E ( p, u)  min p j x j subject to U ( x1 ,...xl )  u
x1 ,...xl
j 1
E(p,u) associates, to every utility level u, the minimal amount
of money required at prices p, to achieve that utility level.
This (expenditure) function is increasing in utility (given prices).
It provides therefore a numerical representation (in money units)
of individual preferences.
Aggregate consumer’s surplus ?
Direct money metric:
 ( p, x )  E ( p, u( x ))
Gives the amount of money needed at prices p to be as
well-off as with bundle x
Indirect money metric:
 ( p, q, w )  E ( p,V (q, w ))
Gives the amount of money needed at prices p to achieve the
level of satisfaction associated to prices q and wealth w .
money metric utility functions depend upon reference
prices
Aggregate consumer’s surplus ?
These money metric utilities are connected to observable
demand behavior
( x1M ( p, w ),...,xlM ( p, w ))  arg max u( x )
xB ( p , w )
Marshallian (ordinary) demand functions
( x1H ( p, u),...,x1H ( p, u))  arg min
( x1 ,...,xn )
l
p x
j 1
j
j
s.t . U ( x1 ,..., xl )  u
Hicksian (compensated) demand functions (depends upon
unobservable utility level)
Aggregate consumer’s surplus ?
Six important identities (valid for every p  +l, w  + and u  ):
E ( p,V ( p, w ))  w
(1)
V ( p, E ( p, u))  u
(2)
H
xM
(
p
,
E
(
p
,
u
))

x
j
j ( p, u)
(3)
x Hj ( p,V ( p, w))  x M
j ( p, w)
(4)

V ( p, w ) / p j
 xM
j ( p, w )
(5)
Roy’s identity
E ( p, u)
 xH
j ( p, u )
p j
(6)
Sheppard’s Lemma
V ( p, w ) / w
Aggregate consumer’s surplus ?
( p, w1 ,...,w n ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
   i ( p,Vi ( p, w i ))    i ( p,Vi ( p' , w'i ))
n
  (  i ( p,Vi ( p, w i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
  (  i ( p,Vi ( p, w i ))   i ( p' ,Vi ( p' , w'i ))   i ( p' ,Vi ( p' , w'i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
  (w i  w'i   i ( p' ,Vi ( p' , w'i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (w i  w'i    x Hi
j ( p1 ,..., p j 1 , q j , p' j  1 ,..., p'l ,u'i )dq j )  0
i 1
j 1 p ' j

surplus Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
identity (1)
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
Recurrent application of Sheppard’s lemma
pj
  (wi  w'i   x ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j
Hi
j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,...,wn ) R (q, w'1 ,...,w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p,Vi ( p, wi ))  i ( p, Vi ( p' , w'i )))  0
i 1
n
  ( i ( p,Vi ( p, wi ))  i ( p' ,Vi ( p' , w'i ))  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hij ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p ' j



surplus
Hicksien du consommate ur
A one good, one price illustration
price
pj’
a
Hicksian demand
b
pj
Surplus
= area
pj’abpj
quantity
ni=1xHij(p1,…,p’j-1,pj’,pj+1,…,pl,ui’) ni=1xHij(p1,…,pj-1,pj,pj+1,…,pl,ui’)
Aggregate consumer’s surplus ?



Usually done with Marshallian demand
(rather than Hicksian demand)
Marshallian surplus is not a correct measure
of welfare change for one consumer but is
an approximation of two correct measures of
welfare change: Hicksian surplus at prices p
and Hicskian surplus at prices p’ (Willig
(1976), AER, « consumer’s surplus without
apology).
Widely used in applied welfare economics
Is the ranking of social states based on the
sum of money metric a collective decision
rule?





It violates slightly the unrestricted domain condition
(because it is defined on all selfish, convex,
monotonic and continuous profile of individual
orderings on +nl but not on all profiles of orderings
(unimportant violation)).
It satisfies non-dictatorship and Pareto
It obviously satisfies collective rationality if the
reference prices used to evaluate money metric do
not change
It violates binary independence of irrelevant
alternatives (prove it).
Ethical justification for Aggregate consumer’s
surplus is unclear
Conclusion: Escape out of Arrow’s
impossibility theorem




Restricting the domain (depends upon the context)
Relaxing Binary independence of irrelevant alternatives (a
lot of work can be done in this area if one can justify
ethically the use of specific methods for numerically
representing preferences and specific aggregation o f this
(ex: de Borda, axiomatized by P. Young, JET 1974,
Aggregate consumer’s surplus, no axiomatization, etc.)
Interpreting individual preferences as reflecting individual
welfare and accepting to measure welfare « precisely »
(welfarist escape)
Putting non-welfare information in the setting (non-welfarist
escape)