welfarist escape out of arrow’s theorem. what are the individual preferences standing for ? what...

Welfarist escape out of Arrow’s theorem

What are the individual preferences standing for ?

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 ? 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) 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) 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 Ri

Can we measure happiness ? (3) 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) 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.




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.




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) Stronger measurement of well-being: cardinal. Suppose U: X and G: X are two measures of well-being. 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)

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) 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) 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) 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 ?

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 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

u1

u1 = u2

utility possibility set

Contrasting utilitarianism and max-minu2

u1

u1 = u2

u

u

u’

u’-1

Utilitarian optimum


u1

u1 = u2

u

u

u’

u’-1

Rawlsian optimum


u1

u1 = u2

Rawlsian optimum

Utilitarian optimum

Best feasible egalitarian outcome

Contrasting utilitarianism and Max-min

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 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 functionalu2

u1

u1 = u2

r=1

r=0


u1

u1 = u2

r=1

r=0

r =-


u1

u1 = u2

r=1

r=0

r =-

r=+


u1

u1 = u2

r=+Max-max indifferencecurve

Extension of Max-min

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

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)

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 well-being 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 well-being 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 well-being 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)

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 well-being 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)


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 ?


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

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.


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. 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.




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.




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. Case 2: (x’,y’) = (y,x).



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. 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. 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. 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. 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. 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

u1

u2

u

Illustration

u1

u2

u

A

u

Illustration

u1

u2

u

A

B

u

Illustration

u1

u2

u

A

B

u

C

Illustration

u1

u2

u

A

B

u

C

D

Illustration

u1

u2

u

A

B

u

C

D

Better thanu by Pareto

Illustration

u1

u2

u

A

B

u

C

D


Worse thanu by Pareto

Illustration

u1

u2

u

A

B

u

D



By NC, all pointsin C are ranked in thesame way vis-à-vis u

Illustration

u1

u2

u

A

B

u

D



a

b

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

u1

u2

u

A

B

u

D



a

b

Illustration

u1

u2

u

A

B

u



a

b

all points hereare also rankedin the same way vis-à-vis u

Illustration

u1

u2

u

A

B

u



a

b

all points hereare also rankedin the same way vis-à-vis u

by Pareto, a and bcan not be indifferent to u(and to themselves)by transitivity)

Illustration

u1

u2

u

A

B

u

C

D

by NC, the(strict) ranking of region Cvis-à-vis u mustbe the oppositeof the (strict) ranking of D vis-à-vis u

Illustration

u1

u2

u

A

B

u

C

D

Illustration

u1

u2

u

A

B

u

C

Dd

c

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 = (u2

i-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

u1

u2

u

A

B

u

C

D



Illustration

u1

u2

u

A

B

u

Worse

Better



Illustration

u1

u2

u

A

B

u

Worse

Better

Individual 1is the dictator

Illustration

u1

u2

u

A

B

u

C

D



Illustration

u1

u2

u

A

B

u

Better

Worse



Illustration

u1

u2

u

A

B

u

Better

Worse

Individual 2Is the dictator

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

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

III

I

IIIV

V

VI VII

VIII

IX

X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

III

I

IIIV

V

VI VII

VIII

IX

X

u’ and u’(.) areindifferent byanonymity

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

III

better than u’ (and than u’(.) by Pareto

IIIV

V

VI VII

VIII

IX

X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

III


IV

V

VI VII

VIII

IX

X


u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1


IV

V

VI VII

VIII

IX

X



u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

V

VI VII

VIII

IX

X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

VIII

IX

X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

all pointsin this zoneare rankedin the sameway vis-à-visu’ (and u’(.) by transitivity)

IX

X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

aIX

X

b

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

aIX

X

b

b1

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

aIX

X

b

b1

b2

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

aIX

X

b

b1

b2

a1

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse

aIX

X

b

b1

b2

a1

a2

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.

u1

u2

u’1

u’2

u’2

u’1

u2 = u1

aIX

b

b1

b2

a1

a2

u1

u2

u’1

u’2

u’2

u’1

u2 = u1

aIX

bb2

a1

b1 = a2

b1 = a2

u1

u2

u’1

u’2

u’2

u’1

u2 = u1

aIX

bb2

a1

b1 = a2

b1 = a2

g(x)

Remark: we use the fact that we are free to use any increasing function

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse


X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse


X

by anonymity same is truefor this zone

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse X

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worse X

by Pareto, these points can not be indifferent tou’ (and by transitivity, to themselves)

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worseranking of those points vis-à vis u’ is the opposite than those in the yellow zone


u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

IV

worseranking of those points vis-à vis u’ is the opposite than those in the yellow zone


Same is true for those points

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

worse

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

worse

Hence we are left with two possibilities

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

worse

Max is the dictator

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1

better

worse

Min is the dictator

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 2-5, 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 two-individuals 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 ?

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1Continuity ?

better

worse

better

worse

We go continuously from the better…

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1Continuity ?

better

worse

better

worse

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1Continuity ?

better

worse

better

worse

to theworse

u1

u2

u’(.)

u’1

u’2

u’2

u’1

= u’(1)

= u’(2)

u’

u2 = u1Continuity ?

better

worse

better

worse

Without encounteringindifference

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 well-being if and only if it is utilitarian.

Proof: Diagrammatic again

u1

u2

a2

b1

u2 = u1

IX

b

c

b2

a1

c

a

b

-1

2c

u1

u2

a2

b1

u2 = u1

IX

b

c

b2

a1

c

a

b

-1

2c

Indifferent byanonymity

u1

u2

a2

b1

u2 = u1

IX

b

c

b2

a1

c

a

b

-1

2c

a must be rankedvis-à-vis c the same way than c is vis-à-vis b

u1

u2

a2

b1

u2 = u1

IX

b

c

b2

a1

c

a

b

-1

2c

a must be rankedvis-à-vis c the same way than c is vis-à-vis b(because of CUC)

u1

u2

a2

b1

u2 = u1

IX

b

c = a1 + c-a1

b2

a1

c = a2

a

b

-1

2c


+c–a2

u1

u2

a2

b1 = 2c-a1

u2 = u1

IX

b

c = a1 + c-a1

b2= 2c

a1

c = a2

a

b

-1

2c


+c–a2

–a2

u1

u2

a2

b1 = 2c-a1

u2 = u1

IX

b

c = a1 + c-a1

b2= 2c

a1

c = a2

a

b

-1

2c

The only possibilitycompatible with indifferencebetween a and b isindifference between c anda, or b.+c–a2

–a2

u1

u2

a2

b1 = 2c-a1

u2 = u1

IX

b

c = a1 + c-a1

b2= 2c

a1

c = a2

a

b

-1

2c

Hence all utility vectorson the isoutility line are socially indifferent

+c–a2

–a2

u1

u2

a2

b1 = 2c-a1

u2 = u1

IX

b

c = a1 + c-a1

b2= 2c

a1

c = a2

a

b

-1

2c

By Pareto, all utilityvectors on the northeast of the isoutility lineare better than any pointon the line+c–a2

–a2

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 iNiUi(x) iNiUi(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 Von-Neuman 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 Pareto-indifference, 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 well-

being 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): Pigou-

Dalton principle of equity A social welfare functional W satisfies the Pigou-

Dalton 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)

u

g(v)

v

g(u)

aIX

b

g(u) + (1-)g(v)

u+(1-)v

g(x)

g(u+(1-)v)

Concavity?

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.


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




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)




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




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:




Proof:. Uh(x) = Uh(z) for all h i, j, and Uj(y) = v < Uj(x) = (v +u)/2 = Uj(x) < u = Uj(y).




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)




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.




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


n satisfies Pareto-indifference, 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 ?


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 = nn

for all u X, n(u) is the dimension of u (number of people)

How should we compare these vectors ?


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))

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


Classical utilitarianism : Generates the repugnant conclusion (Parfitt, reason and persons, 1984). For any positive level of well-being , 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.

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