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Welfarist escape out of Arrow’s theorem

Delve into the nuances of individual preferences, happiness, and well-being in economic theory. Learn about the challenges of measuring and comparing happiness across individuals and the limitations of ordinal measurements in assessing well-being.

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Welfarist escape out of Arrow’s theorem

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  1. Welfarist escape out of Arrow’s theorem

  2. 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 ».

  3. 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 ?

  4. 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 • Uiis such that, for every x and y in X, Ui(x)  Ui(y)  x Ri y • Ordinal measure of happiness

  5. 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) = eU(z1,z2) = elnz1elnz2 = z1z2 • H(z1,z2) = -1/G(z1,z2) = -1/(z1z2) all represent numerically Ri

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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:  .

  11. 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

  12. 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))

  13. 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.

  14. 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).

  15. 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.

  16. 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 ?

  17. 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 nthe 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 ?

  18. 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

  19. Contrasting utilitarianism and max-min u2 utility possibility set u1 = u2 u1

  20. Contrasting utilitarianism and max-min u2 u’ -1 u1 = u2 Utilitarian optimum u u1 u u’

  21. Contrasting utilitarianism and max-min u2 u’ -1 u1 = u2 Rawlsian optimum u u1 u u’

  22. Contrasting utilitarianism and max-min u2 Utilitarian optimum u1 = u2 Rawlsian optimum Best feasible egalitarian outcome u1

  23. 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

  24. 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.

  25. Mean-of-order r functional u2 r=0 u1 = u2 r=1 u1

  26. Mean-of-order r functional u2 r=0 u1 = u2 r=1 u1

  27. Mean-of-order r functional u2 r =- r=0 u1 = u2 r=1 u1

  28. Mean-of-order r functional u2 r =- r=0 u1 = u2 r=1 u1

  29. Mean-of-order r functional u2 r =- r=0 u1 = u2 r=1 r=+ u1

  30. Mean-of-order r functional u2 u1 = u2 Max-max indifference curve r=+ u1

  31. 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))

  32. 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 ?

  33. 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)

  34. 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)

  35. 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)

  36. Information used by a social welfare functional • 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 ?

  37. Information used by a social welfare functional • 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 ?

  38. 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)

  39. 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 Nimplies 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)  DUand 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 xR y  x R’ y where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n))

  40. Welfarist lemma: If a social welfare functional W satisfies 2, 3, 4b and 5, there exists an ordering R* on nsuch that, for all profiles(U1,…,Un)  DU, xR y (U1(x),…,Un(x)) R* (U1(y),…,Un(y)) (where R = W(U1,…,Un))

  41. 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.

  42. Proof of the lemma • Sublemma (neutrality): If W is a social welfare functional satisfying 2, 3, 4a and 5, then, for all social states x, y, x’ and y’  X and all profiles (U1,…,Un) and (U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i  N implies that x R y  x’ R’ y’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n)) • Proof.Case 1: {x,y}  {x’,y’} = .

  43. Proof of the lemma • Sublemma (neutrality): If W is a social welfare functional satisfying 2, 3, 4a and 5, then, for all social states x, y, x’ and y’  X and all profiles (U1,…,Un) and (U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i  N implies that x R y  x’ R’ y’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n)) • Proof.Case 1: {x,y}  {x’,y’} = . By unrestricted domain, one can find a profile of utility functions (U’’1,…,U’’n)  DU such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) = Ui(y) for all i  N.

  44. Proof of the lemma • Sublemma (neutrality): If W is a social welfare functional satisfying 2, 3, 4a and 5, then, for all social states x, y, x’ and y’  X and all profiles (U1,…,Un) and (U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i  N implies that x R y  x’ R’ y’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n)) • Proof.Case 1: {x,y}  {x’,y’} = . By unrestricted domain, one can find a profile of utility functions (U’’1,…,U’’n)  DU such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) = Ui(y) for all i  N. By the independence axiom, xR y  xR’’ y.

  45. Proof of the lemma • Sublemma (neutrality): If W is a social welfare functional satisfying 2, 3, 4a and 5, then, for all social states x, y, x’ and y’  X and all profiles (U1,…,Un) and (U’1,…,U’n)  DU , Ui(x) = U’i(x’) and Ui(y) = U’i(y’) for all i  N implies that x R y  x’ R’ y’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n)) • Proof.Case 1: {x,y}  {x’,y’} = . By unrestricted domain, one can find a profile of utility functions (U’’1,…,U’’n)  DU such that Ui(x) = U’’i(x’) = U’’i(x) and U’’i(y) = U’’i(y’) = Ui(y) for all i  N. By the independence axiom, xR y  xR’’ y. By Pareto indifference, x’ R’’ y’ and by, independence again, x’ R’ y’.

  46. 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).

  47. 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).

  48. 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)

  49. 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)  zR’’’’ y (Pareto-indifference and transitivity)

  50. 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)  zR’’’’ y (Pareto-indifference and transitivity)  z R’’’ y (independence)

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