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Normative foundations of public Intervention

Normative foundations of public Intervention. General normative evaluation. 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

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Normative foundations of public Intervention

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  1. Normative foundations of public Intervention

  2. General normative evaluation • 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 • Example 1: X= +n (the set of all income distributions) • Example 2: X = +nm (the set of all allocations of m goods (public and private) between the n-individuals. • Ria preference ordering of individual i on X (with asymmetric and symmetric factors Piand Ii). • Ordering: a reflexive, complete and transitive binary relation. • x Ri y means « individual i weakly prefers state x to state y » • Pi= « strict preference », Ii= « indifference » • Basic question (Arrow (1950): how can we compare the various elements of X on the basis of their « social goodness ? »

  3. General normative evaluation • Arrow’s formulation of the problem. • <Ri > = (R1 ,…, Rn) a profile of preferences. •  the set of all binary relations on X •   , the set of all orderings on X • D n, the set of all admissible profiles • General problem (K. Arrow 1950): 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 « x is at least as good as y when individuals’ preferences are (<Ri >)

  4. Examples of normative criteria ? • 1: Dictatorship of individual h: x R y if and only x Rh y (not very attractive) • 2: ranking 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 profiles(<Ri>). 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.

  5. Examples of collective decision rules • 3: Unanimity rule (Pareto criterion):x R y if and only if xRiy for all 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: xRiy}  #{i N:yRix}. Widely used, but does not always lead to a transitive ranking of social states (Condorcet paradox).

  6. the Condorcet paradox

  7. the Condorcet paradox Individual 3 Individual2 Individual1

  8. the Condorcet paradox Individual 3 Individual2 Individual1 Marine Nicolas François

  9. the Condorcet paradox Individual 3 Individual2 Individual1 Nicolas François Marine Marine Nicolas François

  10. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François

  11. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François A majority (1 and 3) prefers Marine to Nicolas

  12. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François

  13. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marine be socially preferred to François

  14. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marinene be socially preferred to François but………….

  15. the Condorcet paradox Individual 3 Individual2 Individual1 François Marine Nicolas Nicolas François Marine Marine Nicolas François A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marine be socially preferred to François but…………. A majority (2 and 3) prefers strictly François to Marine

  16. Example 5: Positional Borda • 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

  17. Borda rule Individual 3 Individual2 Individual1 François Marine Nicolas Jean-Luc Nicolas François Jean-Luc Marine Marine Nicolas Jean-Luc François

  18. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1

  19. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8

  20. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9

  21. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8

  22. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5

  23. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marne 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Nicolas is the best alternative, followed closely by Marine and François. Jean-Luc is the worst alternative

  24. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Problem: The social ranking of François, Nicolas and Marine depends upon the position of the (irrelevant) Jean-Luc

  25. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Marie = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

  26. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Jean-Luc 2 Marine 1 Marine 4 Nicolas 3 Jean-Luc 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

  27. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Marine 2 Jean-Luc 1 Marine 4 Jean-Luc 3 Nicolas 2 François 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

  28. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Marine 2 Jean-Luc 1 Marine 4 Jean-Luc 3 Nicolas 2 François 1 Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

  29. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Marine 2 Jean-Luc 1 Marine 4 Jean-Luc 3 Nicolas 2 François 1 Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

  30. Borda rule Individual 3 Individual2 Individual1 François 4 Marine 3 Nicolas 2 Jean-Luc 1 Nicolas 4 François 3 Marine 2 Jean-Luc 1 Marine 4 Jean-Luc 3 Nicolas 2 François 1 Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 The social ranking of Marine and Nicolas depends upon the individual ranking of Nicolas vs Jean-Luc or Marine vs Jean-Luc

  31. Are there other collective decision rules ? • 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

  32. Five desirable properties on the collective decision rule • 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 (completeness) and the majority rule (transitivity) • 3) Unrestricted domain.D =n (all logically conceivable preferences are a priori possible)

  33. Five desirable properties on the collective decision rule • 4) Weak Pareto principle. For all social states x and y, for all profiles <Ri> D , x Pi y for all i Nshould 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 xRiy x R’i y for all i, one must have xR 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.

  34. Arrow’s theorem: There does not exist any collective decision function C: D   that satisfies axioms 1-5

  35. 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

  36. 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 ?

  37. 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.

  38. Illustration: An Edgeworth Box xA2 B xB1 y 2 z x xA1 A 1 xB2

  39. Illustration: An Edgeworth Box xA2 B xB1 x is efficient z is not efficient y z x xA1 A xB2

  40. Illustration: An Edgeworth Box xA2 B xB1 x is efficient z is not efficient y x is not socially better than z as per the Pareto principle z x xA1 A xB2

  41. Illustration: An Edgeworth Box xA2 B xB1 y is better than z as per the Pareto principle y z x xA1 A xB2

  42. 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 ifor 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).

  43. 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.

  44. 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.

  45. 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 (andon 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

  46. 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.

  47. 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.

  48. 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 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.

  49. Single peaked preference ? Single-peaked left right Nicolas Jean-Luc François

  50. Single peaked preference ? Single-peaked left right Nicolas Jean-Luc François

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