Arrow’s theorem and the problem of social choice.

Arrow’s theorem and the problem of social choice

Can we define the general interest on the basis of individuals interests ? 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 R i a preference ordering of individual i on X (with asymmetric and symmetric factors P i and I i ) x R i y means « individual i weakly prefers state x to state y » P i = « strict preference », I i = « 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 ? = (R 1,…, R n ) 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 of individual preferences a binary relation R = C( ) x R y means that the general interest is (weakly) better served with x than with y when individual preferences are ( )

Examples of collective decision rules 1: Dictatorship of individual h: x R y if and only x R h 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( ) =  for all  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 3: Unanimity rule (Pareto criterion): x R y if and only if x R i 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 R i y}  #{i  N :y R i x}. Widely used, but does not always lead to a transitive ranking of social states (Condorcet paradox).

the Condorcet paradox

Individual 1 Individual 2 Individual 3

the Condorcet paradox Individual 1 Individual 2 Individual 3 Ségolène Nicolas François

the Condorcet paradox Individual 1 Individual 2 Individual 3 Ségolène Nicolas François Nicolas François Ségolène

the Condorcet paradox Individual 1 Individual 2 Individual 3 Ségolène Nicolas François Nicolas François Ségolène François Ségolène Nicolas

the Condorcet paradox Individual 1 Individual 2 Individual 3 Ségolène Nicolas François Nicolas François Ségolène François Ségolène Nicolas A majority (1 and 3) prefers Ségolène to Nicolas

the Condorcet paradox Individual 1 Individual 2 Individual 3 Ségolène Nicolas François Nicolas François Ségolène 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 Individual 3 Ségolène Nicolas François Nicolas François Ségolène 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 Individual 3 Ségolène Nicolas François Nicolas François Ségolène 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 Individual 3 Ségolène Nicolas François Nicolas François Ségolène 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 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 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1

Borda rule Individual 1 Individual 2 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Sum of scores Ségolène = 8

Borda rule Individual 1 Individual 2 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Sum of scores Ségolène = 8 Sum of scores Nicolas = 9

Borda rule Individual 1 Individual 2 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Sum of scores Ségolène = 8 Sum of scores Nicolas = 9 Sum of scores François = 8

Borda rule Individual 1 Individual 2 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Sum of scores Ségolène = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Marie = 5

Borda rule Individual 1 Individual 2 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 Individual 3 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 Individual 3 Ségolène 4 Jean-Marie 3 Nicolas 2 François 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 Individual 3 Ségolène 4 Jean-Marie 3 Nicolas 2 François 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 Individual 3 Ségolène 4 Jean-Marie 3 Nicolas 2 François 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 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 ? 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 1) Non-dictatorship. There exists no individual h in N such that, for all social states x and y, for all profiles, x P h y implies x P y (where R = C( ) 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 4) Weak Pareto principle. For all social states x and y, for all profiles  D, x P i y for all i  N should imply x P y (where R = C( ) (violated by the collective decision rule coming from an exogenous tradition code) 5) Binary independance from irrelevant alternatives. For every two profiles and  D and every two social states x and y such that x R i y  x R’ i y for all i, one must have x R y  x R’ y where R = C( ) and R’ = C( ). 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) 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 P i y for all i  G and y P h x for all h  N\G implies x P y (where R = C( )) Analogously, say that group G is decisive on x and y if x P i y for all i  G implies x P y Clearly, decisiveness on x and y implies semi- decisiveness 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 P i y P i b for all i in G and y P h x and y P h 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 P i y P i b for all i in G and y P h x and y P h 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 P i y P i b for all i in G and y P h x and y P h 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 P i y P i b for all i in G and y P h x and y P h 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 P i y P i b for all i in G and y P h x and y P h 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 P i y P i b for all i in G and y P h x and y P h 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 P i x P i y for all i in G and y P h x and a P h 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 P i x P i y for all i in G and y P h x and a P h x for all h  N\G. Since G is semi- decisive 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 P i x P i y for all i in G and y P h x and a P h x for all h  N\G. Since G is semi- decisive 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 P i x P i y for all i in G and y P h x and a P h x for all h  N\G. Since G is semi- decisive 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 P i x P i y for all i in G and y P h x and a P h x for all h  N\G. Since G is semi- decisive 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 P i x P i y for all i in G and y P h x and a P h x for all h  N\G. Since G is semi- decisive 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 quasi- decisive 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 quasi- decisive on x (= b) and a. By case 4), if a group is quasi- decisive 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 quasi- decisive 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 quasi- decisive on b and a (=y). By case 4), if a group is quasi- decisive 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, quasi- decisive 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, quasi- decisive on x and b. But by case 2), this implies that G is decisive on a and b.

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 G 1 and G 2 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 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 G 2 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 G 1 and G 2 and let x, y and z be 3 social states. By unrestricted domain, consider a profile where x P h y P h z for all h in G 1, y P i z P i x for all i in G 2 and z P j x P j 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 G 1 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 G 2 is quasi-decisive on y and x. By the Field expansion lemma, it follows that either G 1 or G 2 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 A B xB2xB2 xA1xA1 xA2xA2 xB1xB1 x y z 22 11

A B xB2xB2 xA1xA1 xA2xA2 xB1xB1 x y z x is efficient z is not efficient

Illustration: An Edgeworth Box A B xB2xB2 xA1xA1 xA2xA2 xB1xB1 x y z x is efficient z is not efficient x is not socially better than z as per the Pareto principle

Illustration: An Edgeworth Box A B xB2xB2 xA1xA1 xA2xA2 xB1xB1 x y z y is better than z as per the Pareto principle

Should we relax the Pareto principle ? (2) Three variants of the Pareto principle Weak Pareto: if x P i y for all i  N, then x P y Pareto indifference: if x I i y for all i  N, then x I y Strong Pareto: if x R i y for all i for all i  N and x P h 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 R i is single-peaked for  if, for all three states x, y and z such that x  y  z, x P i z  y P i z and z P i x  y P i x A profile is single peaked if there exists an ordering  for which all individual preferences are single-peaked. D sp   n the set of all single-single peaked profiles Theorem (Black 1947) If the number of individuals is odd, and D = D sp 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 ? leftright Ségolène François Nicolas Single-peaked

Single peaked preference ? leftright Ségolène François Nicolas Not Single-peaked

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 1Individual 2 Individual 3 Ségolène François Nicolas François Ségolène Nicolas François Ségolène Individual 4 Nicolas François Ségolène Nicolas is weakly preferred, socially, to François François is strictly preferred to Ségolène but Nicolas is not strictly preferred to Ségolène Preferences are single peaked (on the left-right axe)

Domain restrictions that garantees transitivity of majority voting Sen and Pattanaik (1969) Extremal Restriction condition A profile of preferences 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 P i y P i z must imply, for all individuals h for which z P h x, that z P h y P h x. Theorem (Sen and Pattanaik (1969). A profile of preferences 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) x i   + l individual i’s bundle in x R i, 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 w i = x i and y i = z i, x R i y  w R i 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 =(p 1,….,p l ) and wealth w i. B(p,w i )={x   + l p.x  w i } (Budget set) Individual ordering R i on  + l induces the dual (indirect) ordering R D i of all prices/wealth configurations (p,w)   + l+1 as follows: (p,w) R D i (p’,w’)  for all x’  B(p’,w’), there exists x  B(p,w i ) for which x R i x’. U i :  + l  , a numerical representation of R i (U i (x)  U i (y)  x R i y) (such a numerical representation exists by Debreu (1954) theorem; it is unique up to a monotonic transform) V i :  + l+1   a numerical representation of R D i V i (p,w i ) = « the maximal utility achieved by i when facing prices p   + l and having a wealth w i » 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: 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 ? Gives the amount of money needed at prices p to achieve the level of satisfaction associated to prices q and wealth w. Direct money metric: Gives the amount of money needed at prices p to be as well-off as with bundle x Indirect money metric: money metric utility functions depend upon reference prices

Aggregate consumer’s surplus ? Hicksian (compensated) demand functions (depends upon unobservable utility level) These money metric utilities are connected to observable demand behavior Marshallian (ordinary) demand functions

Aggregate consumer’s surplus ? Six important identities (valid for every p   + l, w   + and u   ): (1) (2) (3) (4) (5) Roy’s identity (6) Sheppard’s Lemma

Aggregate consumer’s surplus ?

identity (1)

Recurrent application of Sheppard’s lemma

A one good, one price illustration price quantity Hicksian demand pj’pj’ pjpj Surplus = area p j ’abp j  n i=1 x Hi j (p 1,…,p’ j-1,p j ’,p j+1, …,p l,u i ’)  n i=1 x Hi j (p 1,…,p j-1,p j,p j+1,…,p l,u i ’) a b

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)

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Arrow’s theorem and the problem of social choice.

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