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 • Ri a preference ordering of individual i on X (with asymmetric and symmetric factors Pi and Ii) • x Ri y means « individual i weakly prefers state x to state y » • Pi = « strict preference », Ii = « indifference » • An ordering is a reflexive, complete and transitive binary relation • The interpretation given to preferences is unclear in Arrow’s work (influenced by economic theory)

Can we define the general interest on the basis of individuals interests ? • <Ri > = (R1 ,…, Rn) a profile of individual preferences •  the set of all binary relations on X •   , the set of all orderings on X • D  n, the set of all admissible profiles • Arrow’s problem: to find a « collective decision rule » C: D   that associates to every profile <Ri > of individual preferences a binary relation R = C(<Ri >) • x R y means that the general interest is (weakly) better served with x than with y when individual preferences are (<Ri >)

Examples of collective decision rules • 1: Dictatorship of individual h: x R y if and only xRhy (not very attractive) • 2: ranking of social states according to an exogenous code (say the Charia). Assume that the exogenous code ranks any pair of social alternatives as per the ordering  (x  y means that x (women can not drive a car) is weakly preferable to y (women drive a car). Then C(<Ri >) =  for all <Ri >  Dis 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 xRiy 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:xRiy}  #{i N :yRix}. Widely used, but does not always lead to a transitive ranking of social states (Condorcet paradox).

the Condorcet paradox

the Condorcet paradox Individual 3 Individual2 Individual1

the Condorcet paradox Individual 3 Individual2 Individual1 Ségolène Nicolas François

the Condorcet paradox Individual 3 Individual2 Individual1 Nicolas François Ségolène Ségolène Nicolas François

the Condorcet paradox Individual 3 Individual2 Individual1 François Ségolène Nicolas Nicolas François Ségolène Ségolène Nicolas François

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

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

the Condorcet paradox Individual 3 Individual2 Individual1 François Ségolène Nicolas Nicolas François Ségolène Ségolène Nicolas François 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 3 Individual2 Individual1 François Ségolène Nicolas Nicolas François Ségolène Ségolène Nicolas François 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 3 Individual2 Individual1 François Ségolène Nicolas Nicolas François Ségolène Ségolène Nicolas François 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 3 Individual2 Individual1 François Ségolène Nicolas Jean-Marie Nicolas François Jean-Marie Ségolène Ségolène Nicolas Jean-Marie François

Borda rule Individual 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1

Borda rule Individual 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Sum of scores Ségolène = 8

Borda rule Individual 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Sum of scores Ségolène = 8 Sum of scores Nicolas = 9

Borda rule Individual 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 1 Sum of scores Ségolène = 8 Sum of scores Nicolas = 9 Sum of scores François = 8

Borda rule Individual 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Jean-Marie 2 Ségolène 1 Ségolène 4 Nicolas 3 Jean-Marie 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 Ségolène 4 Jean-Marie 3 Nicolas 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 Ségolène 4 Jean-Marie 3 Nicolas 2 François 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 3 Individual2 Individual1 François 4 Ségolène 3 Nicolas 2 Jean-Marie 1 Nicolas 4 François 3 Ségolène 2 Jean-Marie 1 Ségolène 4 Jean-Marie 3 Nicolas 2 François 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 <Ri>, x Ph y implies x P y (where R = C(<Ri>) • 2) Collective rationality. The social ranking should always be an ordering (that is, the image of C should be ) (violated by the unanimity and the majority rule; violated also by the Lorenz domination criterion if X is the set of all income distributions) • 3) Unrestricted domain.D =n (all logically conceivable preferences are a priori possible)

Five desirable properties on the collective decision rule • 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.

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 Pi y for all i G and y Ph x for all h  N\G implies x P y (where R = C(<Ri>)) • Analogously, say that group G is decisive on x and y if x Pi y for all i G implies x P y • Clearly, decisiveness on x and y implies 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 (groupcontraction) lemma, we are going to use the first lemma to show that if a group G containing at least two individuals is decisive on a pair of social states, then it contains a proper subgroup of individuals that is decisive on that pair. • These two lemmas prove the theorem because, by the Pareto principle, we know that the whole society is decisive on every pair of social states.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} .

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G. Since G is semi-decisive on x and y, x P y.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G. Since G is semi-decisive on x and y, x P y. Since C satisfies the Pareto principle, y P b and, since the social ranking is transitive, x P b.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G. Since G is semi-decisive on x and y, x P y. Since C satisfies the Pareto principle, y P b and, since the social ranking is transitive, x P b. Now, by binary independence of irrelevant alternatives, the social ranking of x (= a) and b only depends upon the individual preferences over a and b.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G. Since G is semi-decisive on x and y, x P y. Since C satisfies the Pareto principle, y P b and, since the social ranking is transitive, x P b. Now, by binary independence of irrelevant alternatives, the social ranking of x (= a) and b only depends upon the individual preferences over a and b. Yet only the preferences over a and b of the members of G have been specified here.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 1:a = x and b{x,y} . Since C is defined for every preference profile, consider a profile such that x Pi y Pi b for all i in G and y Ph x and y Ph b for all h  N\G. Since G is semi-decisive on x and y, x P y. Since C satisfies the Pareto principle, y P b and, since the social ranking is transitive, x P b. Now, by binary independence of irrelevant alternatives, the social ranking of x (= a) and b only depends upon the individual preferences over a and b. Yet only the preferences over a and b of the members of G have been specified here. Hence G is decisive on a and b.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 2:b = y and a{x,y} .

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 2:b = y and a{x,y} . As before, using unrestricted domain, consider a profile such that a Pi x Pi y for all i in G and y Ph x and a Ph x for all h  N\G.

Field expansion lemma • Lemma: If C is a collective decision function satisfying axioms 2-5, then, if a group G is semi-decisive on two social states x and y, G is in fact decisive on all social states a and b. • Proof: We consider several cases. 2:b = y and a{x,y} . As before, using unrestricted domain, consider a profile such that a Pi x Pi y for all i in G and y Ph x and a Ph x for all h  N\G. Since G is 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 Pi x Pi y for all i in G and y Ph x and a Ph x for all h  N\G. Since G is 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 Pi x Pi y for all i in G and y Ph x and a Ph x for all h  N\G. Since G is 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.

Arrow’s theorem and the problem of social choice