Arrow’s Theory of Social Choice

In this article we will discuss about Arrow’s theory of social choice.

Prof. Arrow, in his monumental work, Social Choice and Individual Values, published in 1951, shows that the task of constructing a social welfare function to reflect the aims and aspirations of a free democratic society is an impossible one. Arrow has proved a general theorem about the impossibility of constructing an ordering for society as a whole which will in some way reflect all the individual orderings of the members who make up the society.

Arrow’s main concern is to consider if a social choice can be satisfactorily derived from individual decisions. The problem is easily solved by dictatorship under which social decisions are made by single individual in small group. In a democratic society, each and every individual will have their idea of social welfare function.

It is therefore difficult to construct a social welfare function which reflects the individual orderings. Therefore Arrow lays down five reasonable conditions which social choices must meet in order to reflect individual’s preferences.

The first condition may be called universality condition. It states that a definite social ordering is derivable from a reasonably wide range of individual orderings. This social ordering must have the properties of connexity and transitivity. By the axiom of connexity, any two alternatives must be related either by preference or indifference.

Thus for any two alternatives X and Y, either X is preferred to Y, on Y is preferred to X or the two are indifferent. By the axiom of transitivity if X is preferred on indifferent to Y and Y is preferred on indifferent to Z, then X must be either preferred on indifferent to Z. These two axioms constitute the foundation of modern choice theory.

The second condition is called Responsiveness condition. It states that social ordering is positively related with the individual orderings. On simply social choices must move in the same direction as individual choices.

The third condition is called by Arrow the Independence of Irrelevant Alternatives. It simply, states that the choice made by a society depends on the orderings of individuals in that environment and not on the orderings of alternatives outside that environment.

Fourth is the non-imposition condition also called the conditions of citizens sovereignty. It required that there should be no external control over a society’s choice. The social welfare function is not to be imposed.

Condition number 5, called, the condition of Non-dictatorship is a part of the condition 4. It permits the construction of social choices by collective methods and not by dictatorial ones. Hence the condition is the social welfare function is not to be dictatorial.

Condition 1 specifies the scope of social welfare function and other four are value judgements. Arrow next considers whether a social ordering can be derived from any set of individual orderings. He demonstrates that impossibility of doing this without violating at-least one of the value judgements as expressed in five conditions. This is his “General Possibility Theorem”.

General Possibility Theorem:

Arrow first considers a simple case where there are only two alternatives and shows that in this case the method of majority decision yields a social welfare function satisfying all the five conditions. But when there are three or more alternatives difficulty emerges and no valued social welfare function can be derived therefore a social welfare function may be either imposed on dictatorial.

Arrows offers three important deductions-consequences 1,2 and 3. The three alternatives are X,Y,Z and there are two individuals. Consequence I states that whenever both individuals prefer X to Y, society will prefer X to Y.

Consequence 2 states that, if in four given choice the will of individual 1 prevails against the opposition of 2, then individual 1 prevails against the opposition of 2, then individual 1’s view will prevail if 2 is indifferent on if he agrees with 1. Consequence III states that, if two individuals have opposing interests, then the society will be indifferent between the two alternatives.

Based on these consequences, the General Possibility Theorem is stated in its simplest form. Let there be two individuals and three alternatives X,Y,A. If individual 1 prefers X to Y and individual 2, Y to X, the society is indifferent between the two.

If individual 1 has X,Y,Z and individual 2 has the ordering Z,X,Y. Since individual one prefers Y to Z and individual 2 prefers Z to Y, the society should be indifferent between the two. For both X is preferred to Y and society also prefers X to Y. By the axiom of transitivity society prefers X to Z.

By since individual 1 prefers X to Z and 2 prefers Z to X we are to concluded that the society is indifferent as between X and Z. But this contradicts the earlier conclusion that X is preferred to Z. it can’t be that for society X is both preferred and also indifferent to Z.