Arrow’s Impossibility Theorem | Economics Optional Notes for UPSC PDF Download

Introduction

Bergson in his social welfare function showed that a social ranking of alternative economic situations can be made only by making interpersonal comparisons of utility that are incorporated in such a function. But the question arises whether it is an imposed function reflecting the tastes of a dictator or the preferences of the society in some democratic way. K.J. Arrow in his Social Choice and Individual Values has demonstrated the impossibility of obtaining the social welfare function even if individual preferences are consistent. He suggests five minimum conditions or criteria which social choices must satisfy in order to reflect preferences of individuals.

They are as follows:

1. Collective Rationality:

• All possible alternatives must be derived from social choices which, in turn, must be based on rationality. The rule for making a social choice can be derived from an ordering of all possible alternatives open to society. This ordering must obey two conditions, of consistency and transitivity.
• Consistency refers to the requirement that individuals have preferences that are fully defined, i.e. every alternative is ranked in relation to every other. Second, social choices must satisfy the condition of transitivity. If an individual prefers X to Y and Y to Z, then he must prefer J to Z. Thus social preferences, like individual preferences, must be completely ordered.

2. Responsiveness to Individual Preferences:

• Social choices must be directly related to individual preferences. It implies that social choices must change in the same direction as individual choices. Individual choices must be derived within the society. But it is not possible to derive such alternatives which affect the socially desirable alternatives.

3. Non-imposition:

• Social choices must not be imposed by customs or from outside the society. It must be derived from individual preferences. For instance, if the majority of individuals do not prefer to B, then the society should not follow it.

4. Non-dictatorship:

• Social choices must not be dictatorial. They must not be imposed by one individual within the society. In other words, social choices must not be based on any single individual’s ordering.

5. Independence of Irrelevant Alternatives:

• Social choices must be independent of irrelevant alterna­tives. In other words, if any one alternative is excluded, it will not affect the ranking of other alternatives. Arrow demonstrates that it is not possible to satisfy all these five conditions and obtain a transitive social choice for each set of individual preferences without violating at least one condition.
• In other words, social choice is inconsistent or undemocratic because no voting system allows these five conditions to be satisfied. This has come to be known as the Arrow Impossibility Theorem. To illustrate Arrow’s general impossibility theorem, suppose there are three individuals A, B and C in a society. They are asked to rank three alternative situations X, Y and Z They vote by writing number 3 for their first choice, 2 for their second choice and 1 for their third choice. Suppose the voting pattern is as shown in Table 1.

• The table shows that each individual has consistent preferences. A prefers X to Y, and Y to Z and hence X to Z. B prefers Y to Z, and Z to X, and hence Y to X. C prefers Z to X, and X to Y and hence Z to X But the majority voting leads to intransitive social patterns.
• Two individuals A and C prefer X to Y. Two individuals A and B prefer Y to Z. Nonetheless, B and C prefer Z to X. Hence the majority prefers X to Y and Y to Z, but it also prefers Z to X. This is illustrated in Figure 5 which shows a multiple-peaked pattern. This explains the paradox of the majority rule which is inconsistent with those of individuals composing the majority.
• Thus Arrow shows that the use of the democratic process of voting leads to a contradictory welfare criterion. “This ‘voting paradox’ explored by Prof. Arrow, comes as a shock to one’s faith in electoral democracy,” according to Prof. Musgrave. “Fortunately the paradox does not imply that majority rule cannot work. Rather, the conclusion is that for majority rule to give non-arbitrary results, the preference structure of individuals must be typically single-peaked.” By single-peaked preference patterns, Musgrave means, where there is an absence of voters with “extremist” preference patterns.

It’s Criticism:

Arrow’s general impossibility theorem has been criticised by Samuelson, Little and other welfare econo­mists on the following grounds:

1. Not related to Social Welfare Function:

• According to Little, Arrow’s negative conclusions have no relevance in welfare economics. His impossibility theorem relates to a decision making process and not to a social welfare function.

2. No Solution of Interpersonal Comparisons:

• Mishan opines that Arrow fails to solve the problem of interpersonal comparison of utility in his search for a satisfactory social welfare function. Rather, his method of majority rule involves interpersonal comparisons. If a majority prefers X to Y, then the majority decision in favour of X means that X is preferred to Y only if the aim is to maximise utility and the choice of one individual has equal utility with that of another.

3. Mathematical Politics:

• Samuelson is of the view that Arrow has proved the impossibility of “a political constitution function” which would be able to resolve any interpersonal differences brought to it while at the same time satisfying certain reasonable and desirable axioms. Thus the Arrow result is a basic theorem of what Samuelson calls “mathematical politics”.

4. Social Choice not the only Alternative.

• Baumol shows that “Arrow’s requirements are more strict than they seem at first view and that inconsistent or “undemocratic” social choice-making is not really the only alternative.

5. Majority Voting Pattern Unrealistic:

• Moreover, the Arrow theorem is based on the assumption of a majority voting pattern which does not take into consideration the possibility of a voting system that requires unanimity and permits buying and selling of votes.

Conclusion

Despite these criticisms, J.M. Buchanan holds that “the alternative to majority voting may be failure to make collective choices and there is no reason to believe that this is a preferred course of action. Voting is a means by which collective choices are made, and imperfections in the voting mechanism hardly seem an adequate reason to discard the system.”