Arrow's Impossibility Theorem demonstrates that it is impossible to create a social welfare function that aggregates individual preferences into a collective social preference in a consistent, democratic manner when there are 3 or more alternatives. Arrow identified 5 criteria that any social welfare system should satisfy: collective rationality, responsiveness to individual preferences, non-imposition, non-dictatorship, and independence of irrelevant alternatives. However, his theorem showed that no voting system can simultaneously satisfy all 5 criteria. This finding challenged the notion that majority-rule voting can consistently translate individual preferences into a social ranking.

1. Arrow’s Impossibility
Theorem

2. • 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.

3. • 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.
• 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.

4. • 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. • 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.

6. • 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

8. • 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.

9. • 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.

10. Criticism
• 1. Not related to Social Welfare Function
• 2. No Solution of Interpersonal Comparisons
• 3. Mathematical Politics
• 4. Social Choice not the only Alternative.
• 5. Majority Voting Pattern Unrealistic: