Arrow’s Impossibility Theorem
In an attempt to construct a consistent social ranking of a set of alternatives on the basis of individual preferences over this set, Arrow obtained:
1) an impossibility theorem;
2) a generalisation of the framework of welfare economics, covering all collective decisions from political democracy and committee decisions to market allocation; and
3) an axiomatic method which sets a standard of rigour for any future endeavour.
Prof. Arrow pointed out that the construction of social welfare function, which reflects the preferences of all individuals comprising the society, is an impossible task. His main contention is that it is very difficult to set up reasonable democratic procedure for the aggregation of individual preferences into a social preference for making a social choice. Arrow has proved & general theorem according to which it is impossible to construct a social ordering which will in some way reflect the individual ordering of all the members of society.
While constructing his argument, Arrow has maintained that individual’s ordering of social states does not depend exclusively upon the commodities consumed but also on the amounts of various types of collectives such as municipal services, parks, sanitation, erection of statues of famous men, etc. In other words, an individual solely on the basis of her consumption cannot evaluate welfare results of collective activity; instead, individual ordering of social states will depend on her own consumption as well as on the consumption of others in a society. Individual ordering of alternative social states reflects her value judgments, which are also called simply ‘values7 by Arrow. According to him, it is ordering of social states according to the values of individuals as distinct from the individual tastes, which should be determined for the construction of valid social welfare function.
The theorem’s content, somewhat simplified, is as follows: A society needs to agree on a preference order among several different options. Each individual in the society has a particular personal preference order. The problem is to find a general mechanism, called a social choice function, which transforms the set of preference orders, one for each individual, into a global societal preference order. This social choice function should have several desirable (“fair”) properties:
- Unrestricted domain or universality: the social choice function should create a deterministic, complete societal preference order from every possible set of individual preference orders. (The vote must have a result that ranks all possible choices relative to one another, the voting mechanism must be able to process all possible sets of voter preferences, and it should always give the same result for the same votes, without random selection.)
- Non-imposition or citizen sovereignty: every possible societal preference order should be achievable by some set of individual.preference orders. (Every result must be achievable somehow.)
- Non-dictatorship:the social choice function should not simply follow the preference order of a single individual while ignoring all others.
- Positive association of social and individual values or Monotonicity: if an individual modifies her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. (An individual should not be able to hurt an option by ranking it higher.)
- Independence of relevant tenatives: if we restrict attention to a subset of options, and apply the social choice function only to those, then the result should be compatible with the outcome for the whole set of options. (Changes in individuals’ rankings of “irrelevant” alternatives [i.e., ones outside the subset] should have no impact on the societal ranking of the “relevant” subset.)
Arrow examined the problem rigorously by specifying a set of, above requirements that should be satisfied by an acceptable rule for constructing social preferences from individual preferences, which can be simplified as the conditions of social choice as follows:
- Social preferences should be complete in that given a choice between alternatives A and B, it should say whether A is preferred to B, or B is preferred to A or that there is a social indifference between A and B.
- Social preferences should be transitive, which implies, if A is preferred to B, and B is preferred to C, then A is also preferred to C.
- If every individual prefers A to B, then socially A should be preferred to B.
- Social preferences should not depend only upon the preferences of one individual; i.e., the dictator (not in the pejorative sense of the word).
- The last condition asserts that the social preference of A compared to B should be independent of preferences for other alternatives.
According to Arrow, “if we exclude the possibility of interpersonal comparisons of utility, then the only method of passing from individual tastes to social preferences which will be satisfactory and which will be defined for a wide range of sets of individual ordering are either imposed or dictatorial”.
The democratic procedure for reaching a social choice or group decision is the expression of their preferences by individuals through free voting. Social choice will be determined by the majority rule. But Arrow has demonstrated through his impossibility theorem mentioned above that consistent social choices cannot be made without violating the consistency or transitivity condition. The social choice on the basis of majority rule may be inconsistent even if individual preferences are consistent. Arrow first considers a simple case of two alternative social states and proves that in this case group decision or social choice through a majority rule yields a social choice, which can satisfy all the five conditions. But when there are more than two alternatives, majority rule fails to yield a social choice without violating at least one of the five conditions. Thus, Arrow’s theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social choice function that satisfies all these conditions at once.
Various economists have tried to explain Arrow’s impossibility theorem in very different ways, but we will illustrate the proof of the theorem with the help of the table given below.
Figure: Ranking of Alternatives by Individuals and Social Choices
In this table three individuals A, B and C who constitute the society have been shown to have voted for three alternative social states, X, Y and Z by writing 3, against the most preferred alternative, 2 for the next preferred alternative and 1 for the least preferred alternative. As shown in the table, individual A prefers X to Y, Y to Z and therefore X to Z. Individual B prefers Y to Z, Z to X and therefore Y to X. and individual C prefers Z to X, X to Y and therefore Z to Y. It is clear that two individuals A and B prefer Y to Z and also two individuals A and C prefer Z to X. Thus, the majority (two of the three individuals) prefers X to Y and also Y to Z, and therefore, Z to X. But majority also prefers Z to X. Thus, we see that majority rule leads to inconsistent social choices because on the one hand, X has been preferred to Z by the majority and on the other hand, Z has also been preferred to X by majority, which is contradictory or inconsistent.
On the basis of five conditions as mentioned above, Arrow has derived three consequences to explain his impossibility theorem. Let us analyze these consequences in the case of three alternatives X, Y and Z available to the two individuals, A and B. According to Consequence I, whenever the two individuals prefer X to Y, then irrespective of the rank of the third alternative Z, society will prefer X to Y . According to Consequence II, if in a given social choice, the will of individual prevails against the opposition of individual B, then the will of A will certainly prevail in case individual B is different or agrees with A. According to Consequence III, if individuals A and B have exactly conflicting interests in the choice between two alternatives X and Y, then the society will be indifferent between X and Y. It is interesting to note that the simple proof of the impossibility theorem follows from Consequence III. For instance, if individual A prefers X to Y and individual B prefers Y to Z and if society opts for X, then A will be a dictator inasmuch as her choice will always be a social choice. Thus, Arrow’s theorem says that ‘the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social choice function that satisfies all these conditions at once ‘. Arrow, therefore concludes that it is impossible to derive a social ordering of different conceivable alternative social states on the basis of the individual ordering of those social states without violating at least one of the value judgments as expressed in the five conditions of social choices. This is in essence his impossibility theorem.