Kenneth Joseph Arrow(1921–2017)
In contemporary societies, a pre-eminent position is held by democracy in the political domain and by the market in the economic domain. Kenneth Joseph Arrow’s contributions regarding both democracy and the market are of such fundamental importance that any worthwhile academic discourse would be impossible on either without referring to his works.
Kenneth Joseph Arrow, regarded by most as thegreatest economist of the second half of the 20th century, Nobel laureate at the age of 51, passed away on 21 February 2017 at the age of 95. His influence spans almost every area of economics. A large number of his contributions, particularly those that were made in the 1950s and the 1960s, were foundational and resulted in the development of new areas and directions.
The Impossibility Theorem
Among his many seminal contributions, possibly the greatest is what is now known as the Arrow Impossibility Theorem, which he proved in his doctoral dissertation. It resulted in the creation of a new sub-discipline called Social Choice Theory. Arrow Impossibility Theorem says that there does not exist any rule that can aggregate individual rankings into social rankings satisfying some extremely appealing conditions. These conditions are:
(i) For every profile of individual rankings, the rule must assign a unique social ranking. That is to say, the rule must work regardless of what individual preferences are. If a rule does not satisfy this condition then it would work in some situations and not work in other situations.
(ii) If some alternative x is preferred to another alternative y by everyone in the society, then in the social ranking as well, x must be preferred over y.
(iii) The rule must be non-dictatorial. A person is called a dictator if it is the case that whenever he prefers an alternative x to some other alternative y, so does the society. The non-dictatorship condition requires that there be no dictator.
(iv) The rule must be such that the social preferences over any set of alternatives must depend solely on the individual preferences over those alternatives. In particular, the social preferences over any set of alternatives must remain unaffected if individual preferences over those alternatives have not undergone any changes but have undergone changes over other alternatives.
These four conditions are known as conditions of unrestricted domain, weak Pareto principle, non-dictatorship, and independence of irrelevant alternatives, respectively. Arrow’s monograph Social Choice and Individual Values,>1 by now a classic, contains the statement, proof, and exposition of the Impossibility Theorem.
The framework within which Arrow set out his Impossibility Theorem is very general and can be interpreted in several different ways. Two of these interpretations are particularly important. Under the first interpretation, the Arrow Impossibility Theorem says that no “satisfactory” voting rule can be designed, the term “satisfactory” meaning that the above-mentioned four conditions are fulfilled.
Under the second interpretation, the theorem says that there is no satisfactory way of evaluating social welfare on the basis of individual welfares.
Paradox of Majority Voting
The famous paradox of majority voting illustrates the Impossibility Theorem. Consider three candidates x, y, and z. Let the total number of voters be a multiple of three. Let one-third of the voters prefer x to y and y to z (and therefore, x to z); one-third of the voters prefer y to z and z to x (and y to x); and one-third of the voters prefer z to x and x to y (and z to y). Then x defeats y in a majority vote as two-thirds of the voters prefer x over y as against only one-third preferring y over x; y defeats z in a majority vote as two-thirds of the voters prefer y over z and only one-third prefer z over y; and z defeats x in a majority vote as two-thirds of the voters prefer z over x and only one-third prefer x over z. It is obvious that majority voting satisfies conditions (ii)–(iv); it, however, fails to satisfy condition (i). From Arrow’s theorem, it follows that the problem is not confined to majority rule only. Every voting rule would violate at least one of the four conditions.
The voting paradox associated with the majority rule was discovered at the end of the 18th century by the French philosopher Marquis de Condorcet. Arrow, however, was not aware of Condorcet’s work and rediscovered the paradox independently. Although the interest in voting rules goes as far back as Roman times,>2 the systematic study of voting rules became possible only after Arrow’s creation of the social choice theoretic framework.
In a personal reminiscence, Arrow recounted how he came to suspect and then prove the Impossibility Theorem (1991). In 1949, philosopher Olaf Helmer who was on the staff of the Rand Corporation at that time, was finding it difficult to comprehend how one could apply game theory to the case of countries. While it makes sense to assume that individuals have well-defined preferences, what meaning could one possibly ascribe to countries having preferences given that countries have individuals with heterogeneous preferences? When Arrow told Helmer that this problem had been solved by Abram Bergson (1938) in terms of utilities and could easily be translated in terms of preferences, he was requested by Helmer to write it up. It was in the process of writing up this exposition that Arrow realised there was no satisfactory way of arriving at collective preferences on the basis of individual preferences.
On Market Mechanism
Arrow’s contributions regarding market mechanism are also of fundamental importance. Competitive markets are characterised by large numbers of participants who take decisions in ignorance of the decisions of others. The ﬁrst important question that arises is how one can be sure that such uncoordinated decisions on the part of a large number of participants would not lead to a chaotic situation. In one of the most elegant results of economic theory, Arrow in a paper co-authored with Gerard Debreu, showed that under certain conditions, such uncoordinated decisions would be compatible with each other in the sense that an equilibrium would exist (Arrow and Debreu 1954). The existence of equilibrium was independently proved by Lionel W McKenzie (1954).
An important question that one can ask about any institution or mechanism is whether the outcomes that result under it have some speciﬁed desirable property. In economics, Pareto-efficiency is considered a very desirable property. An outcome is Pareto-efficient or Pareto-optimal if it is not possible to make some individuals better-off without making any individual worse-off. There are two fundamental theorems of welfare economics which relate the market with Pareto-efficiency. The ﬁrst fundamental theorem says that under certain conditions, the outcome that results under the market mechanism is Pareto-efficient. The second fundamental theorem says: provided that, by making the required redistributions, the initial endowment is appropriately chosen, then under certain conditions, almost any Pareto-efficient allocation can be obtained through the market mechanism. Although the history of these theorems is long, their ﬁrst rigorous proofs were provided by Arrow (1951).
Some Other Contributions
The number of areas in which Arrow made fundamental contributions is quite large. Special mention, however, needs to be made of his 1962 paper, “The Economic Implications of Learning by Doing”3 and his 1963 paper, “Uncertainty and the Welfare Economics of Medical Care.”4 The former engendered the voluminous literature on endogenous growth more than two decades after its publication, while the latter is credited with giving rise to the sub-discipline of health economics.
It is not widely known that Arrow’s ﬁrst publication was in Meteorology. His involvement with Meteorology came about as part of his military service. The United States Air Force had received a proposal regarding how to use the winds when navigating. Arrow thought that it was not the right question. Rather, the right question was how to get an airplane from one place to another as fast as possible, if you know the winds. This in turn resulted in an interesting mathematical problem. He read the relevant literature, which was mostly in German, made an important innovation and that resulted in the Meteorology paper.
Arrow’s Intellectual Life
Although Arrow devoted a considerable part of his intellectual life researching markets, he was not a votary of unregulated markets and supported state intervention for their regulation. He was critical of the viewpoint that equates freedom with the market. He was concerned from an early age about issues like poverty, income inequality, and racial discrimination. In his later years, he was deeply involved with issues relating to climate change.
Arrow was a voracious reader. He had learnt to read even before going to school, possibly even before he was four years old. He had a great reputation as someone possessing encyclopaedic knowledge not only of economics but even of things far removed from the discipline of economics. There are many anecdotes about his encyclopaedic knowledge and sharpness of intellect that are in circulation.
One anecdote that has been around for quite some time, and recounted recently in a write-up that appeared in the New York Times after Arrow’s death, goes something like this: Arrow’s junior colleagues in a good-natured conspiracy to get the better of him decided to read up on the breeding habits of gray whales and gather on a date and at a place where Arrow was sure to come. When Arrow came, they all started talking about Turner’s theory explaining how gray whales found the same breeding spot year after year. While leaving Arrow is said to have muttered that he thought that Turner’s theory had been discredited by Spenser. In an interview with Jane Hibbard conducted in 2011 (as part of the Stanford University Oral History Project), Arrow had this to say regarding this anecdote:
In a discussion on the breeding habits of Paciﬁc gray whales, I am supposed to have pointed out that a recent article had refuted the statements others were making. I’ve been assured that I said it. I cannot really remember it; doesn’t even seem credible to me.
Regardless of whether this anecdote is true or apocryphal, there is no doubt that it very aptly conveys the breadth and depth of knowledge that he possessed.
In contemporary times, among systems, the pre-eminent position is held by democracy in the political domain and by the market in the economic domain. Arrow’s contributions regarding both democracy and the market are of such fundamental importance that it is inconceivable any worthwhile academic discourse could be possible on either without reference to his works.
1 New York:Wiley,1951; secondedition: 1963.
2 Pliny the Younger in an interesting letter to a fellow senator discusses, in the context of voting in the senate on a particular issue, as to how, on the one hand, if they vote in favour of a more preferred alternative, and against a less preferred alternative in every vote division, then the eventual outcome would be highly undesirable from their perspective. On the other hand, the eventual outcome would be less undesirable, if in some vote divisions, they vote in favour of a less preferred alternative, and against a more preferred alternative (Pliny the Younger 2006).
3 Review of Economic Studies, Vol 29, No 3, pp 155–73.
4 American Economic Review, Vol 53, No 5, pp 941–73.
Arrow, K J (1949): “On the Use of Winds in Flight Planning,” Journal of Meteorology, Vol 6, pp 150–59.
— (1951): “An Extension of the Basic Theorems of Classical Welfare Economics,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J Neyman (ed), Berkeley and Los Angeles: University of California, pp 507–32.
— (1991): “The Origins of the Impossibility Theorem,” History of Mathematical Programming: A Collection of Personal Reminiscences, J K Lenstra, A H G Rinnooy Kan, A Schrijver (eds), Amsterdam: North-Holland.
Arrow, K J and G Debreu (1954): “Existence of Equilibrium for a Competitive Economy,” Econometrica, Vol 22, No 3, pp 265–90.
Bergson, A (1938): “A Reformulation of Certain Aspects of Welfare Economics,” Quarterly Journal of Economics, Vol 52, No 2, pp 310–34.
McKenzie, L W (1954): “On Equilibrium in Graham’s Model of World Trade and Other Competitive Systems,” Econometrica, Vol 22, No 2, pp 147–61.
Pliny the Younger (2006): Complete Letters, Trans P G Walsh, New York: Oxford University Press.
EPW looks forward to your comments. Please note that comments are moderated as per our comments policy. They may take some time to appear. A comment, if suitable, may be selected for publication in the Letters pages of EPW.