Kenneth Arrow and Formal Modeling as a Form of Criticism, Part 1


arrowKenneth Arrow died on Feb. 21 at the age of 95. I am not a scholar of Arrow’s work, per se, but inasmuch as I’ve studied him in the context of my broader work, I’ve always found him to be a thoughtful and intriguing person. My book, Rational Action, even gives him the last word.

My point, channeled through Arrow, is that the people who developed fields like operations research and decision theory and who formalized economics were not advocating an exotic, revolutionary, or naive concept of rationality and governance. Rather, they worked to understand and explicitly describe rationality as it exists in the world and to use and improve on that rationality so as to improve decision making and policy. In 1957, Arrow described building formal (i.e., mathematics and logic-based) models of decision making as striving toward a final destination that could never be reached. But, drawing on Goethe’s Faust, he regarded the very act of striving as offering a chance at intellectual salvation. (“He who ever strives, him can we save” / Wer immer strebend sich bemüht, Den können wir erlösen.)

But to what end was Arrow actually striving? I would argue that, certainly early in his career, it was not primarily toward more faithful descriptions of reality—his craft remained far distant from that destination. Rather, his paramount interest was to use models to build an improved critical understanding of cutting-edge concepts and ideas—their presuppositions and logical consequences, their possibilities, and their limits. In this, Arrow was not so different from the humanistic (literary, historical, or philosophical) critic. Yet, his methods were, of course, very different.

“Mathematical Models in the Social Sciences” (1951)

In this essay, Arrow described directly what he felt to be the value of formal modeling, and defended it against common objections. The essay appears in an eclectic volume called The Policy Sciences: Recent Developments in Scope and Method, edited by Harold Lasswell. I don’t happen to know the history behind the book or Arrow’s contribution to it off hand, but it’s a fascinating survey of the meditations, ambitions, and debates sweeping the post-World War II social sciences in the United States.

Arrow began the essay by arguing against the view that social phenomena can only be evaluated with qualitative language reflecting the judgment of the investigator. He asserted, that, as a “language” itself, mathematics is not only valid but highly valuable on account of “its superior clarity and consistency.” He explained:

If the intuition of the investigator is reliable, it will yield the same judgments every time it is confronted with the same set of facts. But any such unique correspondence can always be represented by a mathematical relation of sufficiently complicated form. Hence, any intuitive knowledge can always be reduced to mathematical terms. Apart from this, there is the general presupposition that scientific knowledge should be interpersonally valid and transmittable and hence expressible in an objective, consistent language.

Arrow also explained what he regarded to be some of the most common and potent arguments against using formal representation:

Every mathematician realizes what a small part of all the potentially available mathematical knowledge is actually grasped at the present time. The usual objection of the ‘literary’ social scientist when confronted with a mathematical system designed as a mode of reality is to assert that it is ‘oversimplified,’ that it ‘does not represent all the complexities of reality.’ In effect, he is saying that the symbolic language in which the mathematical model is expressed is too poor to convey all the nuances of meaning which he can carry in his mind. What happens is that the very ambiguity and confusion of ordinary speech give rise to a richness of meaning which surpasses for the social scientist the limited resources of mathematics, in which each symbol has only one meaning. It is not surprising that there should be a difference between the social and the natural sciences in this regard. Language is itself a social phenomenon, and the multiple meanings of its symbols are very likely to be much better adapted to the conveying of social concepts than to those of the inanimate world. Furthermore, the empirical experience on which one’s understanding of the social world is based consists to a large extent of symbolic expressions of other individuals; one can apprehend these expressions directly because one is himself part of the social world he observes. Such apprehension must inevitably take place on a largely unconscious level unamenable to mathematical expression (which is surely the acme of consciousness). It is precisely in the field of economics, where the individuals studied are engaged in relatively highly conscious calculating operations, that mathematical methods have been the most successful.

While accepting that ordinary language had a closer kinship to humans’ ability to tacitly apprehend the world around them, Arrow nevertheless insisted that the clarity of meaning within mathematics was a superior to languages rife with ambiguity and multiple meanings as a tool for examining ideas and their logical consequences.

How, then, was the developer of formal models to proceed? For Arrow, the answer was not necessarily to elaborate on simple models until one obtains a model complex enough to be used for valid measurement based on statistical data. Arrow, rather, was concerned that formal models should be able to be solved, as he believed the solutions to mathematical models would also yield statements that could be tested against statistical data. He wrote:

It is unfortunately true that it is very easy to formulate theoretical models in which the determination of the optimum statistical methods leads to mathematical problems which have not been solved; in other cases, the resultant problems can be solved in principle, but the computations needed to find the solution in any given case take an impractical amount of time. Here again we must resort to simplification. The customary procedure is to substitute a mathematically practicable theory, as similar as possible to the desired one, and use that as the basis for deriving statistical methods.

In a footnote, Arrow observed that developing computable solutions to measurably realistic economic problems is a formidable task:

An attempt, for example, to estimate statistically the economic laws governing a large modern country on the basis of a relatively simple model (say, a few hundred equations to be fitted) could easily occupy the best computing machines now available for the next five hundred to a thousand years.

Arrow noted that assuming that the solution to a model is itself a valid model also assumes that the model describes actors who behave not only rationally, but optimally. Understanding that such assumptions were controversial, he wrote,

A number of objections have been raised to the usefulness of the principle of rationality: (a) If the complicated nature of the range of choices possible in an actual social situation is even approximately taken into account in the theoretical model, the mathematical problem to be solved in the maximization of utility will become extremely complex, and it will be hard to derive results which have any simple meaning. This objection has been frequently raised, for example, against the Walrasian scheme of general equilibrium in economics. (b) There is no real reason to suppose that individual behavior does conform to the principle of rationality. This argument is partly related to the previous one; it is argued that if the rational choice is too difficult for the trained mathematician to find, it is certainly unreasonable to suppose that the untrained, unreflecting, average individual will be able to locate it. (c) The utility function itself, even if it plays the role assigned to it, is highly unstable over time; hence, for an understanding of social processes, more interest attaches to the determinants of the variation of tastes than to the line of causation from the utility index to the actual decision made. (d) There is a fundamental ambiguity in the concept of rationality in a social situation. An individual will soon realize that his actions, in addition to their other consequences, will alter the obstacles faced by others, thereby affecting their actions and in turn altering the obstacles controlling his choices.

The last two objections are more technical objections, and, concerning (d), it is worth pointing out that Arrow spent much of his essay discussing game theory, which directly addressed that issue. The former two objections are more in line with the realism, complexity, and subtlety objections to mathematical modeling discussed above. Arrow went on to note that alternatives to the rationality assumption generally deploy “ad hoc” propositions “which limit the scope of the principle of rationality.”

Interestingly, Arrow also made a point that is rather similar to sociologists’ concerns about “performativity,” particularly as developed in Donald Mackenzie’s work on the finance theory of the 1970s and 1980s:

One somewhat digressive remark on the principle of rationality may be in order: a rational theory always has a dual interpretation. On the one hand it may be taken as a description of reality to the extent that individuals really are consistent in the sense assumed. On the other hand it may be taken rather as a normative theory, which prescribes what individuals ought to do. Thus, theoretical economics has been used to analyze what the optimum state of economic welfare would be and how to attain it.

Here, Arrow is drawing a difference between a descriptive and a policy-oriented economics. However, the application of normative theories of decision would soon become central to the field of operations research. (It would be a few years before optimization methods became a key feature of OR as it shifted away from its military origins, and Arrow was a contributor to that shift.) It is important to note that, to the end of his life, Arrow held a professorship in both economics and OR.

Arrow’s essay described his view of the uses of formalism in modeling, but it did not actually offer a very rich description of how Arrow himself deployed formalism. My argument is that Arrow used it as a means of establishing the state of understanding of a particular conceptual problem, establishing the validity and limits of that understanding, and suggesting ways to move beyond it. A more detailed explanation of this argument is forthcoming in Part 2 (and possibly beyond).

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