“Mathematical Models in the Social Sciences,” K. Arrow (1951)

I have given Paul Samuelson the title of “greatest economist ever” many times on this site. If he is number one, though, Ken Arrow is surely second. And this essay, an early Cowles discussion paper, is an absolute must-read.

Right on the first page is an absolute destruction of every ridiculous statement you’ve ever heard about mathematical economics. Quoting the physicist Gibbs: “Mathematics is a language.” On whether quantitative methods are appropriate for studying human action: “Doubtless many branches of mathematics – especially those most familiar to the average individual, such as algebra and the calculus – are quantitative in nature. But the whole field of symbolic logic is purely qualitative. We can frame such questions as the following: Does the occurrence of one event imply the occurrence of another? Is it impossible that two events should both occur?” This is spot on. What is most surprising to me, wearing my theorist hat, is how little twentieth century mathematics occurs in economics vis-a-vis the pure sciences, not how much. The most prominent mathematics in economics are the theories of probability, various forms of mathematical logic, and existence theorems on wholly abstract spaces, meaning spaces that don’t have any obvious correspondence with the physical world. These techniques tell us little about numbers, but rather help us answer questions like “How does X relate to Y?” and “Is Z a logical possibility?” and “For some perhaps unknown sets of beliefs, how serious a problem can Q cause?” All of these statements look to me to be exactly equivalent to the types of a priori logical reasoning which appear everywhere in 18th and 19th century “nonmathematical” social science.

There is a common objection to mathematical theorizing, that mathematics is limited in nature compared to the directed intuition which a good social scientist can verbalize. This is particularly true compared to the pure sciences. We have very little intuition about atoms, but great intuition about the social world we inhabit. Arrow argues, however, that making valid logical implication is a difficult task indeed, particularly if we’re using any deductive reasoning beyond the simplest tools in Aristotle. Writing our verbal thoughts as mathematics allows the use of more complicated deductive tools. And the same is true of induction: mathematical model building allows for the use of (what was then very modern) statistical tools to identify relationships. Naive regression identifies correlations, but is rarely able to discuss any more complex relationship between data.

A final note: if you’re interested in history of thought, there are some interesting discussions of decision theory pre-Savage and game theory pre-Nash and pre-Harsanyi in Arrow’s article. A number of interpretations are given that seem somewhat strange given our current understanding, such as interpreting mixed strategies as “bluffing,” or writing down positive-sum n-person cooperative games as zero-sum n+1 player games where a “fictitious player” eats the negative outcome. Less strange, but still by no means mainstream, is Wald’s interpretation of statistical inference as a zero-sum game against nature, where the statistician with a known loss function chooses a decision function (perhaps mixed) and nature simultaneously chooses a realization in order to maximize the expected loss. There is an interesting discussion of something that looks an awful lot like evolutionary game theory, proposed by Nicholas Rachevsky in 1947; I hadn’t known these non-equilibrium linear ODE games existed that far before Maynard Smith. Arrow, and no doubt his contemporaries, also appear to have been quite optimistic about the possibility of a dynamic game theory that incorporated learning about opponent’s play axiomatically, but I would say that, in 2012, we have no such theory and for a variety of reasons, a suitable one may not be possible. Finally, Arrow notes an interesting discussion between Koopmans and his contemporaries about methodological individualism; Arrow endorses the idea that, would we have the data, society’s aggregate outcomes are necessarily determined wholly by the actions of individuals. There is no “societal organism”. Many economists, no doubt, agree completely with that statement, though there are broad groups in the social sciences who both think that the phrase “would we have the data” is a more serious concern that economists generally consider it, and that conceive of non-human social actors. It’s worthwhile to at least know these arguments are out there. (Final version provided thanks to the Cowles Commission’s lovely open access policy)


2 thoughts on ““Mathematical Models in the Social Sciences,” K. Arrow (1951)

  1. About the ” would we have the data, society’s aggregate outcomes are necessarily determined wholly by the actions of individuals”

    You surely know the (in?)famous arguments based on complex theory. Did Arrow discussed something along these lines in the paper (even if he doesn’t use complex theory jargon)?

    • Kevin Bryan says:

      He is not aware of it in this essay, at least. I am not aware of much recent work by Arrow on methodology, so perhaps he’s come around (and, of course, there are other non-complex system arguments for modeling nonhumans as having agency!)

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