Though this site is devoted generally to new research, the essay discussed in this post, I trust, will be new enough to the vast majority of readers. Charles Sanders Peirce is a titan of analytic philosophy, and there is certainly a case to be made that he is the greatest American philosopher of all time. He also has had a fairly well-known indirect influence on economics: Peirce was in some ways rediscovered by the great mathematician Alfred Tarski, who then taught Kenneth Arrow, and in doing so may have introduced Peirce’s relational algebra to the field of economics. (You may be thinking, relational algebra, what is that? But you certainly know what it is: take a set, apply a perhaps partial, often binary ordering with certain properties, then prove results. This surely describes every modern introduction to the theory of preferences, does it not?) But Peirce also has an essay more directly on economics that is fascinating to see in retrospect. This Peirce essay is reprinted in Phil Mirowski’s book “Science Bought and Sold” along with notes on the essay by James Wible which I shall also draw from.
Two final things. First, I note, if only to myself, the following quote from Peirce to be used in a future research paper of my own: “Economical science is particularly profitable to science; and that of all the branches of economy, the economy of research is the most profitable.” Second, check out where this essay was published: the annual report of the U.S. government Coast Survey of 1879! No wonder it has been overlooked. If you know anything of the biography of Peirce, though, there is not much surprising in this odd location. Peirce was supposedly such a nut that, despite obvious brilliance, he was repeatedly blackballed from academic appointments by future colleagues around the country!
Wible claims, and I also know of no earlier such work, that this Peirce essay is the earliest mathematical work on the theory of invention. And given the intellectual history, this seems almost certain to be so. The essay was written right at the cusp of the marginal revolution and mathematical political economy, Peirce is known to have been familiar with the few scraps of earlier mathematical economics like Cournot’s famous 1838 essay, and Peirce is the father of a philosophical school for which selecting the best line of research to examine in order to learn inductively was a pressing concern. If you’ve ever read economics articles from the middle of the 19th century, this one will shock you: in style, I think it is essentially publishable today. It looks like 21st century economics. There are marginal tradeoffs. There is social science done by mathematical manipulation of heavily abstracted concepts. There is even a Marshallian diagram! It’s phenomenal. Since this looks like modern economics, let’s discuss it like modern economics; what does Peirce’s theory say?
As he introduced it, “I considered this problem. Somebody furnished a fund to be expended upon research without restrictions. What sort of researches should it be expended upon?” Essentially, there are some scientific problems which we understand only vaguely; you may think of this purely qualitatively, or as meaning something is measured to within some confidence interval. There are diminishing returns to science, so that while decreasing error can be done at linear cost, the utility gained from such reduction is concave (the inverse is quadratic in Peirce’s formulation). There is a total fixed research budget. What should be worked on first? Note that this paper was first written in 1876: there is no stochastic learning or any such thing, as the mathematics to discuss bandits and related objects was not yet developed. Learning is purely deterministic here.
Solving that constrained maximization problem gives the now-familiar, but then-nonexistent, result that we should compare ratios of MU/MC across different projects. Peirce called this ratio of marginal utility to marginal cost the “economic urgency” of a given line of research. He notes that, given that functional form assumptions, new research fields where we know very little are particularly worthwhile investments: the gains from increasing our knowledge are exponential in ignorance, whereas the cost is linear. As an example, an early chemist with simple vials is able to provide results with more social utility than a thousand chemists working in Peirce’s day with all sorts of modern equipment. Peirce also derives a result concerning sampling which is a bit opaque for modern readers given that it is couched in terms of “accidental probable error” rather than confidence intervals; nonetheless, it is very Wald-esque in that it explicitly argues that optimal sample size in experiments depends crucially on the budget, the costs of sampling and the utility of learning inferences from that sampling. Such considerations are absolutely ignored in a lot of research design even today!
http://books.google.com/books?id=ux79s_IhpFYC (Both Peirce’s original essay and Wible’s commentary appear in “Science Bought and Sold,” edited by Mirowski and Sent. The Google Books Preview is generous enough here for you to read the entirety of both essays; I do not see any other ungated copies of either online.)
A Fine Theorem 
What do you think about prediction markets for science?
I fail to see how introducing another source of bias – financial self-interest – is supposed to counter existent biases in science…
Isn’t it obvious? When you suffer a financial cost for holding false beliefs, you have an incentive to overcome ideology and cognitive biases and to hold correct beliefs. In short, tax false beliefs and we’ll get less of them.