Wednesday, July 16, 2014

Hume and Edgeworth

Or: The Consistency of The English Philosophy!
Hume has been called "one of the most important philosophers in the English language", his skepticism and empiricism inspiring - in positive and negative ways - whole swaths of philosophy starting in the 19th century and continuing to this day. Hume is best known for his attack on the connection between empirics and metaphysics. He argued that we have no way of demonstrating empirically that an event necessarily causes another event. This is of great importance, and both the conclusion and the argument are pregnant with possibilities. He would later clarify that we do have reason to act on the belief that an event causes another event, their (near) constant conjugation instills in a belief in (nearly) necessary cause. This positive solution - often ignored by philosophers - flowered into the associationist school of psychology, game theory, the Bayesian school of statistics.
Francis Edgeworth was an English economist, the author of Mathematical Psychics. In this book, he attempted to take the Jevonian gloss on Utilitarianism a firmer mathematical foundation. Modern economics, with its utility functions etc., is a descendent of this line of thought (which sprung in many places at several times) but is not intrinsically tied to it. Honestly, I haven't read Edgeworth's work in detail, but I have read a little bit of Mathematical Psychics and some of his stats papers for something I wrote on the history of statistics. In Mathematical Psychics, Edgeworth develops an ingenious device for thinking through bilateral trade now called the Edgeworth Box.

I will use this graphical method to illustrate a famous passage of Hume on co-operation. A modern thinker might think that it is obvious that a non-linguistic, behavioral definition of convention and co-operation was possible, and indeed since Lewis it has been standard to found the concept of linguistic meaning on pre-linguistic ideas of co-operation. In Hume's time, it was not so obvious. Hume had to explain "... [C]onvention is not of the nature of a promise: For even promises themselves, as we shall see afterwards, arise from human conventions.". Therefore, Hume gave this as illustration: "Two men, who pull the oars of a boat, do it by an agreement or convention, tho’ they have never given promises to each other.". This is a very important point! Not only do Lewis and Hume, and biologists following him, tell us that this is how meaning got into languages ("In like manner are languages gradually establish’d by human conventions..."), it was immediately used by Hume to explain how property got into society:

"Nor is the rule concerning the stability of possession the less deriv’d from human conventions, that it arises gradually, and acquires force by a slow progression, and by our repeated experience of the inconveniences of transgressing it."

These passages of Hume are pregnant with theory, and the modernity of the theory is sometimes surprising. Hume's theoretical stances - which are those of evolutionary game theory if I may be anachronistic - run deep. Property is, he says, some sort of evolutionary strategy, one with advantages and disadvantages. Hume, obviously, believes the advantages outweigh the disadvantages. This is a story about property that can extend beyond humanity (Hume was wrong to deny this), and it has been used - by Maynard Smith and others - to examine the phenomena of nesting in animals.

Let's analyze one of these pieces, with the more modern equipment of an Edgeworth Box. Two men, Mr Blue and Mr Green, pull the oars of a boat. They must paddle the same speed in order to avoid moving in a circle. Even if these men do not speak the same language, they can and will co-ordinate. We will ask more than Hume does explicitly here (he makes more assumptions implicitly elsewhere), we will ask that the men understand that you cannot go faster than the slower paddler (we don't assume that they know the others strength). The possible speeds they can go are a set of real numbers, setting up an axis:

That gives the following box as the range of possibilities:

Our assumption about their preferences gives them Leonteif indifference curves. For Mr Blue, his indifference curves are:

And for Mr Green:

Putting these together, we get:

Let's say they just start rowing at some speed. That means they get something like the following:

Anywhere inside the square of which that dot is the corner is better for both! A simple way to think about it is that the faster rower knows to slow down, but not slower than the slower rower, and the slower rower knows to speed up, but not faster than the faster rower. Eventually, the rowers will come to a corner on both curves. Here, neither rower can improve by himself. This is a stable situation! Here is one possible solution:

Edgeworth pointed out that this is not the only solution. In fact, there is a continuum of solutions called the "contract curve" or "core":

This is the Edgeworth analysis of Hume. Hume's correctness is not in doubt in this manner of thinking. Hopefully this shows both the depth of Hume's thinking and how it relates to modern ideas. I wouldn't mind if it helped one understand the modern ideas a little better too. There are more extensions that can be made (what happens as the quantity of rowers climbs? What if they can only imperfectly measure the others speed? What kind of equilibrium did we obtain?). Notice that Hume didn't make any explicit assumptions about the nature of their preferences, yet the Edgeworth explanation explicitly assumes convex preferences. Can non-convexity be made sense of here? What other interpretations of Hume are possible - do any of them attack the substance of this translation?

Incidentally, I had a devil of a problem making the images for this post. Matlab decided some of the lines I drew just weren't good enough for her. Awful thing it is, when I turned off the axis, the invisible axis was over the lines I actually drew. Goddamn thing. Some of these images are corrected, some not. The ones which were not may change if I come back later.

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