Sunday, October 4, 2015

If I Have Mine and You Have Yours, Do We Have Ours?

Let's play pretend. Let's say you're an earnest young music producer and you're going to make a star out of an earnest young talent. She's really good, Joni Mitchell or Kate Bush level. How do you capture her genius? Do you hire strings, really go balls out? Or is it more beautiful to capture her pure and unrestrained? Well, here's one piece of advice:

That's not helpful is it? Alright, well, here's what we think mathematically. This is a problem of "Multi-Objective Programming".

Single objective programming, or simple optimization, is pretty simple to understand (though in practice it can become quite difficult). You want to do ... something and you want to do your best. You wanna make the most money, or move the most people, or be the purest, or rock the hardest, or be the funniest, etc., etc. It was recognized by Newton that for smooth payoff functions, finding maximums and minimums was equivalent to finding the roots of derivatives (he wouldn't have phrased it like this). So all you have to do is have a root-finding technique, and there are very efficient ways of doing that. It's simple enough that reducing a design decision to an optimization problem is equivalent to solving it. I've actually done some work in this area, engineering stuff by optimization techniques.

Multi-Objective Programming is more complex. It's satire rather than just comedy - now you want to be as funny as possible and reach as many people as possible. Occasionally you're going to run into some trade-offs. How long are you willing to stray from the message for a laugh? How many laughs are you willing to sacrifice to get to the point? There's no longer any notion of a simple best, there's real trade-offs. How can we solve such a problem?

The notion of a "solution" to a problem with multiple objectives dates back to Hume, but has its first clear enunciation in the work of economist Vilfredo Pareto. Pareto was a very interesting person. He was at first a dedicated Free Market Liberal type and dedicated much of this part of his life to simplifying and extending the economic equilibrium theories of Leon Walras. He later decided that economic theories from Hume to himself left out the "irrational" aspects of human behavior. He believed that human nature was fundamentally non-logical, that political power was found in the vicious struggle of elites and that empirical study of sociology is what is best in life. By the end of his life, he was calling for releasing the hounds on anyone to the left of Mussolini - of whom he was very, very big fan. Hey, I said he was smart, not wise.

Well, Pareto's other beliefs aside, his proposal for the solution of multi-objective problems was very important innovation. A solution is called "Pareto" if improving in one aspect must mean detracting another. For the Joni Mitchell example, it would mean going beyond tasteful, jazzy backing to syrupy Mantovani backing (I don't hate Mantovani, just note that he'd be wrong for Mitchell). This isn't an optimum in the original sense -it can't be an optimum in the original sense. There's no longer a clear notion of best. It's a notion of stability of decisions. Imagine that Joni Mitchell had two producers, one who wanted bigness and the other honesty. Of course, much of what she does can please both producers - write a beautiful melody with lots of hooks and nice lyrics. But at some point, adding/subtracting some more strings will please one and not the other. Pareto's idea is not perfect. For instance, if Joni decides to completely listen to one producer and not the other, that idea is perfectly Pareto - even if it is wrong. To put it another way, let's say Mom bakes us a pie. If I take the whole pie, you can't get any without taking some from me. The distribution is Pareto - but not good.

The most important application of this idea, and Pareto's original application, was to the distribution of goods in a static economy. In this post, I discussed about how a highly unequal distribution of goods could develop from trade economy. Notice that at a Pareto stable point, trade effectively stops - Chamberlin stays the richest person. Nobody can make themselves better off without hurting someone (not necessarily Wilt, mind you). This has powerful implications. In fact, an economist named Francis Edgeworth developed a highly innovative iterative argument where trade people negotiate and renegotiate their contracts. Since only the set of Pareto equilbria are stable (this is a mathematical sticking point), they must land on one. This was, arguably, the first argument that an invisible hand could actually achieve a stable distribution, rather than the Walras/Pareto demonstration that one exists. Of course, this argument is also present in a purely verbal form in Hume. I've written about this before.

Well, I'm having trouble with my laptop battery, so we'll have to get into Pareto and dynamic economies some other time. See you then!

Uninteresting Note: I originally wanted to name this post "Economies Wobble But Do They Fall Down?" but that's about the stability of equilibria and not what an equilibrium is.

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