calling all mathematicians and physicists: please fix economics
I just took a couple hours to work my way through this talk by Eric R. Weinstein on Gauge Theory in Economics. It’s a really good talk; however, it is a math talk, so if you aren’t comfortable with vector spaces being the simplest object in view, I’m not sure how much you’d enjoy it. If you do, though, check it out! A phrase for temptation purposes: “A Rosetta Stone for economics, physics, and geometry.”
But even if you’re not into advanced mathematical brain fry, the underlying point of the talk is very important. According to Weinstein, there is an obviously incorrect economic assumption about human behavior, expressed in this quote:
“…tastes neither change capriciously nor differ importantly between people. On this interpretation one does not argue over tastes for the same reason that one does not argue over the Rocky Mountains—both are there, will be there next year, too, and are the same to all men.”
—Gary Becker and George Stigler, 1977, De Gustibus Non Est Disputandum
…and that wrong assumption has affected the discipline of economics as described in this quote:
“That the problems [of changes of taste] have remained central and largely unresolved for twenty-five hndred years no doubt makes some economists think it wise to define them out of the discipline, at whatever cost in realism and relevance.”
M.S. McPherson, ‘Changes in tastes’, entry in The New Palgrave: A dictionary of Economics, 1987, pp 401-403.
In short, economics uses, as a fundamental axiom, the obviously wrong idea that we all prefer the same things to the same degree for our entire lives.
Srsly!
However, I sort of can’t blame any economist who would hope like hell this assumption, while obviously incorrect, would be unimportant. I can’t blame them because, when you assume the obviously wrong idea is in fact wrong, the mathematics behind fundamentally important and useful economic concepts become, to use a technical phrase, really fucking hard. I have compassion for anyone who doesn’t want to go anywhere near an infinite-dimensional function space bundle!
But, still. A crazily wrong axiom has to go. Suddenly, there’s all kinds of low-hanging theorems just waiting to be picked up by mathematicians and physicists and anyone else who is willing to touch a fiber bundle. So, who’s with me?
[ETA: There's a whole interesting undercurrent of the extension of that idea of 'the unreasonable effectiveness of mathematics', and I think that's really interesting from a calculus education standpoint, too. I got too excited about the economics part to mention it!]
Tags: axioms, economics, gauge theory
Great stuff. A number of folks I’ve passed this onto have raised the objection that mathematicians “caused the problem in the first place.”
Being a cocky primate, I immediately corrected them but I wonder if I’m correct. I was under the impression that the “Quants” were speaking high-level bullshit, and other math heads have serious objections/reservations about Quant models. (I seem to recall reading a book-length example of exactly that called “The Black Swan.”)
Historically, my impression is that Quants found an audience in Wall Street because they were useful. They didn’t need to be right, they just needed to be smart.
How accurate are my impressions, according to your own impressions?
That’s precisely the sort of pushback I was expecting. I don’t know enough about the experimentally discredited models the quants were using to do more than speculate. (I’d be delighted to find one of these models and pick apart its definitions but I wouldn’t know where to look. Now you all know what to get me for Christmas.)
Mathematical modeling of anything at all is a balance between compression (“Can we use simplifying assumptions to make tractable the untractable…”) and utility (“…without simplifying so much that we don’t learn anything real?”) When you hit the sweet spot in between, you get, say, airplanes. They are a real thing, but they are understood in terms of models of varying levels of complexity depending on what you’re trying to do. Hyperinflation is also a real thing. It has real effects. (As you may know, these effects are Very Bad.) But if it’s going to be understood, it’s going to be understood mathematically, because it’s a real, abstract thing.
Now, since this is my blog, I’m going to start with the speculating! The important bit about this Weinstein talk is that it is foundational. If you give a mathematician a wrong assumption, anything he or she builds on top of it is nonsense. It seems entirely plausible that the quants were speaking very respectable mathematics built on top of other mathematics built on top of incorrect axioms. So yeah: if the quants looked at a given mathematical model and figured out they could do something cool with it, they might not have noticed that the given model was axiomatically fucked.
(Aside: Einstein once referred to ‘ignoring an axiom’ as a big part of his physics insights. The axiom in this case was, ‘Time flows the same everywhere’, which is intuitively obvious, and also wrong. Axioms are tricky because they are often invisible.)
So, my justification for this post’s title is this: a foundational change means everything has to be redone, from first principles. The foundational change leads immediately to mathematics requiring calculus on a vector bundle. The population of people who respond to phrases like “calculus on a vector bundle” with a hearty, “Yar! Sounds like fun!” is limited to mathematicians and physicists. So if economics is a real thing with real effects (like, you know, hyperinflations that lead to Nazi takeovers), then it’s time to call in the nerd cavalry.
The model deemed to have done the most damage is known as the Gaussian copula, which ultimately concluded that all stocks and their covariances followed a multivariate normal distribution. This model became really popular, very quickly because it was very fast to compute and seemed to work perfectly well. It’s problem was that it traded reality for simplicity and as a consequence did not anticipate the huge risks that a single negative shock can cause to a financial system.
This may not be the single cause of the whole disaster but it’s certainly one of the major causes. It was a mathematician’s folly for suggesting this model but it was the industry’s folly for accepting it wholesale without asking any questions or examining the basics first.
From your post and comment, it seems this is the kind of thing you were talking about. I’ve just started life as a financial engineer (just started a PhD here in Switzerland) so this is almost certainly the kind of thing I’m going to be working on for the next couple of years at least.
Tom – I’m not 100% sure what you’re asking/proposing here.
Are you asking for mathematicians and physicists to create a a mathematical model that accounts for taste to counter Weinstein’s axiom?
Justin – The problem with claiming that “mathematicians caused the problem in the first place” or “mathematicians need to save us from this mess” is thinking that mathematicians agree on solutions.
Math has a long walk through this list just on the micro level:
http://en.wikipedia.org/wiki/List_of_cognitive_biases
It wasn’t even the quants who caused all the harm. In finance there is a lot of work that goes into finding “arbitrage” (or Arb in the argot) opportunities, which are mis-pricings of factors and differences in risk among sets of interdependent or related securities. Arb ops come in multiple forms and mean that you have information the market doesn’t. Jurisdictional and regulatory aribtrage is a good example, where a trade that is not profitable under one regime due to regulations and taxes, becomes very profitable in another. Statistical arbitrage (stat arb) exploits errors in pricing models and the pricing of risk in a set of trades. Basically, given the price of this option or hedge, how much will it cost whom if it is wrong? Convertible arb is when securities of one sort that are convertible into another (equity and debt) and the information about the price change in one is not expressed in the other is not instant. There are tons of other kinds, as if you can think of an attribute of a security or trade, there is an arb op, the question remains, however, is it profitable and for how long? THe guys over at Renaissance Technologies – mentioned in the lecture about Jim Simmons figuring this out and becoming a billionaire in rhode island – at least initially took another approach. Simmons’ practical background was in signal processing, which is all about FFTs, calculus, information theory and comp.sci, which meant that he had a huge advantage over economists who were essentially playing in a sandbox in the 80s.
The way people really made money in the hedge fund business, however, was not necessarily about real returns. They used “quants” as a marketing tool. The issue they posed to institutional investors was, “who are you going to trust with your money, some old guys and their punk nephews or my room full of escaped russian scientists?” Nobody was ever fired for blaming rocket science, and an industry was born.
So, yes, there are are great opportunities to saunter into economics with some real math and to maybe even meet the King of Sweden, but it is just an arb op like any other. Information in physics did not trickle out to economics and there is a bundle to be made before everyone piles into the trade.
If you are at all quantitatively inclined, as a physicist, mathematician, hacker or economist,
[...] Tom’s post on it here. [...]
really fucking hard
…is an understatement. What happens when you relax these assumptions is that you either have to make other assumptions that are even more broken or get very poor results (sufficient conditions that are not necessary, necessary conditions that are not sufficient, assumptions about the underlying functions and spaces that are unreasonable and/or untestable).
Economists have tried using mathematical models to explain everything from addiction to choices about education. Addiction is an interesting problem because it deals with a taste for one good that is so powerful it overwhelms all other consumption. Education is interesting because if everyone chose their majors based on benefit/cost analysis, we’d have lots of engineers and almost no English majors. The fact that we have the opposite is impossible for an economist to explain.
The resulting models have been interesting but haven’t given as much insight as we’d like.
A mathematician might have better tools for solving these problems. I’d like to see more cooperation between good mathematicians and talented economists, but I don’t know if throwing a mathematician into the deep end of the economics pool and shouting “Swim!” would be a good idea.
Perhaps not a good idea, but possibly a fun idea!
Very interesting. I’m very curious what the ‘then’ part of that last hypothetical is…