Posts Tagged ‘maths’

I hope that P does not equal NP

Monday, August 9th, 2010

And, further, I hope that this Vinay Deolalikar guy proved it. I don’t normally will mathematics to behave a certain way, but in this case I make an exception for a few reasons. (For a quick primer on the P=NP problem, check out Wikipedia).

First, I like hearing about epic math problems that have been solved. And learning about how they were solved, and who solved them, and who came up with those problems in the first place, and everything that was going on at the time that led up to the discoveries. I find this stuff fascinating. It’s like Edmund Hilary scaling Mount Everest, except with maths. In college my friends and I watched a documentary on Fermat’s Last Theorem and Andrew Wiles’ pwning the hell out of it. We had a glorious time. Oh, I also read a book about Godel’s incompleteness theorems. That was similarly awesome.

Second, there is a paper which proves markets are efficient iff P=NP (“iff” is pronounced “if and only if”). Now, I’m pretty sure markets aren’t efficient. It’s my opinion that a belief in the efficient markets hypothesis is correlated with being more of a jerk. Fightin’ words, I know, but economists in general (and market fundamentalists in particular) could use a little more humility. But I digress. I’m not sure how much play the “P=NP <=> efficient markets” paper got, or if anyone proved it; and I remember Tyler Cowen didn’t think much of the claim, but I hope it’s correct. These two papers, then, would prove mathematically that markets aren’t efficient. I would feel vindicated, and people like Eugene Fama and his followers would be provably incorrect.

Third, I am a big fan of the general approach Vinay Deolalikar used in his proof.  I can’t find a good summary online, so here goes: basically he drew from a whole bunch of fields, saw the conceptual similarities among a bunch of otherwise very narrow branches of maths, and stitched together a diverse, “multidisciplinary” proof (all while filling in the nitty gritty details, of course). I will always be a fan of the general over the specific; in general I think a broad view is more valuable than a narrow one. If a problem of epic proportions, like P=NP, can be solved by taking a step back and tying what we already know together, rather than going deeper and deeper into sub-sub-subfields that have probably been mined of intellectual value long ago, it will make me very happy.

High point

Saturday, January 23rd, 2010

Yesterday was unequivocally the high point in my graduate school career to date. The big event was our first Micro 2 class, in game theory. Micro was the only class we hadn’t had yet, and my expectations were high: Econometrics is typically dry and exceedingly difficult, and our Macro class is shaping up to be intense, courtesy of our new professor. I was hoping that Micro could be the class to keep me sane this semester.

Luca Anderlini is our professor for Micro. He’s the new Director of Graduate Studies too, so my performance in Micro serves the dual role of not failing out of the program and not embarrassing myself in front of the guy running things. I had seen him present a paper last semester, and this gave me high hopes. He had a sense of humor, an entertaining manner of lecturing, and a way of making the topics at hand seem relevant.

Let me cut to the chase: my hopes were realized. The lecture was interesting, but most important, something crucial happened, something I have been waiting my entire time at Georgetown to hear someone admit. Before Professor Anderlini got into the meat of the lecture, he made a caveat. He expressed to us, in no uncertain terms, that math is not the point of what we’re doing. While, he explained, he enjoys math a great deal, and even considered a career in math, he stressed that math is a just a tool to clarify our thinking. Anyone can reason, he argued, and make a convincing case. The key is that math is a rigorous formal language to express our ideas,  so that we can make sure we are not just deluding ourselves with words. Again, math is not the end, it is only the means.

That was the breath of fresh air I needed.

FOSS crashes economy?

Thursday, November 19th, 2009

Not to be alarmist or anything, but Free and Open Source Software (FOSS) is probably to blame in bringing the global financial system to its knees. A few months ago I came across an intriguing article in the New York Times about fat tails and gaussian copulas. It was a pretty good piece, worth at least a glance.

The interesting part begins on page six. Long story short, JPMorgan developed a way of using maths to quantify financial risk into a dollar value. Value at Risk, or VaR, as it was abbreviated, was a useful tool internal to JPMorgan. Then they did something totally bonkers: they gave VaR away. Anyone who wanted to learn and implement VaR could do it, and JPMorgan would help you out. Why would they just give away such a valuable piece of proprietary technology? This quote sums it up nicely:

As Guldimann wrote years later, “Many wondered what the bank was trying to accomplish by giving away ‘proprietary’ methodologies and lots of data, but not selling any products or services.” He continued, “It popularized a methodology and made it a market standard, and it enhanced the image of JPMorgan.”

The story ends with a score of financial firms coming to rely too heavily on VaR, then they overextend themselves, get lulled into a false sense of security, and finally fat tails come in and kick everyone’s asses. Also the economy exploded.

I can’t help but wonder if it was the tactically-superior give-it-away model of FOSS that allowed JPMorgan’s mathematical monstrosity to consume the world’s financial sector in a blaze of nihilist glory.

Why I like economics

Monday, October 19th, 2009

I’m an intuitive person: I’m more concerned with broad-reaching theories than with any particular instantiation of fact. I scored a solid N on the Myers-Briggs Type Indicator. That’s not to say I don’t like facts. On the contrary, theories need to be generalized from somewhere, and starting with the known state of the world is common sense. But I don’t operate in Sensing-land — I need to be able to clearly see the general pattern in order to understand something.

Framed this way, I like economics for the same reason I like engineering. The extensive maths used in both fields are a means of grounding one’s intuition in something substantial, something provable. Maths allow one to derive, from a set of axioms and real-world data, a bunch of consistent theorems, equations, etc. that describe the system the axioms (and data) generate. If your intuition clashes with your results, either you’re wrong or you made a math mistake. Here you have a very solid check to balance out your intuition.

This allows me to satisfy two competing objectives: first, I don’t want to spend all my time in math land; second, I don’t want my intuitions to be wildly off base. So I ponder my intuition, I learn the requisite maths, and then I can go write a bunch of equations to ensure that my intuition is correct.

The tide is high

Tuesday, September 1st, 2009

The past two weeks have been a tizzy. Moving to a new city along with six hours per day of maths all but overwhelmed my ability to keep things organized. I suppose I could have swam harder against the tide, but that’s honestly not my style. Times like those I’m happy to keep my head above water.

This week begins my normal, much more open, schedule — three classes, no more than 4hrs/day, no class before 10am, no class on Fridays. And I am going to get on top of things again, and it is going to be awesome. I have a lot of emails, a lot of blog posts in my Reader, a lot of drafts and emails and phone calls I owe people. But that’s all going to come under control soon. I know that by the end of the week I’ll be back in command. Getting Things Done always has my back.

I’m not a GTD-wizard yet, but it keeps me sane.

Malleable preferences

Tuesday, July 21st, 2009

I think a lot about peoples’ preferences and how individuals make decisions to achieve their desired ends. Microeconomic theory touches on these sorts of questions, which I think is why I’m drawn to it. But while micro theory does some good explaining, it doesn’t go the whole way. I don’t just want a descriptive framework that maybe works good enough in most cases. I want some heuristic models that I can apply to my everyday life.

So I’m going to bastardize microeconomics and use it as I see fit. Here goes:

A rational decision maker will be better off the more malleable his preferences are. Imagine a continuum of control over one’s preferences. At one end, individuals are endowed with perfectly stable, static preferences at birth. At the other end, individuals can change their preferences however they see fit, so that only if they starve to death will their utility be less than infinite.

Now, I’m pretty sure standard utility theory presupposes stable preferences. With malleable preferences, you’re essentially optimizing two things at the same time: the mix of goods and services you purchase, and how you feel about the goods and services you purchase. That would probably lead to some pretty ridiculous maths assuming a tractable solution exists.

Behavioral economics has shown us that, no, preferences are not always pre-formed and stable. For a very cool paper on this point, try reading “Tom Sawyer and the construction of value” by Ariely, Lowenstein, and Prelec. I mean, sometimes our preferences are well-defined, like if I know I dislike strawberries. But to me the interesting case is when we are in a new situation and don’t have our preferences defined. For example: Do I like guava? I’m not sure. Allow me to assume that I do. Some other behavioral economics studies have shown that you can actually frame experiences so that you are more inclined to like (or dislike) them.

Obviously if your preferences are malleable enough, you will be happy with pretty much anything you consume. On first face this may seem like a trivial point. But I think the question “What shall I choose to consume to maximize my happiness?” is one step too far. The question instead should be “How shall I best maximize my happiness?”. Utility theory should help answer that question, I think, in as rigorous a manner as possible.

I wonder if I can formalize this idea using a neoclassical utility framework.

Nerditry checklist

Thursday, July 9th, 2009

Living in a pretty little bubble for the summer, I’ve begun to miss the intellectual stimulation of being at a university surrounded by nerds. So, while I’d rather the summer not end too quickly, I am looking forward to being part of a horde of academics come Fall.

It struck me, talking to a friend from RPI, that the types of nerds I encounter at Georgetown will be markedly different from the nerds found at Rensselaer. While (I imagine) it will be easier to find econ nerds, I have a feeling that there will be far fewer techies. And, you know, internet geeks and movie buffs and mathematicians.

So I’m trying to develop a list of  “necessary topics for nerditry,” at least inasmuch as I myself define nerditry. Note that they are necessary conditions but not sufficient conditions, since there is a certain je ne sais quoi about nerds that can’t be defined. And I’m not looking for  “A nerd is X” or “A nerd does Y” either — think of this as: “if I wanted to emulate a nerd, what would I need to know?”

Nerds need to be familiar with at least one and preferably many esoteric academic topics very well. So the first condition isn’t a particular topic, it’s any topic, provided you can speak with authority and are actually interested in it. Next you need to know a bunch of trivia. Again the particular topic isn’t important, though movie quotes and internet memes are sure bets. A nerd should be comfortable with maths. If someone starts talking about sets or normal distributions or Laplace transforms, you should know what they’re saying. Maths provide a useful language for discussion, much like economics, so learning the terminology of your nerd group is vital.

I suppose all nerds are different. I wonder if a set of necessary conditions for nerditry could ever be formulated. This is harder than I had anticipated.

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This work is licensed under a Creative Commons Attribution 3.0 Unported.