# A smoking gun linking Economics to Physics?

1. Apr 26, 2008

### Dcase

http://en.wikipedia.org/wiki/Pareto_distribution

Economist Vilfredo Pareto in about 1906 developed the Pareto principle [80-20] later expanded to Pareto distribution, a probability density function.

Mathematician John Nash developed Noncooperative Game Theory [NCGT] with Equilibria using Pareto optimality as one of four criteria. NCGT is employed in economics, engineering, biology and social sciences.

Imagine my surprise when I read “... as k -> oo the distribution approaches ... the Dirac delta function”.
There is a demonstration graph for k=1,2,3,oo.

The Dirac delta is related to the Kronecker delta.

This may usher in an era of economical physics that may not only lead to GUT / TOE, but unification of various Nobel Prize categories?

Statistical economics, statistical physics [mechanics], statistical GUT / TOE could become a dynamic mathematical field of study?
See AMS 2000 Mathematics Subject Classification, 37-xx Dynamical systems and ergodic theory and related topics.
http://www.ams.org/msc/

2. Apr 26, 2008

### BenTheMan

Well, it's all based on differential equations.

For example, a friend of mine now works in finance, after getting his PhD in theoretical physics. He showed me a (highly simplified) example of the type of problem that he works on.

Suppose you want to buy a stock---a LOT of the stock. Once you start buying the stock, you increase the demand for the stock. When this happens, the price increases, so that you have to pay more for each additional share of stock. The question is, what is the optimum rate for you to buy the stock, so that you maximize your holdings, but minimize your expenditures?

If you are a physicist, you can do this problem in a few minutes, because it's the type of problem that we solve all the time. (Bonus points to anyone who can solve it!) So there is no real smoking gun''---it's something that people have realized all along. Note also that some hedge funds were started by physicists or mathematicians, and the whole concept of hedge fund was INVENTED by a mathematician. (Trivia: He was also the first guy to crack'' the game of blackjack.)

3. Apr 27, 2008

### shoehorn

Not true. One of the fundamental assumptions of mathematical finance is that the market you're trying to model is liquid; in other words, you can enter and leave (buy and sell) the market at any time, and your actions in the market don't affect prices. Without this assumption you lose vast swathes of the results that comprise mathematical finance including, for example, everything based on the Black-Scholes model.

So, regardless of whether you're a physicist or not, you can't "do this problem in a few minutes."

To the OP, lots of interesting work being done on the interface between theoretical physics and economics is based on contact geometry, the odd-dimensional analogue to symplectic geometry. Unfortunately, as with all such things, detailed results rarely get published, meaning you've often got to rely on person-to-person contact to find out what the current state of the art is. If you're interested though, there's a very readable (non-contact geometry) exposition of the relation between gauge theory and mathematical finance in Ilinski's Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing. Lots (most?) of his results are entirely non-rigorous, but it's a pretty fun read.

4. Apr 27, 2008

### BenTheMan

Well, I disagree. I didn't say anything about external forces in the problem. In fact, the external forces are probably why my friend gets paid so much.

I was just pointing out an idealized situation where one could apply a simple set of differential equations. If you want to include some liquidity in the whole thing, then surely the modeling is more difficult.

5. Apr 27, 2008

### shoehorn

With respect, you don't sound as though you quite understand what I'm trying to say. The entire point is that essentially everything we know about modeling financial markets is based on the assumption that the markets are liquid and that our actions don't move the market. Once you throw out the assumption of not moving the market you can't predict a thing with any reasonable level of confidence. You can't predict spot prices, you can't predict payoffs, and you can't predict whether any options or futures you hold will be in-the-money.

Essentially, being able to accurately model the effects that your actions have on a market is the holy grail of mathematical finance, rather like quantum gravity is in physics. That's why I'm disputing the claim that a physicist can "do this problem in a couple of minutes"; it's obvious that this is untrue if you've ever spent time studying math finance or working back-office at a trading desk.

6. Apr 27, 2008

### BenTheMan

Ok---I didn't mean to trivialize a whole field, which is aparently what I've done. Apologies around.

I should have been more explicit about the problem that I was talking about.

7. Apr 27, 2008

### Fra

I for one very much appreciate the conceptual game theoretic angle to physics, it seems to merge well conceptually with the information view. The concept of "players" basing their actions on to them available information, is at least for me personally one of the better abstractions and source of intuition for reflecting over fundamental physics I've found.

Could the actions of a physical system interacting with the environment, be understood as a kind of rational decision maker? Ie. could it be that the laws of physics does actually share similarities with what one would expect from rational actions? I think this is a deep an interesting question. However it seems like the concept of what is a rational action and what is irrational is hard to defined objectively. It seems to be a relative thing. I think this type of reasoning of relative actions, and emergent rationality connects remotely to some of the ponderings on evolutionary ideas in physics.

I definitely think there is a deeper connection here, on the level of the foundations of statistics and probability theory applied to a scientific method. And I expect more interesting connections at this level in the future. I think the connection goes far beyond beeing based on differential equations. The game perspective can apply well do discrete models as well, and applies not only to a theory but also to theory building - the game of building theories.

/Fredrik

8. Apr 27, 2008

### shoehorn

The difficulty is that the "rational participant" approach to markets (and to economics more generally) that was so popular during the sixties/seventies has now thoroughly been discredited. The simple fact is that participants in a real market do not behave rationally. For a striking example of this, remember that it was long feared that the price of oil would go over $100 a barrel. In fact, this didn't actually happen for a long, long time; in the jargon of traders and economists,$100 was a very strong point of "resistance" for the spot price of oil. However, back at the start of January a single trader went into work determined that he would be the first person to pay over \$100 for a barrel of oil, so he bid significantly above the asking price on a very small contract in order to ensure that he was paying above the magic one hundred dollar level.

Why did he do this? Vanity. He wanted to be able to tell his grandkids that he was the first man to pay more than a hundred dollars for a barrel of oil. Behaviour based on vanity, pride, and so on, is hugely common in all markets and cannot, in my opinion, be modeled accurately.

From my perspective some of the most interesting connections between physics and "real world" systems like markets can be found in information geometry. Vast amounts of standard differential geometry can be applied to developing geometries of probability distributions; indeed, you see some beautiful geometry associated even with very simple probability distributions. And the links with physics are myriad: renormalization group flows in QFTs, equilibrium and non-equilibrium thermodynamics on contact manifolds, and so on. In particular, black hole thermodynamics (and hence black hole mechanics) becomes beautiful when it's described using contact forms and contact manifolds.

9. Apr 28, 2008

### Fra

This is certainly true! and it IMO connects to the fact that who is the judge to determined what is rational and what is not? IMO, I see no way of defining a certain universal objective measure of rationality, but there can still be an emergent effective measure.

Also in reality, everbody are players, there are no referees. So you don't even have the game defined, so part of the "game" is to try to figure the game itself out. I associate this to background independence - there is no given universal objective background (IMHO).

I didn't mean to say that classical game theory solves are problems. I meant to say only that the perspective of game theory to physics can be, to me at least, a very interesting and rich source of intuition.

I agree. Information geometry is an interesting area, although I am far from sure I like to take the the continuum approach as a starting point. I see it as the limit of a combinatorical approach. But IMO that angle can work fine in parallell to the conceptual game theoretic angle.

Regarding rationality measure itself I also see this as beeing dependent on confidence. Irrational decisions that does not have much negative effects really isn't that irrational after all. This is allowed if you see rationality is indeterministically emergent, rather than exact. Rationality seems to be more like tha expectation value of the actions, but I don't think it's so to speak "rational" to expect zero variation.

/Fredrik

10. Apr 28, 2008

### Micha

It is not exactly that you did trivialize a field. It is that parts of mathematical finance can be called a pseudo science, pretending an accuracy which is not there.

In physics you can talk about the free fall without friction, it makes sense. You can do even experiments by letting something fall in the vacuum.

In mathematical finance, friction, which is subjectiveness of the actors, is at the core of the system.

"The alchemy of finance" by George Soros makes a good read.

11. Apr 28, 2008

### Dcase

Hi shoehorn, Fra and BenTheMan,

The 2001, 300 page reference to Kirill Ilinski,
‘Physics of finance: gauge modeling in non-equilibrium pricing’
is very intriguing.
I intend to add it to my reading list, but probably will not get to it until June.

I could not do this on Google books, although this text is listed.

The most interesting entry was the last in the appendix on the Ito Lemma from his Stochastic Calculus used across many physical and social science disciplines.
Kiyoshi Ito in 2006 was awarded the initial Carl Friedrich Gauss Prize for Applications of Mathematics, granted jointly by the International Mathematical Union and German Mathematical Union.
Ito essentially expanded upon Brownian motion, moving from the randomness of Einstein to a probability model.

Ito Calculus
http://en.wikipedia.org/wiki/Itō_calculus

12. Apr 29, 2008

### Fra

What I like about these associations is not to use off-the-shelf game theory to explain physics, or use off-the-shelf physical theories to explain economics. It's more the observation that whatever what you are trying to learn something about, physical systems or economical systems, there are some deep common denominators on howto proceed, that touch the foundations also of statistics and probability theory. After all history is shared. Von neumann made formalisations of both game theory as well as quantum mechanics.

This may be interesting for serveral reasons, it seems to me that it's a alternative possible source of funding for fundamental research. Because fundamental research in physics, economy and general scientific methodology has sufficient in common that cross funding seems to me at least to be a possibility.

The other aspect is the source of intuition and the source of observational testing. Is the only possible chance to test new theories, to construct larger accelerators? Or can we look at other systems in nature (ie. large complex system), and test our theories here? Could tuning the models there, help making predictions on other areas?

/Fredrik

13. May 5, 2008

### Dcase

Hi Fra,

John von Neumann also made an important contribution to Ergodic Therory:
"mean ergodic theorem, holds in Hilbert spaces"
and
the other primary, initial contributor George David Birkhoff:
Ergodic theorem, in part;
"... a measure-preserving transformation on a measure space ...",
"... One may then consider the "time average" of a well-behaved function ...",
"... The "time average" is defined as the average (if it exists) over iterations ...",
"... we can consider the "space average" or "phase average" ...",
and
"In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere."
http://en.wikipedia.org/wiki/Ergodic_theory

This topic has been expanded in the AMS 2000 Mathematics Subject Classification as
37-xx Dynamical systems and ergodic theory and related topics.
Subcategory 37Bxx Topological dynamics even links Cellular automata of John Conway game of life and Stephen Wolfram New Kind of Physics.

14. May 7, 2008

### Dcase

I obtained a copy of the book recommended by shoehorn through the interlibrary loan program sooner than I expected.

Kirill Ilinski, ‘Physics of Finance’ gets to the mathematical relationship of economics and physics on p3-5.

1900, Loius Bachelier, Sorbonne, mathematical thesis, ‘The Theory of Speculation’ has this function: delta_S=sigma*sqrt(t).

1905, Albert Einstein, "On the Motion Required by the Molecular Kinetic Theory of Heat of Small Particles Suspended in a Stationary Liquid", Annalen der Physik 17: 549-560, has this function: delta_x=D*sqrt(t).

Ilinski comments: “We can add the underlying mathematics is similar too, if not identical.”

Illinski is apparently now in finance;
PhD physics, St Petersburg;
Post Doctorate Research Fellow, School of Physics and Space Research, Birmingham;
Senior Researcher, Institute of Spectroscopy, Russian Academy of Sciences, Moscow;
credited with 26 ArXiv papers.
http://arxiv.org/find/cond-mat/1/au:+Ilinski_K/0/1/0/all/0/1