Tuesday, March 20, 2007

Economics, complexity and feedback

There are two fundamental reasons why planned economies in general (with a few specific exceptions) are a Very Bad Idea.

The first reason is that whoever plans the economy is also part of the economy. They're going to always plan the economy to favor what they like in general, and all sorts of mechanisms are necessary to keep them from simply redefining the economy to give themselves all the wealth and turn the rest of us into slaves. From the planners' personal point of view, that's the most efficient economy possible.

The second reason is more severe. Even if we could somehow guarantee the absolute personal disinterest of the planners, there's simply too much information to ever competently manage in a central location.

We can see a fair analog in the way that the internet works: There is no single central location which receives every single packet and sends it to its correct physical location. Rather, all the routing information is distributed into millions of individual routers, each of which makes part of the decision of where to send a packet. There are no small few technical details that are suboptimal, but the overall design really is a work of genius. The internet has become several (6ish?) orders of magnitude larger than its original design.

The problem is that the routing of each packet is arbitrary: There is no a priori way to determine to whom (and from whom) any particular user wishes to send a piece of data. That means that the complexity of the routing system is factorial[1]. Any time you have polynomial or greater complexity, you benefit to some degree from partitioning, breaking up the problem more-or-less arbitrarily into two two problems and then solving them separately: nk + mk < (n + m)k. There's always a cost to actually creating the partition, but usually partitioning can be done in linear time.

In just the same way, the determination of value (as opposed to actual cost and actual price), the sine qua non of economics, is a factorial-complexity problem, because it depends on each person's arbitrary, subjective evaluation. Every time a person buys, for example, a DVD player, he is making a subjective decision that the value of the DVD player is at least a little higher than its price. Furthermore, every time a person doesn't buy a DVD player, he is making a subjective decision that the value of the DVD player is at least a little lower. The value of a particular DVD player is a statistical function of all of these individual, arbitrary votes, for and against.

Value is not only subjectively relative, it's also internally relative: We can't just put a number on the value of an individual item, we must always (at the end of the day) express the value of something relative to the value of something else: the value of a potato is less than the value of a DVD player. It doesn't mean anything to say with no context that a DVD player has a value of \$89.95: We must compare that dollar value to the dollar value of a potato, or a 60" TV, or a romantic dinner with one's spouse to get any meaningful sense of the actual value of the DVD player.

So a planned economy is right out. It's actually computationally infeasible for entire human race to analytically solve the value problem even for a relatively small population and a relatively small number of items. The only way to manage an economy is to distribute the decision making.

The above argument, though, does not entail that an absolute laissez-faire economy is necessarily the correct answer.

Although we must partition the economic problem, the partitioning itself is less complex than the economy as a whole, and a good partitioning can make the overall problem many orders of magnitude more efficient (good partitioning and other sorts of lesser-complexity optimizations can yield five or more orders of magnitude more efficiency in computing very complex functions such as the Ackermann function). While central planning is impossible, central optimization is not only possible but desirable.

More importantly, economics is dynamic. Every economic transaction not only is a vote to establish or determine a value, but also a request to change the value of something. When a lot of people buy DVD players, they not only set the value, but they are ipso facto requesting that the cost be reduced, which changes the relative value of the DVD player. It's a bit counter-intuitive, since setting a high value on something entails a request to lower that value, but that's the basic story of the law of supply and demand.

Any time you have any sort of dynamic process or system, you have two ways the system changes: positive feedback and negative feedback. You can get a quick example of positive feedback by putting your microphone next to your speakers and listening to the ensuing squeal. Your thermostat is an example of negative feedback: When the room gets too cold, the thermostat turns on the furnace, making the room hotter. When the room gets too hot, the thermostat turns off the furnace, making the room colder.

Any dynamic system which has only positive feedback will quickly explode, and pure individual material selfishness is a positive feedback mechanism. In any finite economy, the rich get richer and the poor get poorer, until (absent other mechanisms, such as violent revolution) one individual has all the wealth and everyone else is his or her slave. If an economy is to have any other outcome, we need to impose negative feedback mechanisms (e.g. the graduated income tax and monetary inflation) by non-economic (i.e. persuasive or coercive, usually coercive) means.

The classical liberal political-economic system, where individuals and more-or-less private companies set value on a distributed, free-market basis; and the government addresses itself to partition optimization and negative feedback mechanisms, is plausibly optimal.

[1] The order of complexity goes from constant (simplest) to linear, logarithmic, polynomial, exponential to factorial (more complex); in O-notation, where n is the size of the problem and O(n) is the time (or cost) to solve that problem (c and k are constants): O(c), O(n), O(n*log(n)), O(nk), O(kn), O(n!). There are even greater orders of complexity, found in such as Ramsey theory.

1. My brain broke at "factorial."

I've long been of the opinion that regulation and state control isn't the answer to the ills of capitalism, but a government empowered to monitor and be punitive when necessary. Activist government needn't be a micro-manager. I'm a big fan of some of Arend Lijphart's work on how democratic governments best function.

Economics was more honest in the 18th century when it was "political-economy."

2. The factorial of n (n!) is 1 * 2 * 3 * ... * n. It's basically the number of ways you can shuffle a deck of n cards.

There are two ways of shuffling a 2 card deck. There are 24 ways of shuffling just the 4 aces. There are about 6 thousand million ways of shuffling just one suit of 13 cards. There are 80 million million million million million million million million million million million ways of shuffling a full 52 card deck.

Just to put these sorts of numbers in perspective, if you throw in the two jokers, there are more ways of shuffling a 54 deck of cards than there would be subatomic particles if every subatomic particle were a universe the size of our own.

The cost of analytically computing the "correct" price of a loaf of bread in a closed society of 54 people is about of this order of complexity.

3. Ahhhhhhh! Algebra! Brain... shutting... down... Must... return... to social sciences!

4. Good grief, James. Algebra is not only not that hard, it's the language of reality. It's just as important to be numerate as it is to be literate.

5. I'm kidding. I'm fine with most lower-order mathematics, and I do a lot of statistics in my work. But calculus... I failed it twice. It was like taking a course in ancient Greek without a translator. I simply couldn't get it.

6. Factorials are fundamental in statistics.

Calculus is easy: It's all about how to multiply and divide by zero and/or infinity. Nothing to it!

But complexity theory is just straight algebra, not even statistics.

7. Yes, I have Post-Mathematics Stress Disorder. You should see how I react when someone says "binomial."

Please pick a handle or moniker for your comment. It's much easier to address someone by a name or pseudonym than simply "hey you". I have the option of requiring a "hard" identity, but I don't want to turn that on... yet.

With few exceptions, I will not respond or reply to anonymous comments, and I may delete them. I keep a copy of all comments; if you want the text of your comment to repost with something vaguely resembling an identity, email me.

No spam, pr0n, commercial advertising, insanity, lies, repetition or off-topic comments. Creationists, Global Warming deniers, anti-vaxers, Randians, and Libertarians are automatically presumed to be idiots; Christians and Muslims might get the benefit of the doubt, if I'm in a good mood.

See the Debate Flowchart for some basic rules.

Sourced factual corrections are always published and acknowledged.

I will respond or not respond to comments as the mood takes me. See my latest comment policy for details. I am not a pseudonomous-American: my real name is Larry.

Comments may be moderated from time to time. When I do moderate comments, anonymous comments are far more likely to be rejected.