The Quandaries of Quantifying Complexity

My good friend and computer scientest Kyle Burke has recently started a highly interesting blog on his research field: combinatorial game theory. The idea of this field is to use games as a tool for studying issues of complexity. Though his blog is only a month old, some important foundational ideas have begun to rear their heads, one of which I'll explore in this post.

Understanding complexity is important for almost any human endeavor, but defining it in rigorous terms is notoriously difficult. For example, which is the more complicated game, chess or tic-tac-toe? Almost anyone would say chess, but suppose you had a computer that was designed only to play chess. In fact, this computer has no capacity for calculation; it simply has the best move for any given chess position hardwired into its architecture. To get this computer to play tic-tac-toe, you would have to program it to translate each tic-tac-toe position into an analagous chess position, so it could then find the best chess move and translate this move back into tic-tac-toe. This computer would certainly find chess an easier game to play.

Computer scientists have a way around this paradox: instead of looking at individual games or problems, they look at classes of problems. Each problem in the class has a certain size, and they look at how complexity increases in relation to size.

For example, you could easily imagine playing tic-tac-toe on boards of various sizes. Computer scientists can analyze how the complexity of tic-tac-toe varies with the size of the board. (Chess, on the other hand, doesn't generalize as easily to larger sizes, which makes it difficult to talk about its complexity.)

Unfortunately, if we are faced with a real-world issue (such as how to provide for the needs of a large population), we will want to know the complexity of the specific problem at hand, not how the complexity might theoretically scale with problem size. Part of the reason that complexity issues are so often ignored (to the detriment of many well-meaning policies and programs) is that defining and quantifying complexity is so unavoidably slippery.

Further reading

The Criminalization of Poverty

Barbara Ehrenreich had an excellent article in yesterday's New York Times on the many ways that being poor can land you in trouble with the law. One striking example:

In just the past few months, a growing number of cities have taken to ticketing and sometimes handcuffing teenagers found on the streets during school hours.

In Los Angeles, the fine for truancy is $250; in Dallas, it can be as much as $500 — crushing amounts for people living near the poverty level. According to the Los Angeles Bus Riders Union, an advocacy group, 12,000 students were ticketed for truancy in 2008.

Why does the Bus Riders Union care? Because it estimates that 80 percent of the “truants,” especially those who are black or Latino, are merely late for school, thanks to the way that over-filled buses whiz by them without stopping. I met people in Los Angeles who told me they keep their children home if there’s the slightest chance of their being late. It’s an ingenious anti-truancy policy that discourages parents from sending their youngsters to school.

The column was based on a report by the National Law Center on Homelessness and Poverty, which finds that the number of ordinances passed and tickets issued for crimes related to poverty has grown since 2006.

Hey, no one likes poverty, right? Let's pass a law!

The Evolution of Bad Ideas

It is by now common wisdom that our current financial crisis is due in large part to misplaced incentives in our financial system. Analysts and fund managers were rewarded for short-term thinking and risk-taking. If we can rework our financial system to reward long-term, careful planning, it is often argued, we can avoid collapses like this in the future.

While I agree that misplaced incentives were a fundamental problem, the question of how to change this is rather more deep and complex than I think many people realize.

Our economy is, of course, an evolutionary system. Successful businesses grow in size and their practices are imitated by others; unsuccessful businesses vanish. This process has led to many good business practices, even in the financial sector.

However, evolution does not always yield the best outcomes, in biology or in economics. Our recent crisis illustrates two key limitations of evolutionary systems, limitations which allow bad ideas to evolve over good ones.

The first problem has to do with time lags. Suppose Financial Company A comes up with an idea that will yield huge sums of money for five years and then drive the company to bankruptcy. They implement the idea, obfuscating the downside, and soon the company is rolling in cash. Investors line up to give them money, magazines laud them, and other companies begin imitating them.

Not so Company B. Company B believes in long-term thinking, and can see this idea for the sham it is. They persue a quiet, sound strategy, even when their investors begin pulling money out to invest in A.

We would like to think that in the end, Company B will be left standing and reap them benefits of their foresight. But there is a fundamental problem of time-scales here: by the time A folds, B may already be out of business, due to lack of interest from investors. In theoretical terms, there is a fundamental problem when the evolutionary process proceeds faster than the unfolding of negative consequences. In these situations, good ideas never have a chance to be rewarded, evolutionarily speaking.

One might argue that investors, not to mention government regulators and ratings agencies, should have forseen the flaw in A's plan. But this highlights a second limitation of the evolutionary process: it favors complexity. Simple bad ideas can be detected by intelligent agents, but complex ones have a chance to really stick. If Company A's idea was so complicated that no one aside from a few physicists could figure it out, investors and regulators could easily be fooled.

It's not clear to me how to patch these flaws in the evolutionary system. Increased transparency and oversight will help, but unless we can somehow cap the complexity of financial instruments (difficult) or slow down the evolutionary process (impossible), I'm not sure how we'll avoid similar crashes in the future.