Thursday, May 24, 2007

Mathematical Politics: Part 7

Expecting the Unexpected.
The “super complexity” of human beings means that they are capable of throwing up unexpected “anomalies” and by “anomalies” I don’t mean phenomena that are somehow absolutely strange, but only something not covered by our theoretical constructions. Just when you think you have trapped human behavior in an equation, out pops something not accounted for. These anomalies strike unexpectedly and expose the limits of one’s analytical imagination. They can neither be treated statistically, because they are too few of them, or analytically because the underlying matrix from which they are sourced defies simple analytical treatment.

Take the example I have already given of the supermarket check out system. This system can, for most of the time, be treated successfully using a combination of statistics, queuing theory and the assumption that shoppers are “rational and selfish” enough will look after the load-balancing problem. But there is rationality and rationality. For example, if there is a very popular till operator who spreads useful local gossip or who is simply pleasant company one might find that this operator’s queue starts to lengthen unexpectedly. The simplistic notions of self-serving and ‘rationality’ breaks down. Clearly in such a situation they is a much more subtle rationality being served. What makes it so difficult to account for is that it taps into a social context that goes far beyond what is going on in the supermarket queue. To prevent these wild cards impairing the function of the check out system (such as disproportionately long queues causing blockages) the intervention of some kind of managerial control may, from time to time, be needed.

In short, laissez faire works for some of the people for some of the time, but not for all the people all of the time.

To be continued.....

Sunday, May 13, 2007

Mathematical Politics: Part 6

Mathematical IntractabilityRandomness is a complexity upper limit – size for size nothing can be more complex than a random distribution generated by, say, the tosses of a coin. A sufficiently large random distribution configurationally embeds everything there possibly could be. And yet in spite of this complexity, it is a paradox that at the statistical level randomness is very predictable: for example, the frequency of sixes thrown by a die during a thousand throws can be predicted with high probability. In this sense randomness is as predictable as those relatively simple highly organized physical systems like a pendulum or the orbit of a comet. But in between these two extremes of simplicity and complexity there is vast domain of patterning that is termed, perhaps rather inappropriately, “chaotic”. Chaotic patterns are both organised and complex. It is this realm that is not easy to mathematicise.

We know of general mathematical schemes that generate chaos (like for example the method of generating the Mandelbrot set), but given any particular chaotic pattern finding a simple generating system is far from easy. Chaotic configurations are too complex for us to easily read out directly from them any simple mathematical scheme that might underlie them. But at the same time chaotic configurations are not complex enough to exhaustively yield to statistical description.

The very simplicity of mathematical objects ensures that they are in relatively short supply. Human mathematics is necessarily a construction kit of relatively few symbolic parts, relations and operations, and therefore relative to the vast domain of possibility, there can’t be many ways of building mathematical constructions. Ergo, this limited world of simple mathematics has no chance of covering the whole domain of possibility. The only way mathematics can deal with the world of general chaos is to either simply store data about it in compressed format or to use algorithmic schemes with very long computation times. Thus it seems that out there, there is a vast domain of pattern and object that cannot be directly or easily treated using statistics or simple analytical mathematics.

And here is the rub. For not only do humans beings naturally inhabit this mathematically intractable world but their behavior is capable of spanning the whole spectrum of complexity – from relatively simple periodic behaviour like worker-a-day routines, to random behaviour that allows operational theorists to make statistical predictions about traffic flow, through all the possibilities in between. This is Super Complexity. When you think you have mathematicised human behaviour it will come up with some anomaly....

To be continued.....

Thursday, May 03, 2007

Mathematical Politics: Part 5

The Big But
Complex system theory, when applied to human beings, can be very successful. It is an interesting fact that many measurable human phenomena, like the size of companies, wealth, Internet links, fame, size of social networks, the scale of wars, etc are distributed according to relatively simple mathematical laws - laws that are qualitatively expressed in quips like “the big get bigger” and “the rich get richer”. It is an interesting fact that the law governing the distribution of, say, the size of social networks has a similar form to the distribution of the size of craters on the moon. It is difficult to credit this given that the objects creating social networks (namely human beings) are far more complex than the simple elements and compounds that have coagulated to produce the meteors that have struck the moon. On the other hand, there is an upper limit to complexity: complexity can not get any more complex than randomness and so once a process like meteor formation is complex enough to generate randomness, human behavior in all its sophistication cannot then exceed this mathematical upper limit.

The first episode of the “The Trap”, screened on BBC2 on 11th March 2007, described the application of games theory to the cold war (a special case of complex system theory). The program took a generally sceptical view (rightly in my opinion) of the rather simplistic notions of human nature employed as the ground assumptions in order that games theory and the like are applicable to humanity. To support this contention the broadcast interviewed John Nash (he of “Beautiful Mind” and “Nash Equilibrium” fame – pictured) who admitted that his contributions to games theory were developed in the heat of a paranoid view of human beings (perhaps influenced by his paranoid schizophrenia). He also affirmed that in his view human beings are more complex than the self-serving conniving agents assumed by these theories.

Like all applications of mathematical theory to real world situations there are assumptions that have to be made to connect that world to the mathematical models. Alas, human behaviour does, from time to time, transcend these models and so in one sense it seems that human beings are more complex than complex. But how can this be?

To be continued....