### Is There a There There?

The question of whether there is a real "there" for law and complexity theory was raised by my mathematician friend Paul Edelman. In a way, it's also touched upon by J.B.'s most recent posting.

Here's the question that Paul raised:

I get more and more frustrated reading about the supposed implications of CAS to law when no one will be precise about any of the definitions. I suppose it is just the mathematician in me, but until I see some specification of something that can be analyzed in technical way I remain skeptical that there is any there there.This is obviously a very cogent and significant question -- as you would expect from the world's only joint appointment in mathematics and law.

Maybe you can help me in this regard by elaborating on your second post. You note that, as an empirical matter, that the probability of a quake given as a function of its intensity is best modeled by a power law. OK. You then go on to say that this makes a difference in the analysis of what to do in preparation. Why? The distribution is what it is. We can compute the expected damages under assorted hypotheses and do the cost-benefit analysis. What is it about power laws that make any difference to how we think about the question? It may affect the computation, but is there some independent significance to the realization that the power law applies?

You seem to indicate that people underestimate the expected damage because power laws have longer tails. But people are notoriously bad at estimating probabilities even when they are governed by binomial distributions. Are they that much worse in this circumstance. How about massive floods? Are they governed by power law or poisson? I don't know, but I don't know that we are any better there. How about nuclear power disasters? Power law or Poisson? And what about the distribution of hurricanes (whatever that means?) Is it power law or Poisson? If it is Poisson does that mean that everything is now OK with respect to hurricane preparedness?

So what work is all this CAS/power law distribution doing? We are bad at making the correct estimates. We should do better. If we know exactly what the distribution is then maybe we can do a better job, whatever that distribution looks like. What am I missing here?

Rather than tackling the larger question, let me explain why I think power laws specifically are relevant. To begin with, a lot of economic analysis of the kind used in environmental regulation assumes that variances are a second-order consideration -- the main issue is expected loss. For power distributions, variances can be very large compared with means, so risk aversion issues loom large. I was shocked by the very idea that probability distribution could have an infinite variance; I don't have any reason to think that this is an empirically important case, but it does make you realize that variance issues can be critical in risk assessment.

Second, rather than consider the full distribution, analysts often use what they consider to be the median estimate of loss (for example, by picking what they think is the central estimate of risk based on various studies). You see this all the time in risk analysis, where it is also often paired with criticism of EPA for being conservative in its risk estimates. Power laws typically have a mean that's a lot higher than the median, however. So , assuming that we want to use a point estimate in the analysis because using the full distribution would be too unmanageable, we may want to use much more conservative risk estimates even if we know that the risk is probably (more than 50% of the time) going to turn out lower.

In addition, the high degree of variance and the "fat" right tail of the power distribution have some implications for institutional design. In designing institutions, we can't just plan on the typical incident and then add a small allowance for the unexpected. Instead, we have to assume that outcomes may be much worse than "typical."

Finally, to get away from power laws for a minute, the "butterfly effect" aspect of complex systems has led to some interesting rethinking of ecosystem management. J.B. and others, such as Brad Karkkainen, have discussed the need for managers to monitor and adapt to system changes rather than trying to identify an equilibrium strategy.

At the end of the day, will these turn out to be important insights? I am tempted to paraphrase Keynes and say that at the end of the day we are all dead. In the near term, however, I think it's worth making a small investment in the project of applying complexity theory to law.

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