Tuesday, September 06, 2011

local minima and maxima

I'm sure it counts as trite to mention that humans aren't great at coping with complexity. (Computers can but only if the complexity doesn't require adaptability or comprehension.) One example is the oversimplified dichotomy between systems: 1) few pieces with highly organized interconnections and controlled variances among the pieces, 2) numerous similar pieces with little oversight that nevertheless mostly act the same and have few decisions to make. An engine is in #1. An ant colony is in #2. A projectile is in #1. A contained cloud of gas particles is in #2. In #1 systems, analysis is rather easy because all the pieces are ordered to accomplish parts or stages of a defined objective. In #2 systems, analysis is rather easy because the actions of all the pieces are generalizable into "overall/average forces". In #1 systems, statistics consist of a series of well-determined numbers. In #2 systems, variances and aggregates are tameable by modeling the population distribution.

The problem is that as useful as these two categories are, reality can often be more complicated. Loosely-connected systems could consist of many unlike pieces. Or the pieces could be alike yet affected in a nonuniform manner by an external disturbance. Or each piece could individually respond to five factors in its immediate neighbor pieces. Or pieces might have ephemeral subgroups which act in ways that loners don't. The possibilities are abundant and stymie attempts to classify the system as #1 or #2.

Consider a minimum or maximum quantity that represents a system. In a #1, that quantity is calculable directly by finding the corresponding quantities for each piece. In a #2, that quantity is an equilibrium that all the pieces yield through collective activity. Either way, the system has one minimum or maximum, and it's reached predictably.

However, this conclusion breaks down when a system is of a "more complicated" kind. Those systems contain pieces that, taken one at a time, are easily understood, but whose final effect is difficult to fathom. As a representation of that system, the minima and maxima could be messy. For instance, a specific constraint is strongest at the lower end of a range but a second constraint is strongest at the higher end. Under such circumstances, the system has more than one minimum or maximum. To the extent that the description works, the "forces" of the system then push toward the local minimum or maximum, whichever is closest.

From the viewpoint of an uninformed observer trying to cram a complex system into #1 or #2, the apparent failure to reach the absolute furthest (global) maximum could be mystifying. If it's caught in the grip of a local maximum, then the failure is more intelligible. The system "rejects" small changes that result in an immediate "worse" outcome regardless of whether or not it's on the path to an ultimately "better" outcome. In short, a wildly intricate system occasionally gets stuck in a pothole of inferiority. And for that system, that state is as natural as anything else.

Hence knowledge of local minima and maxima provides greater nuance to human interpretation. Reasoning about the national economy is a ripe area. The temptation is to reduce discussion into the relative merits of a #1 system, in which the economy is like a tightly-directed machine operated by government, compared to a #2 system, in which the economy is like a spontaneous clump of microscopic participants. This is a discussion about nonexistent options. The economy isn't solely a dangerous beast that needs strict supervision. It isn't solely a genie that showers gifts on anyone who sets it free. It's beyond these metaphors altogether.

An economy that has run aground on local minima or maxima can't be adjusted successfully by treating it as a #1 or a #2 system. "Freeing" it won't erase the slope toward the local minimum. "Ordering" it to stop misbehaving also won't. The economy doesn't always accomplish every desired purpose. On the other hand, government can't completely override the economic transactions of the entire populace (nor should it try). What government can do, potentially, is help nudge the economy out of a local minimum by bending the system. Of course, the attendant risk is that excessive bending by the government might set up a new local minimum in the economic system...

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