Wednesday, November 12, 2008

Bang-Bang You're Complex

In many economic problems, feedback loops can be used to optimally control a system. For example, the inputs to an economic system might be government spending, taxes, and the federal funds rate, and, given values for these variables the system will produce outcomes for other variables in the system such as output and prices. The goal in these problems is to find a feedback rule for changing the policy variables that best achieves policy objectives.

Take a simpler problem where the system has just one input, the federal funds rate, and one outcome, the level of output. Given a goal such as minimizing the variability of output around a target value, the objective is to find a rule for setting the federal funds rate that minimizes variance in output. That is, if the policy rule is linear and of the form ff = a + b(y-y*), i.e. a Taylor rule, then the solution finds the values of a and b that minimize the loss function (which in this case is the variance of y). The mathematics is very similar to what an engineer might use to control a system with a feedback rule, except that in economics you have to consider how the economic actors respond to actual or expected changes in the feedback policy rule, and that turns out to be much more than a minor complication.

In some classes of these problems, the solution is known as "bang-bang". With bang-bang solutions, the control variable takes just two values, e.g. all the way on or all the way off. The heater in your home probably controls the room temperature with a solution of this form (though it may not be optimal - it depends). The heater is either all the way on or all the way off depending upon the room temperature, and this moderates the variation in the room temperature. The outcome - the temperature - is monitored, and there is a feedback rule that turns the heater on or off when the deviation from the targeted temperature exceeds some set amount. Such solutions are quite common (see here for technical details).

So common, in fact, that according to this research evolution has also discovered bang-bang solutions, and it uses them to regulate evolutionary change, i.e. to "correct any imbalance imposed on them through artificial mutations..." This allows the system to self-correct when random mutations throw it off course, and helps to explain how organisms can "be so exquisitely complex, if evolution is completely random":

Evolution's new wrinkle, EurekAlert: A team of Princeton University scientists has discovered that chains of proteins found in most living organisms act like adaptive machines, possessing the ability to control their own evolution.

The research, which appears to offer evidence of a hidden mechanism guiding the way biological organisms respond to the forces of natural selection, provides a new perspective on evolution, the scientists said.

The researchers ... made the discovery while carrying out experiments on proteins... A mathematical analysis of the experiments showed that the proteins themselves acted to correct any imbalance imposed on them through artificial mutations and restored the chain to working order.

"The discovery answers an age-old question that has puzzled biologists since the time of Darwin: How can organisms be so exquisitely complex, if evolution is completely random, operating like a 'blind watchmaker'?" said Chakrabarti... "Our new theory extends Darwin's model, demonstrating how organisms can subtly direct aspects of their own evolution to create order out of randomness."

The work also confirms an idea first floated in an 1858 essay by Alfred Wallace, who along with Charles Darwin co-discovered the theory of evolution. Wallace had suspected that certain systems undergoing natural selection can adjust their evolutionary course in a manner "exactly like that of the centrifugal governor of the steam engine, which checks and corrects any irregularities almost before they become evident." In Wallace's time, the steam engine operating with a centrifugal governor was one of the only examples of what is now referred to as feedback control. Examples abound, however, in modern technology, including cruise control in autos and thermostats in homes and offices.

The research, published in a recent edition of Physical Review Letters, provides corroborating data, Rabitz said, for Wallace's idea. "What we have found is that certain kinds of biological structures exist that are able to steer the process of evolution toward improved fitness," said Rabitz... "The data just jumps off the page and implies we all have this wonderful piece of machinery inside that's responding optimally to evolutionary pressure."

The authors sought to identify the underlying cause for this self-correcting behavior in the observed protein chains. Standard evolutionary theory offered no clues. Applying the concepts of control theory, a body of knowledge that deals with the behavior of dynamical systems, the researchers concluded that this self-correcting behavior could only be possible if, during the early stages of evolution, the proteins had developed a self-regulating mechanism, analogous to a car's cruise control or a home's thermostat, allowing them to fine-tune and control their subsequent evolution. The scientists are working on formulating a new general theory based on this finding they are calling "evolutionary control." ...

Evolution, the central theory of modern biology, is regarded as a gradual change in the genetic makeup of a population over time. It is a continuing process of change, forced by what Wallace and Darwin, his more famous colleague, called "natural selection." In this process, species evolve because of random mutations and selection by environmental stresses. Unlike Darwin, Wallace conjectured that species themselves may develop the capacity to respond optimally to evolutionary stresses. Until this work, evidence for the conjecture was lacking.

The experiments, conducted in Princeton's Frick Laboratory, focused on a complex of proteins located in the mitochondria, the powerhouses of the cell. A chain of proteins, forming a type of bucket brigade, ferries high-energy electrons across the mitrochondrial membrane. This metabolic process creates ATP, the energy currency of life.

Various researchers working over the past decade, including some at Princeton like George McClendon, now at Duke University, and Stacey Springs, now at the Massachusetts Institute of Technology, fleshed out the workings of these proteins, finding that they were often turned on to the "maximum" position, operating at full tilt, or at the lowest possible energy level.

Chakrabarti and Rabitz analyzed these observations of the proteins' behavior from a mathematical standpoint, concluding that it would be statistically impossible for this self-correcting behavior to be random, and demonstrating that the observed result is precisely that predicted by the equations of control theory. By operating only at extremes, referred to in control theory as "bang-bang extremization," the proteins were exhibiting behavior consistent with a system managing itself optimally under evolution. ... "Control theory offers a direct explanation for an otherwise perplexing observation and indicates that evolution is operating according to principles that every engineer knows."

The scientists do not know how the cellular machinery guiding this process may have originated, but they emphatically said it does not buttress the case for intelligent design, a controversial notion that posits the existence of a creator responsible for complexity in nature. ...

Posted by on Wednesday, November 12, 2008 at 12:24 AM in Economics, Science | Permalink  TrackBack (0)  Comments (29)