Subject: Re: Metastable oscillators
From: "Benjamin J. Tilly " <ben_tilly@operamail.com>
Date: Thu, 27 Feb 2003 21:10:30 +0500

This conversation is both interesting and way OT for
fsb.  I am therefore somewhat hesitant to continue it
here indefinitely.  I have decided to - partly because
I don't see it going on for much longer, but if people
object privately to me, I am just as happy taking it
private.

So if you don't want to see this kind of digression,
just ask...

"Stephen J. Turnbull" <stephen@xemacs.org> wrote:
> >>>>> "ben" == Benjamin J Tilly <" <ben_tilly@operamail.com>> writes:
> 
>     ben> Rich Bodo <rsb@ostel.com> wrote:
> 
>     >> Many robots and aircraft, like pencils on their tips, hold
>     >> themselves in unstable positions during operation.  They remain
>     >> unstable, but their inherent instability is handled by fast
>     >> feedback systems.
> 
>     ben> Note that the feedback systems convert the overall system to
>     ben> one which satisfies the description of Le Chatelier's
>     ben> Principle.  Therefore it appears unstable, but is not.
> 
> That assumes that the the feedback system is capable of handling
> extreme conditions.  It typically is not, in fact, unlike the
> "natural" stable system, the controlled system typically responds
> catastrophically (in both the mathematical and vernacular senses) to
> conditions outside of its design range.[1]

I made no such assumption.  I may not have communicated
the lack of that assumption, or the necessary
consequences, but I didn't assume that.

Stable versus unstable is a local criterion.  Stability
does not imply that the system won't fall over if it is
stressed properly, and most stable equilibria indeed can
be forced to fall over.  Mathematically the transition
from one stable equilibrium to another is indeed called
a catastrophe.  Not coincidentally, laypeople looking at
the system being modelled often call the same events a
catastrophe as well. :-)

Note that there is also no real distinction between
natural and artificial equilibria.  Both tend to be only
locally stable, and both routinely go through
catastrophic adjustements.  Just ask the people who went
through the recent earthquake in China for verification.

> Note how your "unicycle" example works.  There, the _goal_ is
> stability.  It's not surprising that a rather stable control system
> results from finding a solution.  However, economic stability of the
> kind that is found in "traditional societies" is clearly not an
> acceptable macroeconomic goal!  Who needs macroeconomics in such a
> society?

I didn't mean to suggest that stability is good in any
normative sense.  Merely that it is important in figuring
out what might work.

FWIW, the thing that I find interesting about the
unicycle example is not that you can make it stable, it
is how non-intuitively simple the solution is.  Our
natural guess is that we need to deal with all possible
details to make it work.  In fact you just set up a
simple dynamic motion, and a fall that is just starting
is corrected more when the motion moves against the fall
than when it moves with.  Voila!  Slight disturbances
away are corrected back and we have stability!

The moral of the story is that identified undesired
instability is not always an indicator for a complex
control regime.  Indeed a complex control regime often
leads to the opposite effect for a wide variety of
reasons.

> BTW: Nash equilibria are fixed points of mappings, not steady states
> of dynamical systems.  The same is true of market equilibria, with one
> important difference in current models.  Market equilibria can fairly
> naturally be embedded in dynamical systems for which the steady state
> condition is equivalent to the equilibrium condition for the market.
> This is not so for Nash equilibria.  There are several generally
> accepted theories of stability for market equilibria, and they
> coincide for the "canonical" case of upward-sloping supply and
> downward-sloping demand (all equilibria of such markets are stable).
> There is as yet not even one generally accepted dynamical model
> embedding Nash equilibria.

Is the lack of a generally accepted dynamical model
evidence that it is hard to produce one that is
mathematically acceptable, or evidence that a dynamic
system has sufficient additional features that it is
hard to agree on how much of the system can be
included/excluded?

>     >> When he talks about feedback loops and the paradox of logical
>     >> indeterminacy I interpret Soros as describing a sort of
>     >> uncertainty principle that makes it difficult to view any
>     >> economic system as stable.
> 
> The analogy is reasonably apt.  Fortunately, your (or Soros's) ability
> to imagine a stable economic system doesn't have anything to do with
> its possibility.  With a few exceptions, markets are analogous to the
> realm of Newtonian, not quantum, mechanics.  The exceptions are
> important, such as "tipping" in markets for products with significant
> network externalities.  But the overall system is remarkably stable.
> Remember, a 5% drop in production is considered disastrous.

The overall stability that already exists is a good
point.  As for overall stability, if control theory can
teach us anything, it should be that the human mind is
very bad at predicting when a system will and will not
be stable, even for very simple systems.

> Prices?  Yes, prices are volatile.  _But that's precisely because they
> are the dual variables to the "real" quantities in the system, which
> are stable._  Consider: the markets where we insist on controlling
> prices (labor, housing, agricultural products) are precisely with ones
> with rather unsatisfactory quantity outcomes.

Actually with agriculture we control both price and
quantity - we guarantee a minimum quantity and then
offer farmers direct support of some kind for the
resulting low prices.  This is deemed necessary
because a modest shortfall in amount is societally
catastrophic.

> Thus, price volatility, in some sense, is exactly the "quantum judo"
> you refer to as "a wonderful thought."  You just don't notice it
> because it's such an everyday miracle.
> 
> And, of course, because in situations like the labor market neither
> price nor quantity volatility should be accepted without a fight.

s/should/will be/

People disagree wildly on the shoulds in this case, and
the position that you take is strongly correlated with
your overall political stance.

>     >> [...]  When everyone recognizes this, markets become inherently
>     >> unstable.
> 
> This is, I suspect, confusing "instability" of static equilibrium with
> "volatility" (periodic or even chaotic) of a dynamic process.  It is
> true that in a system with multiple steady states, volatility can
> drive the system from one basin of attraction to another (aka,
> "economic depression").  However, it is important to recognize that
> this is evidence _for_, _not_ against, stability of the system as a
> whole.  And it is true that "socially optimal trajectories" are
> typically not steady states[2], and thus attempting to force social
> optimality turns the system into a controlled system fraught with
> catastrophic potential if the controls fail.

Um, this makes no sense to me.  Generally speaking,
chaos is not a sign of stability.  For instance the
basic Hadley cells in the Earth's atmosphere give rise
to the permanent Jet Stream.  However the actual
trajectory of the Jet Stream is unstable and chaotic.
It wanders over a restricted range, but slight
perturbations in its basic path are not counteracted,
they grow.  Therefore the Jet Stream is almost never
found predictably circling the globe at the equilibrium
latitude that you would predict based on the Hadley
cell that gives rise to it.

(For those who do not know, Hadley cells are the
predictable cycles where air near the equator rises,
moves to a higher latitude, cools and falls, then goes
back to where it will be heated again.  When combined
with the fact that the Earth rotates at different
speeds at different latitutdes you get phenomena like
the trade winds, the Jet Stream and various major
climate effects.)

> And it is most definitely true that large fluctuations cause huge
> amounts of pain that would be completely unnecessary if the economy
> would only hew strictly to trend.  But the equation of economic pain
> with instability is incorrect.

Depressions are both stable and painful.  Enough said.

>     >> He therefore argues for strong market authorities.  He wants
>     >> control systems in place.  It's not a totally absurd theory.
> 
>     ben> I agree, and I have the same major qualm about
>     ben> macroeconomics.  There are fundamental questions about what
>     ben> solutions are stable, and my feeling is that well-meant
>     ben> attempts to induce stability can itself be the cause of
>     ben> future problems.
> 
> You've been reading Friedman and von Hayek again, I can tell.  :-)
> (That's a joke, I realize you're coming from advanced diff eq.  But
> their intuition maps quite precisely to the mathematical description
> you give.)

Somehow I am not surprised.  Also I note that the
parties being controlled have a strong interest in the
control regime that controls them.  This interest is
generally more focussed and longer lasting than the
counterbalancing interests.  The result is that they
have a strong incentive to undermine the controls to
their personal benefit, and I think everyone here can
find examples where they succeeded.

>     ben> Therefore I wonder whether the government should deliberately
>     ben> attempt to oscillate interest rates - to try to create
>     ben> periodic mild amounts of risk so that businesses avoid things
>     ben> like the dot bomb excess.
> 
> I think maybe they should just let them alone, except for controlling
> inflation.  Inflation does not affect the real economy to a first
> approximation.  Therefore applying negative feedback to control
> inflation can probably be decoupled from nasty side effects in the
> real economy.  Not so if you actually try to control interest rates.

Predictable inflation rates may not affect the real
economy to a first approximation, but boy oh boy do the
higher order effects matter!  Consider the difference
between inflation and deflation to see what I mean.

As for what they should do, I don't know the
macro-economics well enough to have a strong opinion.  I
just wonder based on a personal belief (backed by mild
evidence) that controlled real risks give rise to risky
behaviour that offsets the controls.  (For instance
seat belt laws do not change overall traffic fatalities,
drivers that feel safer drive more riskily, and the
overall result is that risks move outside of the car.
Yes, this explains how SUVs are driven...)  Therefore I
am inclined to let those who tend most towards risky
behaviour to be encouraged to meet the risks they create
before those risks are likely to affect _ME_. :-)

>     ben> Unfortunately the social sciences generally fail of the most
>     ben> basic requirement for scientific progress.  Which is the
>     ben> availability of examinable systems where useful simplifying
>     ben> approximations can be applied.
> 
> Actually, there's a yet more basic requirement, which is separation of
> the observer from the observed.  Even in economics, which mostly deals
> with quantities that can be measured in the most primitive way (one
> dollar, two dollars, three dollars, ...), it is all too easy to feel
> that (to take the example that kicked off this thread) the genuine
> pain felt by the unemployed is something that must be addressed by
> urgent action.

Despite reading reams of verbiage about the need to
separate observer and observed, my understanding of
actual scientific practice does not support that being
a requirement.  For extreme examples, I know of medical
research where the researcher was also the observed
subject.  (Several of the Victorian scientists did this,
for instance Charles Darwin.  More recently in both
hypothermia research and mapping the human genome the
researcher has been their own experimental subject.)
Sure, the possibility of bias is one that you have to
fight, but a possible cause of bias is not necessarily
sufficient to disqualify research.

I suspect that a lot of the focus on separating the
observer and the observed has to do with the Heisenburg
interpretation of QM.  As a fan of Everett, I am more
than willing to accept modelling the observer as part of
the observation process...  (For those who don't know,
the Everett Interpretation assumes that both the observer
and observed system are subject to QM.  He then predicts
that an observer who observes a superposition goes into
a superposition of observers, each of which seems to
have observed a different collapse of the system.  For
known thermodynamic reasons those observers cannot
notice each other from then on.  This interpretation of
QM is also known as "many worlds" and is more often
parodied than understood.)

> Unfortunately, the actions taken are all too often inexcusably
> destabilizing, themselves.  But they are justified on the grounds that
> they _might_ work as projected, and if they do, the immediate pain
> will be dramatically reduced and the economy will be able to "return
> to (putatively natural) trend".  Then, when they blow up (years or
> even decades later), enormous effort (both by policy advisors and
> academic researchers) goes into adjusting the remedies, when really
> the whole line of attack should be scrapped.  It's hard enough to
> "scrap a whole line of attack" when it's "just an operating system";
> when you're talking about (eg) AFDC or the EU's CAP or rent control,
> and the transition is going to inevitably destroy people's current
> lifestyles, well, objectivity goes right out the window.[3]

Arguing for inaction when appropriate actions seem to be
obvious is always hard.  It becomes harder still when
subsequent events demonstrate that action was, after all,
justified.  See British inaction in the face of the Irish
Potato Famine, motivated by the belief that intervention
now would just cause a later, larger famine.  Given the
subsequent history, this conclusion is dubious.

> And if your measurements are of "variables" like "educational
> achievement" or "sexual harassment", well, it's just not within the
> realm of human ability to treat it with mathematical rigor.

Imagine that I had inserted a rant here on grade
inflation and then decided against keeping it.  Imagine
yourself wisely deciding not to prod me further on the
topic.  These imaginings are, of course entirely
hypothetical...

> Footnotes: 
> [1]  Of course natural systems also respond catastrophically to
> extreme conditions.  Eg, an avalanche.  However, the usual goal of
> control systems is to push things toward optimal solutions, which are
> by definition extremes.

Um.  Control systems are often designed to manage
otherwise predictable natural catastrophes.  Of course
the more stable a system is, the farther away the next
stable equilibrium is likely to be, and therefore the
more catastrophic an eventual catastrophe is likely to
be.  See fire management in the Western US for a
practical example of this theory.

> [2]  Note that by defining "growth rate" as the state variable, a
> steady state need not imply stasis.  A steady state could include (eg)
> a constant rate or constant growth rate of technological "capital".

Indeed in business systems such growth curves are a
common feature.  Christensen's book _The Innovator's
Dilemma_ deals with the predictable catastrophic
consequences as an inferior technology's projected path
of improvement intersects with the needs of the
customers of an established market segment.

The end of such curves usually also comes with
interesting consequences.  A few current ones which
will have to eventually end are Moore's Law, the
continued growth of government, and the rising price of
postsecondary education.

> [3]  And maybe it should.  I disagree, but that's more "religion" than
> "science."

I am bothered by people who are so convinced of the
rightness of their rational models that they can
comfortably disregard the human consequences.  But I am
also wise enough to not lightly disregard their
reasoning...

Cheers,
Ben
-- 
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