Recent inquiries on stability, equilibria, games

[6155GUASTELL@xxxxxxxxxxxxxxx]

From: enilsson@xxxxxxxxxxxxxxx (Eric Nilsson):

>Interesting observation by H. Gintis:

>>. . . systems that are observed to oscillate rapidly
>>and violently over time are likely to be stable, on
>>the grounds that otherwise they would self destruct in a short time.

>Perhaps we might differentiate between stability and fragility.  Some fairly
>stable systems might be fragile (and be destroyed) in the face of large
>knocks.  However, some (but not all) unstable systems might be able to
>survive big bumps after thrashing about for quite a time.

The distinctions among the various types of attractors already exist.
In the Andronov-Pontryagin definition, which is the definition adopted by
mathemeticians, a dynamical system (vector field or map) is structurally
stable if nearby systems have qualitatively the same dynamics. Given that
definition, structurally stable attractors would include fixed point
equilibria, limit cycles, quasiperiodic attractors, and many chaotic
attractors. Herb Gintis was describing the chaotic attractor.

The word "equilibrium" is overly broad in the dynamical systems lexicon,
as it could include any of the foregoing types of attractor. Also in
game theory literature, "equilibrium" is commonly used to describe
saddle points. A saddle point is not strcturally stable because its
dynamic field has vectors goings towards it and vectors avoiding it at
the same time. Saddles are thus fragile, but they are not chaotic.

Nash equilibria, which are typically saddles, are not necessarily
evolutionarily stable states (ESS). ESSs are gaming outcomes that are
found to be stable on repeated plays of the game. For a long time it was
necessary to determine whether a system was ESS on an individual
case basis.


>Adding to this might be the following: generally systems face small bumps,
>and only occasionally large bumps.  But actors within unstable systems might
>try to head off large bumps before they occur because living within an
>unstable system might be very unpleasant when large bumps occur -- thereby
>reducing the likelihood they actually face a large bump.

In applied physics, this principle is known as "control of chaos", and
the major work has been done by Atlee Jackson and Al Hubler at U of Illinois,
Champagin-Urbana since 1989+. Organizational psychologists have picked up
on this theme (Mosskilde et al, Stacey, me) in recent years (1992+).


>However, actors
>within stable systems might take a "what me worry" approach (believing they
>live in a stable system and have, therefore, no concerns) and thereby face
>large bumps more frequently than do unstable systems.

Sounds like the cusp catastrophe model for evolutionary and revolutionary
change in organizations. There are two control parameters: pressure to
change and resistance to change. One might truly reside in a locally stable
situation, but values of control parameters may change slowly. As they
change, nothing appears to be happening, until one day the straw that breaks
the camel's back occurs.

From: "RICHARD P.F. HOLT" <holtri@xxxxxxxxxxxxx>
Subject: instability

>A question: when people are talking about the problem of economic
>instability are you simply referring to inflation and unemployment
>in the short-run and stagnant growth in the long-run? Or other issues
>like income distribution?

Instabilities are created when bifurcation structures are present.
They may take the following forms: (a) gradients whereby the edge of
an attractor basin demarks a place where behavior suddenly changes
into a new, different basin, (b) a convergence of bifurcation gradients
into a very unstable point at which the behavior of a system might
change toward any of the available local attractor basins, (c) a repellor
structure, opposite of an attractor, whereby any vector entering its
basin, known here as a separatrix, is bounced outward into any possible
direction, never to return (d) catastrophic or explosive bifurcations
whereby the structure of an entire dynamic field can change, e.g. when
a repellor increases in size and collides with a limit cycle, producing
a fixed point attractor.

Applications in economics more often revolve around securities and
commodities prices, food chain dynamics, spatial-temporal diffusions
of economic events, competition and cooperation dynamics, to name the
most prevalent. A book on unemployment has JUST come out, I'm still
waiting for my library's copy to arrive. A wide range of applications
are considered in:

Rosser, J.B. JR. (1991). From catastrophe to chaos: A general theory
of economic discontinuity. Boston: Klewer.

More (but more psychologically centered) is scheduled to appear in:

Guastello, S.J. (in press). Chaos, catastrophe and human affairs:
Applications of nonlinear dynamics to work, organizations, and
social evolution. Hillsdale, NJ: Lawrence Erlbaum Associates.


--Stephen Guastello