Not Quite Positive Feedback

[Bob L Eberlein]

Consider the following classical structure:

|---> sales --> revenue --> budget to sales force -|
| |
|------ sales force <---- target sales force <----|

At first glance this appears to be a positive feedback loop.
However, depending on the value of such things as the fraction
of revenue to sales, wage cost and sales effectiveness this
feedback loop can generate growth or decay.

I tend to call loops such as this "contained" positive feedback.
If the gain around the loop is less than 1 (0 for strictly
continuous types), it generates decay, if it is greater than
1 (again 0) it generates growth.

On reflection such terminology is confusing and also seems to be
disagreeable to the best minds in the field. It is simplest
to equate positive feedback with growth, negative feedback with
decay. This is unambiguous for first order systems and means
that the above feedback is either positive or negative depending
on the gain.

The question is what is best pedagogically? The only defensible
position I can see is not to discuss loop polarity until after
simulation, and then use it to tell stories. We all know that
you cannot make inferences about behavior from a causal loop or
stock and flow diagram. Is the early assignment of loop
polarities a dangerous and foolish practice?

The question is not intended to be rhetorical. I would like to
hear the opinions of others who have spent many hours pondering
such issues.

Bob Eberlein
vensim@world.std.com

[norris@ENH.NIST.GOV (Gregory A.]

I believe Bob Eberleins non-rhetorical question about loop polarity is an
important one.

I would add that not only can LOOP polarity change during simulation, but
causal LINK polarity can as well. This is especially true after one tries
to distill a complex-looking, few-stock model into a more transparent
"Structural Essence" diagram with stocks, flows, and the minimum possible
set of "auxilliaries" or "converters."
Late at night I have been trying to develop a graphical convention for portraying
the "Structural Essence" of small (2-4 stock) S.D. STRUCTURES (i.e., small
models, or sub-model) which is sufficiently complete that one CAN infer a
BEHAVIOR MODE from such a picture. Ive been using Alan Grahams
Dissertation as a helpful jumping-off point.

For distilled models, this convention requires indications of whether the
functional relationship in a "distilled link" is zero for a zero input,
monotonically positive or negative, positive only over some range of input
values, etc.

Until someone can define such a graphical method which is easy to comprehend
and communicate with, yet complete enough to provide a basis for inferring
qualitative dynamics at least under known conditions, I would agree with Bob
that
the defensible position is "to not discuss loop polarity until after
simulation, and then to use it to tell stories."

-Greg Norris, NIST, Office of Applied Economics
norris@ENH.NIST.GOV

[GR383@albnyvms.Bitnet (George Ri]

Eberlein asks:
> |---> sales --> revenue --> budget to sales force -|
> | |
> |------ sales force <---- target sales force <----|

In every example of such a structure that I know of there are minor
negative loops -- perhaps hidden in an averaging process (a smooth) or in
the formulation for an outflow rate (shipments = inventory/(delivery
delay). That means (this is significant!) that such structures are not
single positive loops but rather multi-loop structures. I have found it
easy to get across the growth or goal-seeking tendencies of such multi-loop
systems as simply the question of whether the positive loop dominates or
the minor negative loop(s) dominate. Thus the problem of "goal-seeking
positive loops" is really just is special (simple) case of loop dominance.

The paper "Loop Polarity, Loop Dominance, and the Concept of Dominant
Polarity" has such a treatment (System Dynamics Review, 11,1: 67-88).

The concept of "gain" is lurking here -- the question of which loop is
dominant boils down to which loop has the stronger "gain" -- but it seems
easy in practice to avoid talking about "gain." Ive presented Bobs
problem in terms of loop dominance in lots of contexts, including beginning
students and experienced folk, and have found very little difficulty
pedagogically. Everything else Ive tried comes across as pretty smokey or
confused or wrong to those trying to understand.

As a side benefit it seems very "efficient" to get the behavior of such
systems out of loop dominance, which is such a crucial idea for us anyway
in nonlinear cases where we more commonly apply it.

...GPR

.............................................................................
George P. Richardson G.P.Richardson@Albany.edu
Rockefeller College of Public Affairs & Policy GR383@Albnyvms.bitnet
State University of New York at Albany Phone: 518-442-3859
Albany, NY 12222 FAX: 518-442-3398
.............................................................................

["Scholl Greg"]

With regard to Bob Eberleins question about loop dominance:

I agree that causal loop diagrams are most useful, as Bob points out, to tell
a "story" -- problems with using causal loop diagrams for model
conceptualization and formulation are well documented (cf. George
Richardsons article of close to the same name). Therefore, "gain", or loop
dominance, should be implicit in the specific message one is trying to
communicate with the diagram -- the diagram should always have a context and
a purpose.

However, this assumes that the "story" is already known. Considering the
limitations of causal loop diagrams and other pen-and-paper representational
techniques, in the absense of a formal "model", how can one derive an
accurate enough understanding of a system to use causal loop diagrams to
communicate insights about its behavior? How are the limitations of
diagramming overcome when one cannot formally test assumptions about system
structure? In particular, Id be interested in the perspectives of any
system thinkers out there.

----------------------------------------------
Greg Scholl
Booz Allen & Hamilton
scholl_greg@bah.com

If my and my employers opinions coincide, its a coincidence.

["Teiling, Bernard"]

Richardson and Sterman have covered a lot of ground on the subject.
I believe that one aspect of the Eberleins problem remains open.
When the context is complex, a model is a selected set of feedback loops.
For communication (or educational) purposes, the selection has to be simple
and yet meaningful, in terms of structure (the "selection") and, but not
necessarily, in terms of behavior.
Here is the problem: we have a "weak" positive loop which is presented for
its strategic (or educational) value, and (potentially many) minor negative
loops which dominate.
-Should we question the relevance of the weak loop, when minor loops are not
explicitly formulated?
-Should we use a special annotation to indicate that there is a context of
(many) minor negative loops?
-Beyond representation techniques, can a model be credible, when many
specified (and unspecified) "minor negative loops dominate"?
-Is the purpose of a model to open the dialog about a "weak" loop, as we
know that its re-enforcing effect may not (significantly) take place in
reality (or under certain conditions)?
...
Bernard Teiling
usgle000.teilinb@wcsmvs.infonet.com
----------

[jsterman@MIT.EDU (John Sterman)]

RE: Bob Eberleins not quite positive loop.

Bob does not give the stock and flow structure or equations, but I
presume that the sales force adjusts to the target sales force according
to dSALES FORCE/dt = (Target Sales Force - Sales Force)/ADJ TIME. If
so, then the problem is readily solved: The apparent ambiguity about
the relation between the positive loop and the mode of behavior it
generates arises because the loop actually is not one loop but two:
The hidden loop is the negative loop in the adjustment of the sales
force to its goal. The system can generate exponential growth or decay
depending on which of the loops is dominant. There is no ambiguity
about the relationship between loop polarity and behavior, only
ambiguity about which loop is dominant, and this ambiguity can be
resolved by a simple calculation of the open loop steady state gain
(OLSSG) of the "loop" once one knows the parameter values. Thus given
the structure and parameter values one can unambiguously determine
whether it will grow or decay. In Bobs example, the system will grow
if the sales per sales rep * price * fraction of revenue to sales / cost
of sales rep > 1. That is, if each sales person brings in to the sales
department more money (the numerator) than they cost (the denominator),
then there will be a surplus that the organization uses to hire still
more sales people. If not, then the sales force must shrink. If the
parameters determining the OLSSG are constants, then the mode of
behavior will be exponential growth or decay at a constant rate
determined by the OLSSG and the sales force adjustment time. If they
are endogenous variables (as in Forresters market growth model), then
the mode of behavior can shift from e.g. growth to decline as the OLSSG
falls below unity. For an excellent treatment of these issues, see
George Richardsons article "Problems With Causal Loop Diagrams" in the
System Dynamics Review, 2(2), 1986, 158-170. Also see Alan Grahams PhD
thesis (contat Nan Lux at the MIT system dynamics group), and Joel
Rahns article on unity gain positive feedback loops: Rahn, R. J.
(1982). Unity-Gain Positive Feedback Systems. Dynamica (Winter), 96-104.

Of course, I strongly endorse the most important point Bob makes:
People cannot make good intuitive inferences about the behavior of even
apparently trivial systems. It is foolish to think that one can derive
appropriate and correct inferences about dynamics through inspection of
a causal loop diagram alone. Simulation or the ability to find analytic
solutions are required. For all but simple linear and a few special
cases of nonlinear systems, analytic solutions are not known or are
known not to exist; in other words, for all practical purposes with
realistic models, simulation is the only reliable way to relate
structure and behavior.

John Sterman
jsterman@mit.edu