[From Bill Powers (941012.1155 MDT)]
I hate to start another thread while we have a good one going, but this
is relevant to the general idea of mainstream models of behavior.
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Dennis Delprato (941012)--
Your call for EAB types to start exploring PCT (with Bruce Abbott in the
vanguard) is one of those better ideas. One of the problems, however,
will be to convince EABers that they have been doing not modeling, but
curve-fitting. I have a lovely example (Meyerson & Meizin, "The Kinetics
of Choice: an Operant Systems Analysis") in which the authors set up two
"system equations" which are simply the same equation written twice
(they're linearly dependent), and then fit the curves to data thinking
they're solving the system equations. This paper contains several other
rather horrible gaffes of the system-modeling type, but this one is the
worst.
You mention the "generalized matching law." This is an example of the
way EABers have confused manipulating mathematical expressions with
setting up and solving system equations. In fact the matching law is no
law at all; it proclaims a fact that is untrue for most experiments.
Meyerson & Meizin cite it in a two-choice version thus:
B1/(B1 + B2) = R1/(R1+R2)
(B = behavior rate, R = reinforcement rate, or totals)
Now try a little manipulation on this.
Invert both sides:
(B1 + B2)/B1 = (R1 + R2)/R1, or
1 + B2/B1 = 1 + R2/R1, or
B2/B1 = R2/R1, or
B2/R2 = B1/R1
This is exactly equivalent to the original form.
The ratio Bn/Rn is the average ratio of bar-presses per unit time to
reinforcements per unit time, for a particular choice. With unit time in
numerator and denominator, it is also the ratio of total bar-presses to
total reinforcements. It is, in fact, the average schedule of
reinforcements; for fixed-ratio experiments it is exactly the schedule
in bar presses required per reinforcement. If m is that ratio, then the
matching law states that m1 = m2.
Look what the equation says: it says that for any choice experiment, THE
RATIO OF BAR-PRESSES TO REINFORCEMENTS IS THE SAME FOR ALL CHOICES. That
is what that big fat equal sign in the middle means. What the
generalized matching law says is B1/R1 = B2/R2 = ... Bn/Rn: it says that
all schedules of reinforcement in choice situations, in terms of total
bar presses over total reinforcements or rates over rates, are the same.
This is not obvious from the way it is customarily written; you have to
reduce the stated equations to simplest terms before you can see this
"law" as an assertion. And then you can see that the assertion is false
unless all the schedules of reinforcement are in fact the same, in terms
of average ratios.
If you start out with one equation, you can change variable names,
introduce transformations, and go through every permutation of legal
manipulations until Kingdom Come, and you'll still have only one
equation. You can't get blood from a turnip, either. The only way to
handle operant conditioning as a system phenomenon is to write one
equation showing how the reinforcements depend on the behavior, and a
SECOND INDEPENDENT EQUATION showing how the behavior depends on the
reinforcement. You canb't just use the first equation backward or
inside-out. You have to have, or propose, a second INDEPENDENT equation.
You have two variables; you need two independent equations to completely
describe the system. This seems to be a little-known fact in JEAB,
although it's the first thing that electrical engineers learn.
I have seen plots of actual choice behavior against curves representing
the matching law, and they simply don't fit. All sorts of verbal excuses
are given, but the reason they don't fit is obvious: in most choice
experiments the ratios are different, and the matching law is false.
Whatever effects are seen in choice experiments, they are not due to the
matching law.
This failure is covered up by all the complications that people like to
use, particular the variable ratio and variable interval schedules.
These throw a lot of noise into what is otherwise a very simple
relationship, so you can't tell whether a given data set falls on the
matching-law curve without statistical analysis or multiple repeats of
experiments, neither of which EAB types like to use. Herrnstein's
famous matching law is a gigantic goof, and I predict that is it going
to be very hard for people who have cited this law 10^6 times to admit
what a goof it was.
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Best,
Bill P.