[From Bill Powers (951028.1150 MDT)]
Bruce Abbott (951028.1110 EST) --
A+.
The "A" is for stating the theoretical prediction correctly, and the "+"
is for noting correctly the research that remains to be done.
As I was checking out OPCOND5 before sending it to Chris Cherpas, I
realized that the simulation might be able to reproduce the Abbott
Effect. When there is a limit on the error signal, the pressing rate
will remain constant over a wide range of schedules, provided the reward
size is not too large. The error signal simply remains against its stops
until we get to the lowest ratios. But this program isn't well suited to
taking simulated data of that kind; it's set up for display more than
analysis. Also, the model probably needs to be elaborated, because I
don't think it would show the effect for different degrees of
deprivation (i.e., different settings of the reference signal). After
we've checked out the effect with real rats, we can start looking for a
model that will really do everything!
···
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Bruce Abbott (951028.1145 EST) --
Bill, I know it must have been a traumatic experience for you when
we went over this issue nearly a year ago; perhaps that explains
why you are now apparently suffering from Repressed Memory
Syndrome.
Just Second Thoughts Syndrome. I was being careful to base the first
part of that post strictly on fixed ratio schedules -- other kinds lead
to some complex, but I think still valid, arguments, as I tried to show
using general functions.
1. The matching law, when it was proposed, was said to provide a
summary description of the observations, not an explanation for
them, as Boyle's Law does for the pressure, temperature, and volume
of a gas.
What the original matching law for ratio schedules overlooked was that
there is no difference between scheduled and obtained b/r ratios. There
can't be; the ratio is set by the apparatus. If you press at any rate b,
you're going to get reinforced at a rate b/m. No way around it. What
Herrnstein thought he saw in the experiments was expressed by equations
that actually say something different. The equations say m1 = m2. The
reality is that m1 is not, in general, equal to m2. If Herrnstein saw
matching according to his formula when the schedules were not the same,
he was fudging his data.
If the two schedules are not the same, then there is no way that the
outcome can be b1/(b1 + b2) = r1/(r1 + r2), not when b1 = m1*r1 and b2 =
m2*r2 and m1 <> m2. If there is a matching effect, then the equations
Herrnstein used to try to express it were incorrect.
When you try to apply this equation to the case where b2 = 0, you end up
with an expression containing 0/0, which is indeterminate. The
indeterminacy is not evident unless you reduce the equation to its
simplest form first: b1/r1 = b2/r2. If b2 = 0 then r2 = 0 also, and
b2/r2 becomes 0/0. Simply from knowing that b2 and r2 are zero, you
can't deduce m2. This does NOT support the contention made by the
proposed matching equation.
Perhaps we can clear this up if we try to say what the matching effect
is actually supposed to be. Given two keys with different ratio
schedules, the hypothesis was that the greater number of presses (or
rate of pressing, which is the same thing for a fixed-length experiment)
would be seen on the schedule that produced the greater number of
reinforcements per press.
Let total presses = T
presses on 1 = b1
presses on 2 = b2
Ratios = m1, m2
total reinforcements = r
The total reinforcements are thus
r = b1*m1 + (T - b1)*m2, or
r = b1*(m1 - m2) + T*m2
This says that the number of reinforcements is a monotonically
increasing (m1 > m2) or decreasing (m1 < m2) linear function of b1. If
m1 > m2, then the maximum reinforcements occur at b1 = T and b2 = 0. If
m1 < m2, the maximum occurs at b1 = 0 and b2 = T (limited by the fact
that b1 and b2 cannot exceed T).
From what you say, the observed behavior rates are in fact those that
make the reinforcment rate as large as possible. This has nothing to do
with "matching."
For fixed-interval schedules, a similar derivation of total
reinforcement rate as a function of behavior distribution could be
worked out. You would have four cases: b1 > 1/i1, b2 > 1/i2, b1 < 1/i1,
and b2 < 1/i2, in all the possible combinations. I leave it to you to
work out the result, remembering that when b1 < 1/i1, r1 = b1 and when
b1 > 1/i1, r1 = 1/i1, and similarly for b2, r2, and i2. From these
treatments you can find out the value of b1 where the maximum total
reinforcements occur, and I'll bet that is about what is observed.
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If I were modeling this, I would use two control systems, one for total
amount of reinforcement and the other for place of pressing. Whether the
systems would find a maximum of reinforcement would depend on whether
the obtained rate reached the reference level; if not, the systems would
find the maximum rate, at least locally maximum. Otherwise the
distribution would stop changing when the reference rate was achieved.
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Best,
Bill P.