[From Bruce Abbott (951028.1605 EST)]
Bill Powers (951028.1150 MDT)]
A+
Great! Do I have enought credits to graduate? (;->
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!
I've decided to simply videotape the experimental sessions rather than
attempting to automate detection of the rat's position, etc. This means
that I can use the chambers unmodified but will need to construct a new
sound-isolation box having a window on its top surface through which to do
the taping; this shouldn't require much time to make. I'm hoping to borrow
a suitable camera. With everything on tape I should be able to analyze the
action frame by frame. That way we can determine _everything_ the rat does
in this situation, which should provide a good basis for constructing that
"do everything" model.
Bruce Abbott (951028.1145 EST) --
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.
Herrnstein was no fudger of data. As I noted, the matching law first arose
in the context of concurrent variable-interval schedules, and was designed
to summarize the observations, which indicated that th pigeons, to a fairly
close approximation, matched their relative rates of pecking at the two keys
to the relative rates of reinforcment scheduled on those keys:
P1 R1
------- = -------,
P1 + P2 R1 + R2
where P1 and P2 are the peck-rates on the two keys and R1 and R2 are the
reinforcement rates programmed for responding on the same two keys. At this
time R1 and R2 were taken to be the _programmed_ rates. In the actual
experiments the programmed and obtained rates were practically the same so
it really didn't matter which you used, you still got matching, within
experimental error.
Later Herrnstein attempted to extend the matching law to other schedules.
He found that when concurrent ratio schedules were programmed, stable
performance (after the pigeons had enough experience with the consequences
of pecking on the two keys) consisted of exclusive responding on the key
offering the lower ratio of responses to reinforcements. Herrnstein noted
that, if one used _obtained_ reinforcement rate rather than _programmed_
rate, the observed outcome was "trivially consistent" with the matching
law--a degenerate case. For example, if the pecking rate on key 1 is 200
pecks/min, the pecking rate on key 2 is 0 pecks/min, and the schedule on key
1 is FR 50, then
200 4
------- = -----; thus 1 = 1 and matching holds.
200 + 0 4 + 0
Herrnstein fully recognized that this is a trivial result and that, if
programmed reinforcement rates were used in the equation, matching was
violated on concurrent ratio schedules. But remember, the matching law was
_intended_ to describe stable performance on concurrent interval, not ratio,
schedules.
Of course, the ratio analysis in terms of _obtained_ reinforcement doesn't
make much sense unless one views the stable end-point of performance as
resulting from a dynamic process in which response rates affect
reinforcement rates, which in turn affect response rates. (I'm describing
this sequentially but it's a simultaneous, closed-loop process.) One can
then view the outcome (exclusive preference) as the product of an unstable
(positive feedback) process that drives relative response and reinforcement
rates toward exclusive responding on the richer schedule.
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."
Remember, "matching" is an attempted summary description of the
observations, not an explanation. One can say that pigeons match
(trivially) on concurrent ratio schedules _because_ exclusive responding on
the richer schedule maximizes the average rate of reinforcement.
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.
For interval schedules in which responding on both keys is fast enough to
produce all reinforcers at close to their scheduled rates, a wide range of
relative response rates will produce the same (or very nearly the same)
relative reinforcement rate, and the overall rate of reinforcement will not
change much, either. Several theories have been advanced attempting to
account for matching in this circumstance by appeal to a maximization
principle (of overall or of local reinforcement rates, including or not
including "response cost"); none has been completely successful in
accounting for the data. But this is clearly something we can look into
once we are in a position to collect some hard data. One advantage of
programming multiple operants is that it allows one to look at how organisms
allocate their time and effort resources across alternatives. With a
single-operant procedure, alternatives are still there, but usually go
unmeasured.
Meanwhile, have you taken a look at the McSweeney data? If we need to
communicate with Fran, I received an e-mail note from her a couple of weeks
ago informing me that she's finally given in and gotten her own e-mail address.
By the way, what's happened to Rick? I hope he's not back out floating
among the kelp!
Regards,
Bruce