The Opposition

[Fred_Nickols3]

[From Fred Nickols (990505.1740 EDT)] --

Here's another snipped from the LO list. I am particularly interested in
how Bruce A and Bill P will comment on this posting. (By the way, it's
from a fellow who wanted to know more about my point of view and to whom I
recommended joining the CSG list and "lurking" for a while -- so don't get
out the meat axes -- the scalpels of gentle critique will do.)

As a not so aside aside, this thinking, or something like it, is what PCT
must deal with and eventually reconcile with, integrate, subsume, destroy
or whatever...

Those who don't want to bother with this are free to hit the delete key or
press page down...

I would like to start off this contribution by introducing the work of
Richard Herrnstein (who is now deceased), and his matching law for
behavior. Richard Herrnstein was a behaviorist (in the Skinnerian
tradition) who many (in the world of behavior analysis) consider to have
made nearly as large a contribution to behavior analysis as Skinner
himself did.

Herrnstein was able to show in numerous settings and contexts that the
probability of a given behavioral response was directly related to the
availability of reinforcement for the behavior (in relation to the
availability of reinforcement available for competing behavioral
responses). This relationship that he articulated and demonstrated has
been replicated by many other researchers as well, and forms the
foundation of much of what I have written about over the past 6 months.

Herrnstein's matching law, in its general form, can be expressed in the
following equation:

Pr(A) = k( Ra/Ra+Re )

Where

Pr(A) = probability of behavioral response A
k = highest # of possible behavioral responses available (total # of

responses)

Ra = Reinforcement experienced for response A
Re = Reinforcement experienced for extraneous behavior (behavior other
than the response of interest)

This relationship, has been reliably demonstrated to account for over 90%
of the variance in responses. One point of interest, this relationship is
not necessarily a reinforcement maximization rule, and, at times, will
predict patterns of behavioral response that are counterintuitive.

In order to further explain, allow me to illustrate the matching law first
in the simplest of settings. The simplest expression of the matching law,
modeling a choice between 2 possible behavioral responses, is as follows:

Proportion of responses on A = Responses on A/ (Responses on A + Responses
on B) = Reinforcements on A/(Reinforcements on A + Reinforcements on B

or

Pr(A) = Ra/Ra+Rb = rA/rA+rB

[Host's Note: Hmm... I think there should be (..) in the denominators here
and in the similar fractions below? ...Rick]

giving us the proportion of responses which will be accounted for by
response A.

To give an example, we can place two schedules of reinforcement against
each other and see if the relationship expressed in the equation holds
true (it does). So if we had a constrained choice between two equivalent
responses A and B, with choice A being reinforced on an VR3 schedule, and
choice B on a VR6 schedule (see LO20774 for an explanation of schedules of
reinforcement), we can plug these values in the equation and see what
split of responding might be expected.

Pr(A) = (1/3)/(1/3)+(1/6) = .667

So we would predict that choice A would be preferred over choice B by a
ratio of 2:1 (choice A would be chosen 66.7% percent of the time). And,
as it turns out, this is very consistent with what has been demonstrated
empirically. Notice that the prediction isn't in line with a maximization
prediction which would predict that Choice A would receive all of the
responses, resulting in the maximum possible reinforcement.

Another example: Choice A on a VR4 schedule and Choice B on a VR11
schedule

Pr(A) = (1/4)/(1/4) + (1/11) = .25/(.25 + .09091) = .733331

or, Choice A would be chosen 73.3% percent of the time and Choice B chosen
the remaining 26.7% of the time.

This first example allows us to see that the relative frequency of
reinforcement given a response is an important dimension in understanding,
and even predicting, behavior. This is the source for my assertion that
the first dimension which effects a consequences ability to act as a
reinforcer for behavior is the probability (or certainty) of the outcome
to result given the behavior.

One way that this variable can be "controlled" is to change the schedule
of reinforcement, given equivalent responses (for example, in many lab
experiments of the matching law, this has taken the form of bar pressing
behavior for rats and key pecking for pigeons, with each of two different
bars or keys with differing schedules of reinforcement available for
pressing or pecking). However, this variable also gives us a means for
capturing the effect of the relative effort required to perform the two
competing responses. A response requiring greater expenditure of effort
than a competing response, when placed on an identical schedule of
reinforcement, will respond similarly to an identical response placed on a
"thinner" schedule of reinforcement. This line of thinking moves us in
the direction of the flow of units of behavioral energy required to attain
the reinforcer.

But this isn't the whole picture. Another characteristic that impacts the
ability of an outcome to reinforce is the delay of the reinforcement from
the time of responding (we can't forget the decay of delay). In my
previous discussion of the impact of delay, I mentioned that the decay is
not a linear one. Research using the matching law suggests that relative
frequency of responding matches the reciprocals of the delays (given that
the delays are expressed in the same units of time).

Pr(A) = Ra/Ra+Rb = (1/Da)/(1/Da)+(1/Db)

To illustrate, lets consider an example where 2 equivalent responses, A
and B, are both on an equivalent schedule of reinforcement, but, with
Choice A, the delivery of reinforcement (occurrence of the outcome) is
delayed by 2 seconds, and, with Choice B, the delivery of reinforcement is
delayed by 4 seconds. In this situation, we have the following:

Pr(A) = Ra/Ra+Rb = (1/2)/(1/2)+(1/4) = .5/(.5+.25) = .667

Again, according to the matching law, we see that Choice A is preferred
over Choice B 2/3 of the time. And, again, if we were to test this out,
this is essentially what we would find.

The third dimension which effects the ability of a consequence to
reinforce behavior is the magnitude of the reinforcer. In my discussions
of behavior to this point, I have brought up this dimension only in the
form of discussing whether the outcome is perceived to be positive or
negative by the performer. Actually, to be accurate, we would have to
consider to what extent the outcome is perceived to be positive or
negative, relative to the outcomes for other, competing behaviors. The
more positive the outcome, the greater its ability to attract the
behavioral response, and the more negative the outcome, the greater its
ability to repel the behavioral response. This dimension we could call
the magnitude of the available reinforcement (m), giving us:

Pr(A) = Ra/Ra+Rb = Ma/Ma+Mb

controlling for our other dimensions (holding them constant).

In order to illustrate this, we need to have some unit of reinforcement.
This is possible in a lab setting, but it becomes more difficult outside
of that environment (perhaps we might use some form of currency, but even
this relationship is not perfectly linear (perhaps it is on some kind of
logarithmic scale). For our example here, we can again contrast two
equivalent responses, A and B, on identical schedules (cancelling out),
with identical delays (again cancelling out), but with different
magnitudes of reinforcement. Choice A gets reinforced with 2 units of
reinforcement (for example, 2 food pellets, or 4 seconds of pleasureable
electrical stimulation), while choice B gets 1 unit of reinforcement (for
example, 1 food pellets, or 2 seconds of pleasu reable electrical
stimulation).

We would then have the following:

Pr(A) = Ra/Ra+Rb = 2/2+1 = .667

Again, Choice A is preferred 2/3 of the time over Choice B. And we have
the source of the third dimension of a consequence that effects its
ability to reinforce/attract a behavior that I have articulated in
previous posts, the extent to which the consequence is seen as positive
(or, in the case of punishing/repelling, negative). This constitutes the
magnitude of the consequence.

So we have the following three dimensions:

1) A measure of effort in the form of the frequency of reinforcement given
a response (the probability/certainty of reinforcement.

2) The delay of the occurrence of the reinforcement from the occurrence of
the behavior

3) The magnitude of the reinforcer (extent to which it is perceived as
positive or negative).

Giving us the following equation for the matching law given 2 competing
responses, A and B:

Pr(A)= Ra/(Ra+ Rb) =
(rA*(1/Da)*Ma)/(rA*(1/Da)*Ma) + (rB*(1/Db)*Mb)

The matching law was originally articulated in sometime in the 50's or
early 60's, and has been the focal point of an enormous amount of
research. I have tried to illustrate the relationship by providing
examples of its application to 2 competing responses. However, the
research that has been done regarding the matching law has not been
limited to 2 choices, but has extended the relationship to many competing
choices with similar success. And the application of the relationship has
been explored in areas far outside the animal lab, including consumer
choice behavior in internet shopping. The system becomes more and more
complex as we extrapolate into "reality", but the relationship has
continually demonstrated its utility (does this make me a pragmatist? ;-).
Using the matching law, we can then begin to apply the lens to the world
around us and seek to understand many behaviors that are seemingly
"irrational" (which I would propose falls closely in line with a
reinforcement maximization model, as opposed to a matching model).

I hope that I have been able to demonstrate that the ideas conveyed in my
posts over the past 6 months have not been made up by me (I am not nearly
this imaginative), or pulled out of a hat, but rather are based on 50+
years of research conducted by many different researchers in many
different contexts. Most of the information that I have shared regarding
behavior analysis can be read for oneself in any graduate level text on
learning and behavior (should you be curious). If anyone has spotted any
problems with my articulation of these ideas and concepts, please let me
know.

This brings us back to the general form of the matching law that I
originally stated:

Pr(A) = k( Ra/Ra+Re )

and the Digestor model that At articulated in LO21272 in the form of free
energy:

/_\F(un) = -/_\n*mSU*[E(mSY, Msy) - E(mSU, Msu)]

where
F = free energy
n = basic building blocks of system (ion pairs)
m = size of the crystal - quantitative summary in units of lowest orders
of the surroundings (e.g., the number of ion pairs - the basic building
blocks - that have been digested by the crystal, making up its size). The
variable m is expressed in n units.

M = qualitative assessment of the perfection of the lattice structure/the
regular pattern of arrangement of ions in the crystal.

E = energy

I would propose that the matching law is the equivalent of the Digestor
model as it would apply to behavior with the following "translation":

The variable n, in the translation, is the variable k in the general
matching law equation converted into units of behavioral energy (the basic
building blocks of the system of behavioral energy).

The variable m is the proportion of behavioral energy allocated to the
various potential responses (attracted by the competing "crystal seeds" -
mSY and mSU), or the measure of the size of the "behavioral crystal".

The M variable for behavior is a perfection measure of the "bond" between
the behavioral response and its consequences, consisting of the three
dimensions that I have articulated above - ratio of reinforcement to
response, delay of reinforcement, and magnitude of reinforcement.

The combination the m variable and the M variable for behavior form the
Madelung forces which attract behavioral energy towards a potential
response (yes, this explanation is a bit teleological). On an individual
level, the consequences/reinforcement for a given behavioral response form
the "sy" portion of the equation, and the consequences for all other
potential responses (the extraneous reinforcement in the matching
relationship) form the "su" portion of the equation.

Or maybe the Digestor model is the matching law for behavior extrapolated
to the world of crystallography. Is this obfuscation? IMHO, I don't
think so. In both cases, the relationships were uncovered independently.
Which relationship was "discovered" first? What should we do with this?

IMHO, the efforts that At has made to show the difference between
spontaneous changes and nonspontaneous changes and empowered systems, the
attempt that Leo Minnigh made to illustrate how the digestor model might
be applied in an organizational setting (LO21360), and the wrestlings of
Winfried Dressler to reconcile the model with his own thinking are, at the
very least, excellent examples of the learning process that we have been
able to witness, and, at most, important forays into a world that may have
enormous impact on our understanding of organizations, learning, and their
interaction.

With the Digestor model/Matching law, we have a lens through which we may
understand evolutionary change that results from low entropy production in
systems close to equilibrium. We have a lens for understanding CONTINUOUS
changes and improvements. With the Brusselator model, and the parallels
that I have attempted to convey in the relationship between behavior and
its consequences, we have a lens for understanding DISCONTINUOUS,
revolutionary changes and improvements that result from high rates of
entropy production in systems far from equilibrium. To bring this home to
roost with a thread from months ago, these models together give us a means
for understanding the process of CONTINUAL improvement (advocated by
Deming). We really haven't gone that far off the course set for us by
Senge, Forrester, Deming and others. But we are extending the line beyond
what has been drawn before.

Which begs another question: At what point does
generalization/extrapolation of any theory become obfuscation? I'm not
sure that the conclusion can be drawn with foresight. As far as I can
tell, the difference lies in the utility of the model. Maybe after the
fact this differentiation can be made, but if we are indeed after the fact
in this case, and these ideas have been conclusively disproven, please
show me where the evidence is so that I can quit wasting my time.

Hypotheses generated by the theory/model must be articulated in a way that
they can be tested and disproven. From what I have read on both sides,
these ideas have withstood much rigorous examination and been shown to be
robust. I am not an expert in the physical sciences. The Ph.D. that I
hold is not in physics, or chemistry, or biology. But I have read
numerous texts on chaos theory, relativity, quantum physics, and 3 books
by Ilya Prigogine on thermodynamics, entropy and entropy production. The
accounts that At has given of these ideas are representative of what I
have read in these other sources. I have read numerous works by Skinner
and Herrnstein and numerous other behavioral theorists. I have read
Senge, and Forrester and other systems thinkers. I believe that if anyone
else has studied in these areas, they would vouch that my portrayals of
these areas are representative as well.

I, for one, find these parallels to be compelling, and I have invested a
considerable amount of time articulating the connections that I have made
for others to examine. To me, the obvious next step is to see how much
further the model can be extended. Is this really a step into oblivion?

Regards,

Fred Nickols
Distance Consulting "Assistance at A Distance"
http://home.att.net/~nickols/distance.htm
nickols@worldnet.att.net
(609) 490-0095

[Bill_Powers1]

[From Bill Powers (990506.1225 MDT)]

Fred Nickols (990505.1740 EDT)] --

Here's another snipped from the LO list. I am particularly interested in
how Bruce A and Bill P will comment on this posting. (By the way, it's
from a fellow who wanted to know more about my point of view and to whom I
recommended joining the CSG list and "lurking" for a while -- so don't get
out the meat axes -- the scalpels of gentle critique will do.)

I'm afraid I can't reply within your ground rules, because I think there is
a WHOLE LOT wrong with Herrnstein's matching concept. I'll confine myself
to just one point and, if you don't want to discourage your friend, we can
let it drop.

Let's take this version of the matching law:

Proportion of responses on A = Responses on A/ (Responses on A + Responses
on B) = Reinforcements on A/(Reinforcements on A + Reinforcements on B

or

Pr(A) = Ra/Ra+Rb = rA/rA+rB

The equality on the right (if written correctly)says that

           Ra/(Ra + Rb) = rA/(rA + rB)

where r stands for responses and R stands for reinforcements. A and a both
stand for the A key, and so on (the case of the subscript letter is just
cosmetic, while the other is meaningful, showing me that whoever wrote
these equations is not a mathematical whiz).

Multiply both sides by (Ra + Rb), and then by (rA + rB):

           Ra*(rA + rB) = rA * (Ra + Rb)

Expanding:

     Ra*rA + Ra*rB = rA*Ra + rA*Rb

The term Ra*rA appears on both sides of the equal sign and cancels out,
leaving

     Ra*rB = rA*Rb

Divide both sides by rB, then by rA, and we get

     Ra/rA = Rb/rB

Thus the original way of writing the equation turns out to be identical to
saying that the mean ratio of reinforcements to response is the same on all
keys. This also works for the generalized matching law in which there is
any number of alternatives instead of just two.

Obviously this is not true when the actual experiment involves keys with
different schedules on them. However, the data will still fit the equations
plus or minus some variations, so one can say, "See? There's a tendency
toward matching!" In fact, the only case in which the equations correctly
describe the experiment is when the ratios are in fact equal on all keys.
for any other combinations of schedules, the equations are simply false.

Herrnstein is trying to derive behavioral laws relating two variables
(input and output) from a single equation in the two variables.
Mathematically, it can't be done, no matter how many different ways you
find to write that single relationship.

Best,

Bill P.

[Abbott_Bruce]

[From Bruce Abbott (990506.1715 EST)]

Bill Powers (990506.1225 MDT) --

Obviously this is not true [the matching law] when the actual experiment
involves keys with different schedules on them.

Sorry, Bill, but I must remind you once again that the matching law does not
and was never meant to describe the relationship between relative rate of
responding and relative rate of reinforcement on concurrent ratio schedules.

However, the data will still fit the equations
plus or minus some variations, so one can say, "See? There's a tendency
toward matching!"

In the cases for which the matching law was developed, there is NO (I
repeat, NO) necessary relationship between the left and right terms of the
equation, and experimental error is not so great that matching could not be
discriminated from other patterns involving systematic deviations from
matching. It is simply not true that because of experimental variation, one
can always claim that some "tendency toward matching" was obtained.

In fact, the only case in which the equations correctly
describe the experiment is when the ratios are in fact equal on all keys.
for any other combinations of schedules, the equations are simply false.

The above indicates clearly that you are thinking of ratio schedules only,
because you assert that the equations would be false "for any other
combination of schedules." This statement can be true only if, for each
schedule, the ratio of responses to reinforcements is fixed by the schedule,
or if responding takes place exclusively on one key.

Applying the matching law to concurrent ratio schedules is like applying
Boyle's Law to liquids. In either case, no one will be surprised to find
that the law fails to hold.

Regards,

Bruce

[Bill_Powers1]

[From Bill Powers (990506.2-004 MDT)]

Bruce Abbott (990506.1715 EST)]

Sorry, Bill, but I must remind you once again that the matching law does not
and was never meant to describe the relationship between relative rate of
responding and relative rate of reinforcement on concurrent ratio schedules.

But in the post to which I replied that is how it was described. That's
what the equations say, when you reduce them to simplest form. I don't care
what they were "meant" to describe. Many a person has foundered because of
not realizing what his own mathematical manipulations imply. Perhaps in
reality there is some other relationship involved, but my transformations
of the cited equations are exactly correct, and the original set of
equations says the same thing -- identically the same thing -- that my
final version does: Ra/rA = Rb/rB. Unless you can repeal mathematics, no
other conclusion is possible.

The best I can say is that whatever Herrnstein was trying to express, his
equations don't say it.

I think I know what Herrnstein was trying to say. He wanted to say that
given a choice of keys with different schedules, the animal will devote the
most time to the schedules that yield the greatest average ratio of
reinforcement to behavior. He may have thought that his equations were
saying this, but they were not. Instead, his equations say that the yield
of reinforcements per unit behavior is the same on all keys. He simply
didn't know how to write out, mathematically, the idea he was trying to
express. What he did write down says something very different from what he
wanted to say. And evidently he wasn't enough of an algebraist to realize
that his complex-looking forms can actually be reduced to a much simpler
form that make his mistake easy to see.

I suspect that the mistake goes deeper than this, but the proof or disproof
would take more thought than I have given it.

Best,

Bill P.

[Dick_Robertson]

[From Dick Robertson,990507.0747CDT]

[From Fred Nickols (990505.1740 EDT)] --

Here's another snipped from the LO list. I am particularly interested
in
how Bruce A and Bill P will comment on this posting.

I would like to start off this contribution by introducing the work of
Richard Herrnstein (who is now deceased), and his matching law for
behavior.

Herrnstein was able to show in numerous settings and contexts that the
probability of a given behavioral response was directly related to the
availability of reinforcement for the behavior (in relation to the
availability of reinforcement available for competing behavioral
responses). This relationship that he articulated and demonstrated has
been replicated by many other researchers as well, and forms the
foundation of much of what I have written about over the past 6 months.

Is this true for each and every person, or do the results reflect the
averages of group of subjects?
If the latter, what was the range of individual actions that were pooled
in the so-called findings?
Third, is the point being made here substantially different from the
common-sense belief that "people are
more likely to do what they get rewarded for than what they don't get
rewarded for? If so, how?

Herrnstein's matching law, in its general form, can be expressed in the
following equation:

Pr(A) = k( Ra/Ra+Re ) Where

Pr(A) = probability of behavioral response A
k = highest # of possible behavioral responses available (total # of

responses)

Ra = Reinforcement experienced for response A
Re = Reinforcement experienced for extraneous behavior (behavior other
than the response of interest)

This relationship, has been reliably demonstrated to account for over 90%
of the variance in responses. One point of interest, this relationship is
not necessarily a reinforcement maximization rule, and, at times, will
predict patterns of behavioral response that are counterintuitive.

In order to further explain, allow me to illustrate the matching law first
in the simplest of settings. The simplest expression of the matching law,
modeling a choice between 2 possible behavioral responses, is as follows:

Proportion of responses on A = Responses on A/ (Responses on A + Responses
on B) = Reinforcements on A/(Reinforcements on A + Reinforcements on B

or

Pr(A) = Ra/Ra+Rb = rA/rA+rB

[Host's Note: Hmm... I think there should be (..) in the denominators here
and in the similar fractions below? ...Rick]

giving us the proportion of responses which will be accounted for by
response A.

To give an example, we can place two schedules of reinforcement against
each other and see if the relationship expressed in the equation holds
true (it does). So if we had a constrained choice between two equivalent
responses A and B, with choice A being reinforced on an VR3 schedule, and
choice B on a VR6 schedule (see LO20774 for an explanation of schedules of
reinforcement), we can plug these values in the equation and see what
split of responding might be expected.

Pr(A) = (1/3)/(1/3)+(1/6) = .667

So we would predict that choice A would be preferred over choice B by a
ratio of 2:1 (choice A would be chosen 66.7% percent of the time). And,
as it turns out, this is very consistent with what has been demonstrated
empirically. Notice that the prediction isn't in line with a maximization
prediction which would predict that Choice A would receive all of the
responses, resulting in the maximum possible reinforcement.

OK so it becomes clear that so far we aren't talking about people, but
about rats pushing feed levers, and the question is why don't they
always push the lever that gets a food pellet in half the time, right?
Again we
come to the question, do all the rats follow this 2/3 rule?

On the couple of occasions when I have allowed myself to go along to a
slot machine place, I have sometimes put my nickel in one or more
adjoining slots to the one I was feeding mostly, to see whether the
payoff was
better there. Is that similar to, or different from, what the rats
were doing
when they diverged from the "maximiation prediction?"

Another example: Choice A on a VR4 schedule and Choice B on a VR11
schedule

Pr(A) = (1/4)/(1/4) + (1/11) = .25/(.25 + .09091) = .733331

or, Choice A would be chosen 73.3% percent of the time and Choice B chosen
the remaining 26.7% of the time.

How long did it take for a rat to figure out the different payoffs of
the different choices? And what were they doing on the way to that
knowledge?

....

So we have the following three dimensions:

1) A measure of effort in the form of the frequency of reinforcement given
a response (the probability/certainty of reinforcement.

2) The delay of the occurrence of the reinforcement from the occurrence of
the behavior

3) The magnitude of the reinforcer (extent to which it is perceived as
positive or negative).

Giving us the following equation for the matching law given 2 competing
responses, A and B:

Pr(A)= Ra/(Ra+ Rb) =
(rA*(1/Da)*Ma)/(rA*(1/Da)*Ma) + (rB*(1/Db)*Mb)

...

And the application of the relationship has
been explored in areas far outside the animal lab, including consumer
choice behavior in internet shopping.

This might be interesting. Could we see some details?

The system becomes more and more
complex as we extrapolate into "reality", but the relationship has
continually demonstrated its utility (does this make me a pragmatist? ;-).
Using the matching law, we can then begin to apply the lens to the world
around us and seek to understand many behaviors that are seemingly
"irrational" (which I would propose falls closely in line with a
reinforcement maximization model, as opposed to a matching model).

This is not clear to me here; do you mean the maximization model would
seem to be the "rational" one and the matching model would seem to be
the "irrational" one? Also, "seeking to understand many behaviors..."
is a
powerful stimulant to me, whether we are talking about rational or
irrational, but to say that many studies have confirmed that real
organisms follow
the matching law in many circumstances does not qualify as _explanation_
in
my vocabulary. I would call it establishing (a) phenomena (-on), and
that
is well and good. Science begins, I believe, with curiosity about the
underlying workings of interesting phenomena, but _explanation_ requires
the elucidation of how the phenomena work, or in this case, WHY rats or
humans follow a matching rule rather than a maximization rule in the
cases in
question.

Fred Nickols

Best, Dick R.

[Richard_Kennaway2]

[From Richard Kennaway (990507.1439 BST)]

[From Fred Nickols (990505.1740 EDT)] --

Herrnstein's matching law, in its general form, can be expressed in the
following equation:

Pr(A) = k( Ra/Ra+Re )

Where

Pr(A) = probability of behavioral response A
k = highest # of possible behavioral responses available (total # of

responses)

Ra = Reinforcement experienced for response A
Re = Reinforcement experienced for extraneous behavior (behavior other
than the response of interest)

I'm not clear on what these mean. Assuming an experiment involving
lever-pressing to get food-pellets, with each lever yielding identical
pellets, is Ra the rate of delivery of pellets per unit time, or per press?

I think Bill's analysis in [(990506.1225 MDT)] took it as per unit time (in
which case the matching law makes no sense -- Ra/rA is pellets per unit
time/presses per unit time = pellets per press, a property of the lever
independent of the rat).

What's said later in the article suggests Ra is pellets per press. In that
case, Ra/rA has no particular physical interpretation. Ra/rA = Re/rE is of
course still a mathematically equivalent way of writing the matching law.

-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
   School of Information Systems, Univ. of East Anglia, Norwich, U.K.

[Abbott_Bruce]

[From Bruce Abbott (990507.0955 EST)]

Bill Powers (990506.2-004 MDT) --

Bruce Abbott (990506.1715 EST)

Sorry, Bill, but I must remind you once again that the matching law does not
and was never meant to describe the relationship between relative rate of
responding and relative rate of reinforcement on concurrent ratio schedules.

But in the post to which I replied that is how it was described.

Please cite the relevant description. I don't see it. Where does it say
that the matching law was developed to describe performance on concurrent
ratio schedules?

That's
what the equations say, when you reduce them to simplest form. I don't care
what they were "meant" to describe.

_What_ is what the equations say? You're being vague. The equations say
that the ratio of response to reinforcement will be the same across
schedules. _YOU_ (not the equations) state that this cannot happen if the
_schedules_ are different. This is correct only if you assume ratio
schedules, but the matching law does not apply to performance on concurrent
ratio schedules. When applied to, say, concurrent VI VI, what the equations
say not only _can_ happen when the schedules differ, it _does_ happen.

Many a person has foundered because of
not realizing what his own mathematical manipulations imply. Perhaps in
reality there is some other relationship involved, but my transformations
of the cited equations are exactly correct, and the original set of
equations says the same thing -- identically the same thing -- that my
final version does: Ra/rA = Rb/rB. Unless you can repeal mathematics, no
other conclusion is possible.

Yeah, and my transformations of Boyle's law are exactly correct, too, but
that doesn't make the law apply to liquids. The problem isn't with your
math, it's with your assumption that the equations are supposed to say
something about responding on concurrent ratio schedules. They aren't, and
they don't. I don't know how to state that any more clearly. So long as you
persist in assuming that the matching law was designed to account for
responding on concurrent ratio schedules, you are going to draw conclusions
which, though accurate given that assumption, do not hold for those cases to
which the matching law actually applies.

The best I can say is that whatever Herrnstein was trying to express, his
equations don't say it.

The fault isn't with Herrnstein or his equations. The problem is
exclusively with your attempt to apply the matching law where it does not apply.

I think I know what Herrnstein was trying to say. He wanted to say that
given a choice of keys with different schedules, the animal will devote the
most time to the schedules that yield the greatest average ratio of
reinforcement to behavior.

No. What Herrnstein's equations state is that the relative rate of pecking
at a key will match the relative rate of reinforcement delivered on the VI
schedule associated with that key.

He may have thought that his equations were
saying this, but they were not. Instead, his equations say that the yield
of reinforcements per unit behavior is the same on all keys.

That's just another way of saying the same thing.

He simply
didn't know how to write out, mathematically, the idea he was trying to
express. What he did write down says something very different from what he
wanted to say. And evidently he wasn't enough of an algebraist to realize
that his complex-looking forms can actually be reduced to a much simpler
form that make his mistake easy to see.

The only one doing any blundering here is William T. Powers.

I suspect that the mistake goes deeper than this, but the proof or disproof
would take more thought than I have given it.

It doesn't take much thought -- I've already explained what you're doing
wrong (for the second time now in two posts). Are you going to persist in
ignoring that problem? For what purpose?

Regards,

Bruce

[Abbott_Bruce]

[From Bruce Abbott (990507.1010 EST)]

Richard Kennaway (990507.1439 BST) --

I'm not clear on what these mean. Assuming an experiment involving
lever-pressing to get food-pellets, with each lever yielding identical
pellets, is Ra the rate of delivery of pellets per unit time, or per press?

Ra is the rate of delivery of pellets per unit time.

I think Bill's analysis in [(990506.1225 MDT)] took it as per unit time (in
which case the matching law makes no sense -- Ra/rA is pellets per unit
time/presses per unit time = pellets per press, a property of the lever
independent of the rat).

Ra/rA would indeed be a property of the "lever" (schedule associated with
the lever) independent of the rat if the schedule in question were a ratio
schedule, which imposes a particular ratio of lever-presses per pellet
delivery. On other schedules this ratio will depend on the rate of
lever-pressing. For example, on a variable-interval schedule having a
maximum interval of 60 seconds, if the rat pressed at intervals of 60-s or
greater, there would be one press per reinforcement. On the same schedule,
however, if the rat pressed at a rate of several times per second, then
depending on the mix of intervals in the schedule, it might be possible for
the rat to produce dozens or even hundreds of lever-presses per pellet. In
that case the matching law states that the rate of pressing on each lever
would be adjusted by the subject so as to produce an equal ratio of
lever-presses per pellet across the schedules, within the limits of
experimental error.

To answer one of Dick Robertson's questions, the data are reported for
individual subjects (they are not group averages), but do represent average
rates over the duration of a session.

Regards,

Bruce

[Richard_Kennaway2]

[From Richard Kennaway (990507.1628 BST)]

[From Bruce Abbott (990507.1010 EST)]
Ra is the rate of delivery of pellets per unit time.

...

Ra/rA would indeed be a property of the "lever" (schedule associated with
the lever) independent of the rat if the schedule in question were a ratio
schedule, which imposes a particular ratio of lever-presses per pellet
delivery.

Ah. So the matching law does not -- mathematically cannot -- hold for
ratio schedules? For what circumstances is the matching law claimed to
hold? The posted article didn't say.

I have a few other questions of detail: What are the VR3 and VR6 schedules
mentioned in the posted article? (If someone can send me the article
"LO20774" it references, that will do.) Where do the numbers 1/3 and 1/6
in the article come from?

Sorry if this is basic background knowledge to anyone with any acquaintance
with experimental psychology, but I'm not a psychologist.

-- Richard Kennaway, jrk@sys.uea.ac.uk, http://www.sys.uea.ac.uk/~jrk/
   School of Information Systems, Univ. of East Anglia, Norwich, U.K.

[Abbott_Bruce]

[From Bruce Abbott (990507.1525 EST)]

Bill Powers (990507.12026 MDT) --

You will notice that nothing but ratio schedules is discussed. This is the
only kind of schedule mentioned in the whole post. The discussion certainly
does not seem to limit application of the matching law to concurrent VI
schedules. I'm not doubting your word, just noting that the author seems to
be operating under a mistaken assumption if you're right.

Either he's operating under a mistaken assumption, or he's assuming
something he didn't explain in his post. Herrnstein originally formulated
the matching law to _describe_ (not explaiin) a relationship he observed
when pigeons performed on concurrent VI VI schedules. He noted at the time
that the matching law would not apply to concurrent ratio schedules because
on such schedules particular ratios of responses to reinforcements is
enforced by the schedules.

Later experiments showed, however, that the matching law also predicted the
relative rate of responding on the initial link of a current _chains_
schedule. On such a schedule, identical VI schedules are programed
simultaneously on both keys in the "initial links" of the chains, and both
keys are illuminated white. If one of the keys should "pay off," the pigeon
enters the "terminal link" associated with that key: the key changes color
(say, to green), the other key goes dark and becomes inoperative, and
responding on the green key is reinforced on some other schedule, VR-5 for
example. When the requirement of that schedule is satisfied, the pigeon
receives access to grain for some fixed time-period (e.g., 4 seconds). This
completes the terminal link. Both keys become white again, and the pigeon
is returned to the initial link. Something similar happens if the other key
"pays off" in the initial link: the key changes color (say, to red), the
other key goes dark and becomes inoperative, and responding on the red key
is reinforced on yet another schedule, VR-15 for example. When the VR-15
schedule pays off, the bird receives access to grain again (for the same
duration as before), and is then returned to the initial links of the
concurrent chains.

In a concurrent schedule, the relative rate of responding observed at any
moment may be affected by time-related changes in response rate such as are
associated with fixed interval schedules. With identical VI schedules in
the initial links, concurrent chains schedules avoid this problem.

What was observed is that the relative rate of responding on a key _during
the (concurrent) initial links matched the relative rate of reinforcement
associated with the schedules in the terminal links.

Subsequent research has disclosed many situations in which strict matching
is not obtained, thus limiting the applicability of the matching law as a
description of performance on current or current chains schedules. For
example, one can program fixed-interval versus variable-interval schedules
in which the interval parameter is the same (e.g., FI 15-s vs VI 15-s), and
obtain a _bias_ for the VI schedule, despite the fact that on average, both
schedules "pay off" equally often.

Another possibility that the author of the piece we are discussing may have
had in mind is the use of relative _time spent_ engaging in an activity
rather than the relative rate of responding. In that case there is no
_necessary_ relationship between relative rate of keypecking and relative
rate of grain delivery, even on ratio schedules. (The pigeon could spend
more or less time completing a given ratio, and thus could make the relative
time spent standing before a key and pecking at it match the relative rate
of grain delivery.) However, I am no mind reader. It may simply be, as you
suggest, that this author does not realize that matching cannot be obtained
on concurrent ratio schedules when the ratios differ across keys.

Whatever the case, Herrnstein, at least, was aware of the problem.

I should note here that a huge amount of research investigating what goes on
on these schedules has resulted in several competing analyses, some based on
the idea that the pigeons (or other animals) are able to integrate the rate
of occurrence of food over time and respond based on these estimates
("molar" theories) and some based on the idea that responding is sensitive
instead to moment-by-moment events ("molecular" theories). The latter
accounts suggest that matching simply emerges as a byproduct of such
strategies as "momentary maximizing." Furthermore, whether strict matching
actually occurs even under those conditions where it has been assumed to
occur, has been debated. Also, predictions of rival accounts have been
similar enough that it has been difficult to distinguish them empirically,
given the level of experimental error usually present in the data.

The author also says that the equations account for 90% of the variance.
Can someone cast this in terms of RMS prediction error? Accounting for 90%
of the variance might be very good, or very mediocre, depending on what
alternative hypotheses might predict. For example, it's conceivable that a
straight line would fit the data as well as the curve predicted from the
matching law, especially when the data contain large amounts of
deliberately-introduced random noise (the "variable" schedules).

Matching predicts a straight line, not a curve, when formulated as relative
rate of pecking as a function of relative rate of food delivery.
Reformulated as a ratio (P1/P2 = r1/r2), the plot is linear in log-log
coordinates. Accounting for 90% of the variance means that Pearson r,
squared, is .90, implying a Pearson r of about .95. RMS error would be a
bit under 1/3 of the standard deviation of the relative response measure.

The results obtained in such experiments also really ought to be compared
with the results of a simulation using the same schedules but with the
simulated rat pressing different levers at random. That would give us a
baseline against which to evaluate proposals about systematic
relationships. I've proposed this before, but I guess no one is interested
in doing it if I don't do it.

We've talked about this before. If a simulated rat is pressing different
levers at random, then on average, the relative rate of responding will be
.50 (for two levers) regardless of the schedules programmed on the levers.
You won't get matching, except by coincidence (very unlikely).

But be my guest if you need empirical verification.

Regards,

Bruce

[Bill_Powers1]

[From Bill Powers (9905071520 MNDT)]

Bruce Abbott (990507.1525 EST)--

We've talked about this before. If a simulated rat is pressing different
levers at random, then on average, the relative rate of responding will be
.50 (for two levers) regardless of the schedules programmed on the levers.
You won't get matching, except by coincidence (very unlikely).

But be my guest if you need empirical verification.

Not clear whether matching will result on a VI schedule, but if you don't
care, I don't either. End of thread, from here.

Also, I've said all I have to say about coercion. I think I'll just stop
using the word.

New subject:

I notice that Bob Eberlein (Vensim) seems to have disappeared. I have a
question that maybe someone here would care to research. In the advanced
version of Vensim, there is a very nice process for automatically varying
the parameters of a model to get the best fit between model and data. It's
referred to on the screen as the "Powell" method. I would really like to
know how this method works, because it works a whole lot better than my way
of fitting parameters. Can anyone help?

Best,

Bill P.

[Abbott_Bruce]

[From Bruce Abbott (990507.1820 EST)]

Bill Powers (9905071520 MNDT) --

Bruce Abbott (990507.1525 EST)

But be my guest if you need empirical verification.

Not clear whether matching will result on a VI schedule, but if you don't
care, I don't either. End of thread, from here.

It's not that I don't care, Bill, it's that I already know the answer.
Using the values of VI typically explored in these studies (and the response
rates that typically develop on them), the rate of food delivery obtained
very nearly equals the rate implied by the schedule value. Random
responding at the observed rates on the levers will produce equal response
rates across levers in the long run, so P1/(P1+P2) will very nearly equal
.50, yet r1/(r1+r2) will remain close to its specified values. If the
relative rate of reinforcement is not .50, matching will not occur.

This result shows that the matching often observed on concurrent VI VI
schedules is not a trivial consequence of "random responding" on the levers
or keys. Because of the nature of VI schedules, fairly large changes in
P1/(P1+P2) have little effect on the obtained r1/(r1+r2). Even so, the
former value (under the subject's control) changes to match (within
experimental error) the latter value. That is, empirically one observes
matching.

Regards,

Bruce

[Bill_Powers1]

[From Bill Powers (990507.1935 MDT)]

This result shows that the matching often observed on concurrent VI VI
schedules is not a trivial consequence of "random responding" on the levers
or keys. Because of the nature of VI schedules, fairly large changes in
P1/(P1+P2) have little effect on the obtained r1/(r1+r2). Even so, the
former value (under the subject's control) changes to match (within
experimental error) the latter value. That is, empirically one observes
matching.

The reason for suggesting the random pressing was not to show that this
would produce matching (although that would be interesting). It was to give
us a baseline against which to judge how much we gain in going from the
random-pressing hypothesis to the matching hypothesis. If it turned out
that the random-pressing hypothesis accounted for 10% of the variance and
the matching hypothesis account for 90%, the net gain is 80% of the
variance. On the other hand, if the random hypothesis accounted for 60% of
the variance ...

Best,

Bill P.

[Abbott_Bruce]

[From Bruce Abbott (990507.2230 EST)]

Bill Powers (990507.1935 MDT) --

The reason for suggesting the random pressing was not to show that this
would produce matching (although that would be interesting). It was to give
us a baseline against which to judge how much we gain in going from the
random-pressing hypothesis to the matching hypothesis. If it turned out
that the random-pressing hypothesis accounted for 10% of the variance and
the matching hypothesis account for 90%, the net gain is 80% of the
variance. On the other hand, if the random hypothesis accounted for 60% of
the variance ...

Randomly distributing responses across a pair of keys will produce only
random variations from 50-50 responding [P1/(P1+P2) = .50], uncorrelated
with manipulations of relative rate of reinforcement [r1/(r1+r2)].
Therefore, the random hypothesis will account for 0% of the variance ...

Regards,

Bruce

[Bill_Powers1]

[From Bill Powers (990507.12026 MDT)]

Bruce Abbott (990507.0955 EST) --

_What_ is what the equations say? You're being vague. The equations say
that the ratio of response to reinforcement will be the same across
schedules. _YOU_ (not the equations) state that this cannot happen if the
_schedules_ are different. This is correct only if you assume ratio
schedules, but the matching law does not apply to performance on concurrent
ratio schedules. When applied to, say, concurrent VI VI, what the equations
say not only _can_ happen when the schedules differ, it _does_ happen.

Here are some excerpts from the post Fred sent:

···

========================================================================
[From Fred Nickols (990505.1740 EDT)] --

Herrnstein was able to show in numerous settings and contexts that the
probability of a given behavioral response was directly related to the
availability of reinforcement for the behavior (in relation to the
availability of reinforcement available for competing behavioral
responses). This relationship that he articulated and demonstrated has
been replicated by many other researchers as well, and forms the
foundation of much of what I have written about over the past 6 months.

This relationship, has been reliably demonstrated to account for over 90%
of the variance in responses. One point of interest, this relationship is
not necessarily a reinforcement maximization rule, and, at times, will
predict patterns of behavioral response that are counterintuitive.

To give an example, we can place two schedules of reinforcement against
each other and see if the relationship expressed in the equation holds
true (it does). So if we had a constrained choice between two equivalent
responses A and B, with choice A being reinforced on an VR3 schedule, and
choice B on a VR6 schedule (see LO20774 for an explanation of schedules of
reinforcement), we can plug these values in the equation and see what
split of responding might be expected.

Pr(A) = (1/3)/(1/3)+(1/6) = .667

(Note that the author doesn't know how to write equations. As written, the
above gives a Pr(A) of 1.167)

So we would predict that choice A would be preferred over choice B by a
ratio of 2:1 (choice A would be chosen 66.7% percent of the time). And,
as it turns out, this is very consistent with what has been demonstrated
empirically. Notice that the prediction isn't in line with a maximization
prediction which would predict that Choice A would receive all of the
responses, resulting in the maximum possible reinforcement.

Another example: Choice A on a VR4 schedule and Choice B on a VR11
schedule

Pr(A) = (1/4)/(1/4) + (1/11) = .25/(.25 + .09091) = .733331

or, Choice A would be chosen 73.3% percent of the time and Choice B chosen
the remaining 26.7% of the time.

==========================================================================
You will notice that nothing but ratio schedules is discussed. This is the
only kind of schedule mentioned in the whole post. The discussion certainly
does not seem to limit application of the matching law to concurrent VI
schedules. I'm not doubting your word, just noting that the author seems to
be operating under a mistaken assumption if you're right.

The author also says that the equations account for 90% of the variance.
Can someone cast this in terms of RMS prediction error? Accounting for 90%
of the variance might be very good, or very mediocre, depending on what
alternative hypotheses might predict. For example, it's conceivable that a
straight line would fit the data as well as the curve predicted from the
matching law, especially when the data contain large amounts of
deliberately-introduced random noise (the "variable" schedules).

The results obtained in such experiments also really ought to be compared
with the results of a simulation using the same schedules but with the
simulated rat pressing different levers at random. That would give us a
baseline against which to evaluate proposals about systematic
relationships. I've proposed this before, but I guess no one is interested
in doing it if I don't do it.

Anyway, what I am saying is very simple;

Given the statement

(1) Ra/(Ra + Rb) = ra/(ra + rb),

we can show that this statement of equality is exactly equivalent to

(2) Ra/ra = Rb/rb

If R is responses and r is reinforcements (or responses per unit time and
reinforcements per unit time, as Richard Kennaway points out), then
equation (1) says, albeit in a non-obvious way, that the responses per
reinforcement are equal on key _a_ and key _b_. Nothing in this equation
says anything about the type of schedule. If the matching law of equation
(1) applies only to VI-VI schedules, then equation (2) also applies only to
VI-VI schedules, because equation (2) IS equation (1). You can't like
equation (1) and not like equation (2).

Is the matching law simply a statement that animals will adjust the
distribution of their presses or pecks so that the average ratio of
reinforcements per press or peck is the same across all keys? If so, I can
see why ratio schedules are ruled out: the equations can be true only if
the schedules are the same on all keys. The interval schedule is the only
type in which it's possible for the schedule NOT to determine completely
the mean ratio of reinforcements to presses/pecks. The animal on an FI or
VI schedule can emit a lot of extra presses/pecks without changing the mean
number of reinforcements per session [approximately (session length)/I], so
the ratio of presses/pecks per reinforcement can vary.

Best,

Bill P.

[Bill_Powers1]

[From Bill Powers (990508.1006 MDT)]

Bruce Abbott (990507.2230 EST--

Randomly distributing responses across a pair of keys will produce only
random variations from 50-50 responding [P1/(P1+P2) = .50], uncorrelated
with manipulations of relative rate of reinforcement [r1/(r1+r2)].
Therefore, the random hypothesis will account for 0% of the variance ...

But the pressing is causing the reinforcements, via the apparatus, so there
can't be zero correlation.

I prefer to be shown the error of my ways by demonstration. But if you're
sure you're right, I suppose an actual experimental test isn't necessary. I
guess I just don't have your faith in Pure Reason.

Best,

Bill P.

[Bill_Powers1]

[From Bill Powers (990508.1045 MDT)]

Bruce Abbott (990507.2230 EST)--

Randomly distributing responses across a pair of keys will produce only
random variations from 50-50 responding [P1/(P1+P2) = .50], uncorrelated
with manipulations of relative rate of reinforcement [r1/(r1+r2)].
Therefore, the random hypothesis will account for 0% of the variance ..

Thinking this over with half a notion of writing a test program, I realize
I don't understand what you mean. Are you saying that the reinforcements
are "manipulated" independently of the behavior? I thought the
reinforcements were generated as a function of the pattern of presses
acting via the properties of the apparatus. All I'm proposing is that we
substitute an artificial animal pressing randomly among the keys for the
real animal pressing according to whatever principles operate in it, on
concurrent VI-VI schedules, and calculate the degree of equality between
the measured Ra/(Ra + Rb) and the measured ra/(ra + rb) for the artificial
rat.

Or is there some other way in which the matching hypothesis is evaluated?

I'm not familiar with how such VI experiments are laid out -- contemplating
writing a program is a great way to discover details about which one is
hazy. Here are some things I don't know and need to know to write a program.

1. Does the next interval start at the instant the previous interval ends,
or when the reinforcer is actually delivered, or after the animal has
collected the reinforcer and is ready to start pressing again (and how
would that be determined)?

2. If an animal is pressing rapidly and overruns, so there are several
presses after the interval ends and before the food is collected, are the
extra presses or pecks counted as "responses" in the previous interval, or
do they add to the total for the next interval, or are they dropped?

3. In a schedule designated as, for example, VI 5 minutes, how are the
minimum and maximum intervals determined, and what is the distribution
between the limits? It seems to me that not all "VI 5 min" schedules are
comparable without these three additional parameters being specified. What
should I use in setting up a VI schedule? I'll use whatever is customary.

4. It would be nice to have some data from a real rat on the same
concurrent VI-VI schedule used for the artificial rat, just to see how the
matching hypothesis fares in each case. Can you or anyone else out there
supply that kind of information?

5. And finally, how are Ra, Rb, ra, and rb measured?

I could, of course, just go ahead and guess the answers, but it would save
me time to know what the right answers are.

Best,

Bill P.

[R_Marken]

[From Rick Marken (990508.0920)]

Bruce Abbott (990507.1820 EST)

I would like to know why the matching law is considered so
important by behaviorists (if it is). What does matching
tell us about the behavior of living systems? Does it
confirm reinforcement theory? Control theory? S-R theory?

Random responding at the observed rates on the levers will
produce equal response rates across levers in the long run,
so P1/(P1+P2) will very nearly equal .50, yet r1/(r1+r2) will
remain close to its specified values.

I don't undertsand this. What are the "specified values" of
r1/(r1+r2)?

If the relative rate of reinforcement is not .50, matching
will not occur.

Is this an observation or a calculation?

By he way, do you have any data that we could look at? I would
be interested in seeing the average response rates (P1 and P2)
and corresponding average reinforcement rates (r1 and r2) from
experiments using a several different VI VI schedules. I presume
the rate values will satisfy the equality:

  P1/(P1+P2) = r1/(r1+r2)

But it would be interesting (to me) to see what the actual
vaues are.

Best

Rick

···

---

Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken/

[R_Marken]

[From Rick Marken (990507.1350)]

Bruce Abbott (990507.2230 EST)--

Randomly distributing responses across a pair of keys will produce
only random variations from 50-50 responding [P1/(P1+P2) = .50],
uncorrelated with manipulations of relative rate of reinforcement
[r1/(r1+r2)]. Therefore, the random hypothesis will account for
0% of the variance ..

Bill Powers (990508.1045 MDT)]

Thinking this over with half a notion of writing a test program,
I realize I don't understand what you mean.

I agree that Bruce's statement above is puzzling but I am
starting to think (I'm willing to be corrected) that such a
program won't shed much light on the matching law per se. At
the level at which the matching law is described there is no
need (I think) to look at the details of how response patterns
are turned into reinforcement rates. I think just a "molar"
analysis alone can show that the matching law reveals nothing
more about behavior than the fact that matching feedback
functions produce "matching" behavior; that is, I think the
matching law tells us nothing about the organism who shows
matching; rather, it tells us about the nature of the environment
in which that organism is trying to feed itself.

The matching law says that

P1/(P1+P2) = r1/(r1+r2)

(where Pi is response rate on key i and ri is reinforcement
rate on key i). Bruce A. says that this law only applies to VI
schedules. I believe that, at the level this law is described,
it is sufficent to view a VI (or any) schedule as a "black box"
that transforms an input response rate (Pi) into an output
reinforcement rate (ri). We know, from the physical set up of
the operant situation, that an observed reinforcement rate
(ri) is completely determined (somehow) by the corresponding
response rate (Pi). So ri will be an observed proportion of
Pi; this proportion (ri/Pi), I argue, is a sufficient
characterization of the feedback function relating output
(Pi) to input (ri) for the VI (or _any) schedule -- at least
for purposes of analyzing the matching law.

This proportion (ri/Pi) shows up in Bill's equivalent algebraic
representation of the matching law:

r1/P1 = r2/P2.

This means that we will see matching [P1/(P1+P2) = r1/(r1+r2)]
only if the the feedback functions (ri/Pi) on the two keys are
equal; that is, only if the average rate of reinforcement per the
average rate of responding is the same on each key.

I believe that this is a mathematical fact, independent of the
_type_ of schedule (VI, VR, R, I, etc) that determines the
transformation of response rate (ri) into reinforcement rate (Pi).

I think all that a simulation of behavior on a VI schedule can
show (re: the matching law) is that some average rate of responding
(Pi) is transformed into some average rate of reward (ri). That is,
the simulation will show that a particular VI schedule is associated
with some proportional relationship between ri and Pi. This
proportion (ri/Pi) is likely to be different for different VI
schedules. But this seems irrelevant to the matching law because
we already _know_ from the mathematics of the matching law that
there will be matching (that is, P1/(P1+P2) = r1/(r1+r2)) _only_
if the feedback functions (ri/P1) for the VI schedules on the two
keys in the matching procedure are equal (or nearly so); ie.
r1/P1 = r2/P2. So we will only see matching if the VI functions
for the two keys have the same feedback functions (ri/Pi). This
must be true whatever those VI functions are (in terms of average
and variance of intervals) and even if the VI functions are
_different_ for each key. There will be no matching unless the
feedback functions (ri/Pi) for the schedules on the two keys are
_equal_.

I did test this out in a spreadhseet; everything looked OK. But
feel free to fire away.

Best

Rick

···

---
Richard S. Marken Phone or Fax: 310 474-0313
Life Learning Associates e-mail: rmarken@earthlink.net
http://home.earthlink.net/~rmarken/

[Abbott_Bruce]

[From Bruce Abbott (990508.1815 EST)]

Bill Powers (990508.1045 MDT) --

Bruce Abbott (990507.2230 EST)

Randomly distributing responses across a pair of keys will produce only
random variations from 50-50 responding [P1/(P1+P2) = .50], uncorrelated
with manipulations of relative rate of reinforcement [r1/(r1+r2)].
Therefore, the random hypothesis will account for 0% of the variance ..

Thinking this over with half a notion of writing a test program, I realize
I don't understand what you mean. Are you saying that the reinforcements
are "manipulated" independently of the behavior?

The schedule values are varied parametrically over the course of the
experiment to produce different values of r1/(r1+r2). For example, one
might program VI 15-s versus VI 45-s, VI 30-s versus VI 30-s, and so on.
(Here, the overall rate of food delivery programmed by the schedules was
kept constant at 2 deliveries per minute across the two conditions given.)

I thought the
reinforcements were generated as a function of the pattern of presses
acting via the properties of the apparatus.

They are. However, the rate of pressing associated with these schedules is
usually high enough so that fairly large variations in rate of pressing
produce very little variation in rate of food delivery.

All I'm proposing is that we
substitute an artificial animal pressing randomly among the keys for the
real animal pressing according to whatever principles operate in it, on
concurrent VI-VI schedules, and calculate the degree of equality between
the measured Ra/(Ra + Rb) and the measured ra/(ra + rb) for the artificial
rat.

Yes, I know.

Or is there some other way in which the matching hypothesis is evaluated?

Nope.

I'm not familiar with how such VI experiments are laid out -- contemplating
writing a program is a great way to discover details about which one is
hazy. Here are some things I don't know and need to know to write a program.

1. Does the next interval start at the instant the previous interval ends,
or when the reinforcer is actually delivered, or after the animal has
collected the reinforcer and is ready to start pressing again (and how
would that be determined)?

It's been done both of the first two ways; let's assume that the next
interval begins when the pellet is actually delivered.

2. If an animal is pressing rapidly and overruns, so there are several
presses after the interval ends and before the food is collected, are the
extra presses or pecks counted as "responses" in the previous interval, or
do they add to the total for the next interval, or are they dropped?

Typically all responses are counted.

3. In a schedule designated as, for example, VI 5 minutes, how are the
minimum and maximum intervals determined, and what is the distribution
between the limits? It seems to me that not all "VI 5 min" schedules are
comparable without these three additional parameters being specified. What
should I use in setting up a VI schedule? I'll use whatever is customary.

It's been done in a number of ways, including but not limited to both
arithmetic and geometric progressions. Probably the best is to use a
constant probability schedule: you can program this by using the random
function to generate a number between 0 and 1 at each interval dt, and
"setting up" a food delivery if the value is less than or equal to some
value x that is computed based on the desirec average interval size. For
example, if dt = .1 sec, then to produce an average interval size of 30 sec
(for a VI 30-s schedule), x would be 1/(dt*I) = 1/(.1*30) = 1/300.

4. It would be nice to have some data from a real rat on the same
concurrent VI-VI schedule used for the artificial rat, just to see how the
matching hypothesis fares in each case. Can you or anyone else out there
supply that kind of information?

Most of these studies have been carried out using pigeons pecking at keys
for access to grain. I think I know where I can find data from a review and
was published in JEAB a few years ago.

5. And finally, how are Ra, Rb, ra, and rb measured?

ra and rb are the number of food-accesses per minute, per hour, or per
session (it doesn't matter what the divisor is so long as it is the same for
all measures); Ra and Rb are the number of keypecks or, alternatively, the
time spent in each activity (the two measures correlate highly). As most
studies have used number of keypecks, let's stick to that; it's easier to
measure.

I could, of course, just go ahead and guess the answers, but it would save
me time to know what the right answers are.

Those are excellent questions; I hope you found my answers adequate.

Regards,

Bruce