[From Bill Powers (931123.0030 MST)]

I've obtained a paper recommended by Bruce Abbott:

Baum, W. H. (1992); In search of the feedback function for variable

interval schedules. JEAB _57_m 365-375.

This is the same Baum who wrote a letter to Science about my feedback

article in 1973. I corresponded with him for about a year afterward. Our

correspondence ended when I invited him to visit while I was on vacation

in Massachusetts and he declined saying that we had little further to

talk about. He does not cite me or my work.

Baum starts his article this way:

The dynamics of behavior may be understood as the outcome of the

organism and environment interacting together as a feedback system

(Baum 1973, 1981, 1989; Staddon 1983). The organism's functional

relations (or "O-rules"; Baum 1973) link environmental events to

behavioral output. The environment's feedback functions (or "E-

rules"; Baum, 1973) link behavioral output to environmental events

(i.e., consequence). Equilibrium results from the interaction

between functional relations and feedback functions. Any brief

disturbance to the system results in departure from equilibrium, but

once the disturbance is gone, the system stabilizes again. Any

change in a functional relation (e.g., a shift in deprivation) or in

a feedback function (e.g., a change of schedule) results in a new

equilibrium (e.g., Myerson & Hale, 1988; Staddon, 1988). (p. 365)

Very promising start. After going on in an equally promising vein for

another page, Baum comes to the crux of the paper: characterizing the

feedback function for a VI schedule. And here it all falls apart.

One of the "discoveries" Baum announces is that on a VI schedule, the

relation of reinforcement rate to behavior rate for very low behavior

rates is identical to a fixed-ratio 1 schedule. He points out that the

most common equation used for VI schedules,

r = 1/(t+D(B))

does not have this property, and neither do a number of others. He

points out what I noticed when Bruce bought up the Prelec and Herrnstein

equation,

r = B/(tB + a),

which is that the required condition for low behavior rates demands that

a always be 1.

But then matters become confused, because apparently Baum as well as

others are trying to characterize the feedback function in a way that

will explain observed reinforcement and behavior rates on different

schedules as well. So they try to find formulas that will, for example,

"include parameters and a compromise between random and regular

responding." Despite what Baum said about O-rules and E-rules, the

attempt is made to make the E-rules include the O-rules.

As a result, the proposed ways of characterizing the feedback function

for VI schedules become more and more complex, with more and more

parameters: viz:

r = 1/[t + (B/K)(0.5/K + (K-B)/(cB)]

Comparing three models for the observed relation between r and B against

experimental data, Baum finds that all three account for 0.773 of the

variance.

The data to which all three models are fitted show a positive slope of

reinforcements/hr vs responses/min. This suggests that the animals were

behaving in the way I refer to as the left side of the Motherall data in

Staddon's Fig. 7.18. In other words, they are well outside the region of

normal control. The effective ratio of responses to reinforcements is

roughly 60 to 1 and the reinforcement rate from about 3 to 30 per hour.

The long equation above is the Nevin-Baum equation. This is the one Baum

defends on grounds that it takes into account behavior occurring in the

form of alternating pauses and bursts, and other factors, while the

other two equations use arbitrary constants.

Baum puzzles over the very low reinforcement rates (or alternatively,

the too-high response rates) that occur at one extreme. He speculates

that a crossover delay might produce more rational data. He concludes

that we still need theoretically-correct model of the feedback function

for VI schedules.

All this comes about because no model of the behaving organism is

offered: all parties involved seem to think that a single equation can

represent the VI function as well as the organism function. So I must

conclude that Baum did not absorb the meaning of his own introductory

remarks, quoted above.

In fact it is easy to characterize the VI feedback function if we simply

forget about predicting behavior and look for the way in which r depends

on the rate of responding, B. That is completely determined by the way

the scheduling apparatus works. I did this a few weeks ago when we were

talking about constant-probability VI functions. I found, with a fair

degree of confidence, that the average relationship can be expressed as

r = (t/k)(1 - exp(-kB/t).

I think this is what I came up with. This has all the required

characteristics: it becomes equivalent to an FR 1 schedule at low

behavior rates (take the derivative with respect to B and let B approach

0) and approaches an asymptote of t/k at high behavior rates. And it is

derived directly from the properties of a (simple) constant-probability

VI schedule. The constants a and k are completely determined by the

schedule, from basic principles.

Of course to fit behavior and reinforcement to data for a range of

schedules (values of t), it is necessary to have a separate equation

characterizing the organism. That is the step that Baum and all the

others omitted. Without the second equation, finding a curve that will

pass through all the data points becomes an exercise in guessing at

mathematical forms that look as if they have the right shape, and

fitting them to the data. In effect, they are trying to guess at the

solution of the two simultaneous equations, but having offered nothing

at all to represent the organism equation, they are guessing at random.

The proper way to do this is to find a form of the organism equation

such that when it is solved simultaneously with the environment

equation, the resulting values of r and B fit the data for all values of

t (as nearly as possible). If a model of the organism is not proposed,

then we are back to curve-fitting.

This is what I did in simulation for the Staddon data, with fixed-ratio

schedules. I am confident that we will be able to do the same thing for

VI schedules and all the others.

## ···

-----------------------------------------------------------------------

Best,

Bill P.