Adjusting schedules

[From Rick Marken (960403.1810)]

Bill Powers (960403.1330 MST) derived the following transfer function
for an "adjusting schedule" (a schedule where the actual response rate
affects the number of reinforcements/response allowed by the schedule).

Rate | | B = k/m
     > >
     > v
     > R = B *
     > *
     > *
     > *
     > *
     > * <--- R = 0 --------------------->
     > *
      ----------------* * * * * * * * * * * * * * * * * *>>

Before I saw your post (or the one from Chris Cherpas (960403.0911 PT)) I
had written a program for what turns out to be called an "adjusting
schdule" simply to prove to myself that a schedule such as the one Bruce
Abbott described in the April Fools post was actually possible. My HyperTalk
code for the schedule is as follows:

put 1 into countdown

  repeat forever

    put the mouseClick into z
    if z = "true" and countdown <= 0 then
      set the visible of card button scale to true
      wait 20 ticks
      put 1 into countdown
      set the visible of card button scale to false
    end if
    if z = "true" and countdown > 0 then
      put countdown+4 into countdown
    end if

    if countdown > 0 then put countdown -.05 into countdown

  end repeat

The variable countdown corresponds to the variable you call c. The counter
decrement (your k) is .05; what this value is in decrements/second depends
on the speed of the computer. The counter increment (your m*b) is 4 -- again,
what this value is in increments/second depends on the speed of the
computer and the rate of responding.

Based on my observations of the rate of the reinforcer (the reinforcer
is a picture that appears on the screen briefly --in the program, this
occurs when the statement "set the visible of card button scale to true" is
executed) as a function of my response rate, I think the transfer function
(relating reinforcement rate to behavior rate) looks more like this:

Rate |
     > R =B => *
     > * *
     > * *
     > * *
     > * * <--- R = 0 -------------->
     > * *
      -----------------* * * * * * * * * * * * * * *>>
                         Behavior rate

At very low rates of responding the reinforcement rate is about equal to the
press rate (it's an FR 1 schedule); when I start responding too rapidly the
reinforcement doesn't abruptly stop; rather, the rate of reinforcment
actually becomes slower; ultimately I reach a not too fast rate of
responding where the reinforcment is completely inhibited (because the
counter is refreshed faster than it can count down).

I think this "adjusting schedule" provides a GREAT way to compare
reinforcement theory to control theory; I can't see how reinforcment
theory could possibly account for the behavior of a subject controlling
reinforcement rate under such a schedule. But, then, I've never been right
before about what reinforcement theory can and can't account for; so let's
just say a control study using the adjusting schedule would be interesting.