🚀 go-pugleaf

[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.