🚀 go-pugleaf

[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.