Study 9: The Matching Law and Concurrent Schedules of
21 March 1999
Rats will be given a choice between two variable interval
schedules of reinforcement.
Different rats will be given different combinations of schedules.
A matching law analysis will be used to determine whether the
number of responses made on one lever is systematically related
to the relative rate of reinforcement on that side.
Our previous study looked at behavior on a concurrent
VI-VI schedule of reinforcement.
In particular, we examined the changeover behavior of
our rats---how likely they would be to switch from one side
(with a certain schedule of reinforcement) to another (with a
different schedule of reinforcement).
Our focus was on the characteristics of this switching
Left unanswered was the question of what controlled
For example, you may have observed that the mean
interchangeover time of your rat was longer for the side
reinforced by the VI-30 second schedule.
This pattern would indicate that the rat "prefers" the
VI-30 second schedule.
What determines this preference?
Clearly, reinforcement must play a role.
In a landmark article, Herrnstein (1961) suggested that
choice in a situation like ours could be quantitatively related
to the reinforcements earned by the subject.
He proposed a formulation, known as the matching law,
in which the proportion of behavior exhibited depended upon
the reinforcers earned from that behavior.
Specifically, if BL and BR were the amounts of behaviors
exhibited to left and right alternatives, respectively, and RL
and RR were the reinforcers earned from those alternatives,
then the following relation would hold:
BL/(BL+BR) = RL/(RL+RR)
Herrnstein's equation suggests a special view of the way
reinforcement determines choice.
According to what is now termed the molar view of
reinforcement, the overall number of reinforcers (or the
overall rate of reinforcement) is the controlling factor.
In contrast, the molecular view regards the moment-by-moment
probability of reinforcement as the controlling
(Interestingly, Herrnstein has recently developed a
theory of reinforcement, known as melioration theory, that
tends to be molecular.)
The analysis of changeover times can be used to evaluate
molar vs molecular descriptions of concurrent behavior.
For example, Heyman (1979) reported a statistical
analysis of changeover times that supported a molar
(Now that you've had some experience with
interchangeover times, I thought you would be interested to
see how they could be used; I've therefore put Heyman's
article out as the "feature reading" for this week...) For our
study this week, however, we will attempt a simpler
analysis, using a version of the matching law formulated by
Our Sprague-Dawley rats will serve as subjects.
We will be using the six custom-constructed
chambers to test our animals.
Each of these chambers will be fitted with two response
levers that the animal can press.
In addition, we will use electric clocks to
measure time intervals during training.
Reinforcers will consist of chocolate sprinkles
delivered to the food cup in each box after a tap on the
Our lab will again be run in a shift 60 minutes
During that time we will reinforce the rats according to a
concurrent VI-VI schedule.
Rats 1 and 7 will be reinforced with a VI-45
second schedule on each lever (according to Tables 1 and
2; these tables and others will be made available in class).
Rats 2 and 6 will be reinforced on the left lever with a
VI-60 second schedule and on the right lever with a VI-30 second schedule (given in Tables 3 and 4).
Rats 3 and 4 will be reinforced with a VI-80
second schedule on the left and a VI-10 second schedule
on the right (given in Tables 5 and 6).
Use the respective tables to determine the variable
intervals for the two schedules and reinforce your rat
However, for this study, we will not be recording
Instead, we will measure the number of responses made
to each side, and the number of reinforcers earned on that
Because our rats may need some time to adjust to the
schedules and to show stabilized performance, we will only
use the data from the last 30 minutes of the session.
In a full-fledged study, stability criteria would have to be
carefully considered (Killeen 1978).
After the session, weigh and feed your rat.
Baum's analysis generalizes Herrnstein's matching
law to a power law expression relating the ratio of left and
right behaviors to the ratio of left and right reinforcers.
Specifically, Baum's version of the matching law can be
log(BL/BR) = log(k) + a(log(RL/RR))
where a and k are fitting constants.
If k and a are equal to one, then Herrnstein's version fits
exactly; if they are different from one, then there is some sort
of deviation from matching.
To assess whether our results are consistent with
matching, I would like you to plot the class data.
Specifically, for each rat in the class, compute
log(BL/BR) and log(RL/RR).
Plot the eleven data points on a scatter plot.
If Baum's version of the matching law applies, these
points should be on a straight line.
Decide if we obtained such a result.
Estimate, if you can, the values of a and k.
An excellent way to obtain these estimates is to perform
a linear regression analysis on the data.
Baum, W.M. (1974) On two types of deviation from the matching law: Bias and
undermatching. Journal of the Experimental Analysis of Behavior, 22: 231-242.
Herrnstein, R.L. (1961) Relative and absolute strength of response
as a function of frequency of reinforcement. Journal of the Experimental Analysis of Behavior, 4: 267-272.
Heyman, G.M. (1979) A Markov model description of changeover probabilities on
concurrent variable-interval schedules. Journal of the Experimental Analysis of Behavior, 31: 41-51.
Killeen, P.R. (1978) Stability criteria. Journal of the Experimental Analysis of Behavior, 29: 17-25.
Norman, W.D. and McSweeney, F.K. (1978) Matching, contrast, and equalizing in the concurrent lever-press responding of rats. Journal of the Experimental Analysis of Behavior, 29: 453-462.
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