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Study 9: The Matching Law and Concurrent Schedules of Reinforcement

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 behavior. Left unanswered was the question of what controlled switching behavior. 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:


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 variable. (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 interpretation. (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 Baum (1974).



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 chamber wall.


Our lab will again be run in a shift 60 minutes long. 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 accordingly. However, for this study, we will not be recording interchangeover times. Instead, we will measure the number of responses made to each side, and the number of reinforcers earned on that side. 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 written:
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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