A preliminary answer to this question was provided in 1961 by Richard Herrnsteiin. He reported that when pigeons are placed on a concurrent schedule like that described in the example, they tend to allocate their keypecks to the two keys in a highly systematic fashion, so long as steps are taken to prevent the immediate reinforcement of swithing keys. The relationship observed is a particularly simple one: of the keypecks emitted during a session, the proportion delivered to a given key matches the proportion of reinforcers, out of the total earned, that are delivered for responding on that tkey. For example, if 80% of the reinforcers are delivered for responding on the right key, then 80% of the keypecks will be delivered to the right key. Because the two proportions or percentages match, this relationship has been termed **the matching law**.

We can state the matching law a bit more formally in the following equation:

P_{1}/(P_{1} + P_{2}) = R_{1}/(R_{1} + R_{2})

In the equation, P_{1} and P_{2} designate the rate of pecking of the left and right keys, respectively, whereas R_{1} and R_{2} denote the respective rates of reinforcement for pecking at those keys. The values on both sides of the equal sign are proportions; if you multiply each by 100 you get percentages. The matching law states that these two proportions always match (at least, within the limits of experimental error).

Matching can be assessed by plotting the proportion of keypecks emitted on a given key against the proportion of reinforcements delivered for responding in that key. To the extent that matching holds, all points will tend to follow the leading diagonal of the plot, representing all positions on the graph where the two proportions match. Herrnstein's (1961) data for his three pigeons are shown in the figure to right. Note that all points lie fairly close to the diagonal and that there is no apparent systematic deviation from it.

Herrnstein (1961) found that he could obtain approximate matching on these concurrent VI schedules only if he programmed a **changeover delay** (COD) for switching keys. The COD is a period of time immediately following a switch (usually on the order of 1.5 to 2.0 seconds) during which responses cannot be reinforced. Without the COD, pigeons tended to develop a simple "response alternation" pattern, in which they simply pecked left-key, right-key, left-key, right-key. If that happenen then the proportion of left-key responses was always .50 regardless of the proportion of reinforcers delivered for left-key pecking.

Herrnstein reasoned that response alternation developed when there was no COD because of the nature of VI schedules: the longer a pigeon continues to peck at one key, the more likely it becomes that the schedule not being responded on will have set up reinforcement (i.e., the interval it has been timing will be over). Thus, if the pigeon has been responding for a while on, say, the left key, then it is likely that switching to the other key and pecking it will produce immediate reinforcement. This immediate reinforcement of switching behavior would encourage switching, perhaps to the point of response alternation.

By adding a COD, Hernstein made sure that, even if the other schedule had already set up reinforcement, it could not be collected until the COD was over. This meant that switching could never produce immediate reinforcement. Under those conditions, the pigeons did not alternate between the keys, but allocated their keypecks approximately as the matching law predicts.

P_{1} = *k*R_{1}/(R_{1} + R_{2} + R_{O}),

where *k* = a constant and R_{O} ("R Sub-Oh") is the rate of reinforcement for "other" behavior (behavior other than keypecking.

*K*, which has come to be known as **Herrnstein's k**, can be interpreted as establishing how many responses a given subject will expend on average for a single access to the reinforcer. Some subjects will respond more than others for a single reinforcer, so

Herrnstein's equation for absolute rate of responding on a key predicts that as the relative rate of reinforcement for responding on the key increases, the absolute response rate will increase to an asymptotic value. (In fact, *k* represents that asymptotic value.) The *shape* of the function relating absolute rate of responding to relative rate of reinforcement is that of a curve that rises steeply at first and gradually flattens until it reaches the asymptote, and conforms to a portion of the geometric figure called an hyperbola. Because the curve produced by Herrnstein's formula has this form, it has been labeled as **Herrnstein's hyperbola**.

Note that the equation for absolute rate of responding, when used to predict the absolute rate on *each* key, implies the matching law (as it should):

P_{1}/(P_{1} + P_{2}) = [*k*R_{1}/(R_{1} + R_{2})] / [*k*R_{1}/(R_{1} + R_{2})] + [*k*R_{2}/(R_{1} + R_{2})]

The *k*'s and the (R_{1} + R_{2}) terms cancel out, leaving

P_{1}/(P_{1} + P_{2}) = R_{1}/(R_{1} + R_{2})

Having derived his single-key formula, Herrnstein now applied it in a suprising way: to performance on *single* VI schedules.
It was already known that responding increased rapidly at first with increasing *absolute* rate of reinforcement on such schedules, and that the rate of further increase slowed as the curve approached an asymptote. Previous attempts to characterize the curve had resulted in the conclusion that it was well fit by an exponential function. Herrnstein proceded to show that, given the variability in the data, they could be just as well fit by the hyperbolic function if appropriate values for *k* and R_{O} were used. The figure to right shows Herrnstein's hyperbola fit to data from six individual pigeons responding on VI schdules that varied in the rate of reinforcement. The numbers in each panel give the fitted values for *k* and R_{O}.

In fitting his hyperbolic function to the data, Herrnstein was essentially suggesting that single-key responding could be treated as concurrent responding, where the second response option was, not responding on a second key, but rather doing those "other" things that provide their own, intrinsic reward. In effect, Herrnstein was asserting that *all behavior is choice behavior*, involving as it does a choice between doing it and doing something else (anything else) instead. The pigeon responding on a single key for grain reinforcement could do other things besides peck at the key, and to the extent that these other things brought their own rewards, they would take time away from keypecking. It follows that the rate of keypecking *cannot* be predicted *unless you know what other sources of reinforcement are available for doing things other than pecking at the key*!

**Bias** represents a preference for responding on one key more than the other that has nothing to do with the schedules programmed on the two keys. Bias may occur when, for example, one key requires more force to close its contact than the other, so that the pigeon has to peck harder on one key than the other. In that case the pigeon will peck more on the easier key than it otherwise would have, given the relative rate of reinforcement.

Bias shows up on the usual plot of relative rate of responding versus relative rate of reinforcement as a bowing out of the data points away from the leading diagonal that represents matching. Bias often can be prevented or eliminated by making careful adjustments to the apparatus, such as assuring that both keys are equally easy to peck, the keys are illuminated by equally preferred colors, and so on, before running the experiment.

**Sensitivity** refers to the subject's tendency to "overmatch," match, or "undermatch":

**Overmatching**-- the relative rate of responding on a key is*more extreme*(further from .5 or 50%) than predicted by matching. The subject appears to be "more sensitive" to the differences in reinforcement between the two keys than when matching occurs.**Matching**-- the relative rate of responding on a key*matches*the relative rate of reinforcement.**Undermatching**-- the relative rate of resonding on a key is*less extreme*(closer to .5 or 50%) than predicted by matching. The subject appears to by "less sensitive" to the differences in reinforcement between the two keys than when matching ocurs.

The figure to right (from Chung and Herrnstein, 1967) shows a deviation in sensitivity from the matching relationship. Does it show overmatching or undermatching? If you said "overmatching," that is correct. The violation of matching is due to the fact that relative immediacy and relative reinforcement rates were systmatically changing together in the experiment. Because the plot only takes relative frequency of reinforcement into account, the differences in relative immediacy produce a stronger preference for the key offering the greater immediacy than would be predicted from the relative reinforcement rates alone. In other words, they produce overmatching.

The traditional plot of relative reinforcement versus relative response rates can be hard to interpret when systematic deviations from matching appear that may involve both bias and sensitivity effects. It is hard to know how much of the distortion is due to bias, how much to sensitivity, and how much to neither of these. However, a reformulation of the matching law by William Baum eliminated this problem, making it easy to determine the separate influences of bias, sensitivity, and other factors. The new formulation also provided a more general, but less predictive version of the matching law, which has become known as the "generalized matching law."

P_{1}/P_{2} = R_{1}/R_{2}

This new formula can be modified easily to provide two new parameters, *a* and *b*:

P_{1}/P_{2} = *b*[R_{1}/R_{2}]^{a}

*b* represents the **bias**: bias for Key 1 if *b* is greater than 1, bias against K1 if *b* is less than 1, and no bias if *b* equals 1.

*a* represents the **sensitivity**: *overmatching* if *a* is greater than 1, *undermatching* if *a* is less than 1, and *matching* if *a* equals 1.

Note that when both *a* and *b* are equal to 1, the equation reduces to the reformulated matching equation. However, if *a* and *b* are allowed to vary from 1, then the equation describes various conditions that do not meet the strict definition of matching, but do follow the *same form* as the matching law. This is the more general version of the matching law known as the *generalized matching law*.

An important advantage of the generalized matching law is that it allows the values of bias and sensitivity to be derived from the data -- you simply find the values that provide the best fit to the data. When a considerable number of studies reporting matching were reexamined in this way, it turned out that on average these studies demonstrate a slight degree of *undermatching* rather than strict matching. (Whether this signals a real problem for the matching law or a statistical artifact is still being debated.)

If both sides of the generalized matching-law equation are converted to their logarithms, the generalized matching relation can be presented in an especially simple form: the data will follow a straight line, with a slope equal to the sensitivity and an intercept equal to the bias. Strict matching will appear in the form of a line with a slope of 1 (45 degrees) and an intercept of log 1 = 0. Steeper lines indicate overmatching, shallower ones, undermatching. And lines that curve indicate that the generalized matching relation is violated.

The *disadvantage* of the generalized matching law is that it is not as specific as the original, strict matching law. Deviations from strict matching can occur, but there is nothing in the generalized matching law that allow one to predict those deviations from first principles. Instead, the values of *a* and *b* appear as *free parameters* whose values are determined after the fact by the data. Whereas the strict matching law stated exactly what was to be expected if matching is to hold, the generalized matching law can be consistent with a variety of different outcomes, so long as the data conform to the general form of the law. In that sense the generalized matching law is weaker than the law it replaces.

On the one hand, as a way of *testing* the strict matching relationship, the generalized matching equation provides a great way to detect and describe systematic deviations from strict matching. On the other hand, as a replacement rule for predicting behavior, it represents a *loss* of predictive power.