Matching under concurrent fixed-ratio variable-interval schedules of food presentation

Four pigeons were exposed to concurrent fixed-ratio, variable-interval schedules of food presentation. The fixed-ratio requirement was either 25, 50, 75, or 100 responses, with the variable-interval schedule parameter held constant at 4 minutes. A delay time was imposed between a changeover from one schedule to the other and subsequent food availability. The delay time was varied at each ratio requirement over four values; no delay, 0-second delay, 1.5-second delay, and 5.0-second delay. As the fixed-ratio requirement or the delay time increased, a greater proportion of the total responses and time spent responding occurred under the variable-interval schedule relative to the proportion of food deliveries under that schedule. Neither relative overall response rate nor relative time spent responding equalled the relative frequency of food presentation, as would be predicted by a linear “matching” model. Rather, these data were described by power functions with slopes of approximately 1.0 and intercepts greater than 1.0. In the terms of Baum's (1974) analysis, these deviations from linear matching represent bias in favor of responding under the interval schedule. Bias, as reflected in the intercept of the power function, was greater for the ratio of time than the ratio of responses.     

Undermatching: a reappraisal of performance on concurrent variable-interval schedules of reinforcement

The extant data for pigeons' performance on concurrent variable-interval schedules were examined in detail. Least-squares lines relating relative pecks and time to the corresponding relative reinforcements were obtained for four studies. The between-study group slopes for time and pecks and five of seven within-study group slopes from individual studies were less than 1.00. This suggested the generality that pigeons respond less to the richer reinforcement schedule than predicted by matching. For pecks, a nonparametric test for distribution of points also supported this concept of undermatching (to the richer reinforcement schedule). In addition, using mean squared error as the criterion, a cubic curve fit the peck proportion data better than any line or other polynomial. This indicates that the relation between peck and reinforcement proportions may be nonlinear.     

Sensitivity to reinforcement in concurrent arithmetic and exponential schedules

The generalized matching law states that the logarithm of the ratio of responses emitted or time spent responding in concurrent variable-interval schedules is a linear function of the logarithm of the ratio of reinforcements obtained. The slope of this relation, sensitivity to reinforcement, varies about 1.0 but has been shown to be different when obtained in different laboratories. The present paper analyzed the results from 18 experiments on concurrent variable-interval schedule performance and showed that response-allocation sensitivity to reinforcement was significantly smaller when arithmetic, rather than exponential, progressions were used to produce variable-interval schedules. There were no differences in time-allocation sensitivity between the two methods of constructing variable-interval schedules. Since the two laboratories have consistently used different methods for constructing variable-interval schedules, the differences in obtained sensitivities to reinforcement are explained. The reanalysis suggests that animals may be sensitive to differences in the distribution of reinforcements in time.     

An invariant relation between changing over and reinforcement

Although concurrent schedules may arrange reinforcers irregularly, relatively large numbers of reinforcers are obtained when an animal changes from one schedule to the other. This paper proposes a quantitative relation that predicts the proportion of reinforcers obtained when an animal is working on a schedule and the proportion when the animal changes over to a schedule. Basically the relation states that the number of reinforcers obtained while an animal works on a schedule varies directly with the relative amount of time spent working on that schedule; and the number of reinforcers obtained when an animal changes to a schedule varies directly with the relative amount of time spent on the alternate schedule. An important aspect of this relation is that when relative reinforcement rates are less than .50, more reinforcers are obtained just after an animal changes to a schedule than at all other times when this schedule is engaged. Data obtained both from a stat-bird and a live pigeon were in close agreement with the quantitative predictions. The relation between changing over and reinforcement held across several procedural changes that included changes in relative reinforcement rate, changes from independent to interdependent scheduling procedures, and changes in the variable-interval reinforcement distributions. The results are discussed in terms of the effects of the local distribution of reinforcement on responding. The local reinforcement distribution can affect local response rates and affects the resulting matching relation. This arrangement has implications for explanations of choice.     

Multiple and concurrent schedule performance: independence from concurrent and successive schedule contexts

Six pigeons were trained on multiple variable-interval schedules and performance was measured in the presence or absence of another variable-interval schedule (the common schedule) arranged concurrently with both components. Manipulations included varying the rate of reinforcement on the common schedule, leaving the common schedule unchanged while the components of the multiple schedule were varied, varying the multiple schedule components in the absence of the common schedule, and varying one component of the multiple schedule while the other component and the common schedule were unchanged. The normal rate-increasing and rate-decreasing effects of reinforcement rate increase were found, except that changing one multiple schedule component did not affect the response rate in the successively available common schedule component. Both concurrent and multiple schedule performance undermatched obtained reinforcement-rate ratios, but the degree of undermatching in multiple schedules was reliably greater. Allocation of responses between multiple schedule components was unaffected by the concurrent availability of reinforcement, and allocation of responses between concurrent schedules was unaffected by the successive availability of different reinforcement rates.     

Matching and maximizing with concurrent ratio-interval schedules.

Animals exposed to standard concurrent variable-ratio variable-interval schedules could maximize overall reinforcement rate if, in responding, they showed a strong response bias toward the variable-ratio schedule. Tests with the standard schedules have failed to find such a bias and have been widely cited as evidence against maximization as an explanation of animal choice behavior. However, those experiments were confounded in that the value of leisure (behavior other than the instrumental response) partially offsets the value of reinforcement. The present experiment provides another such test using a concurrent procedure in which the confounding effects of leisure were mostly eliminated while the critical aspects of the concurrent variable-ratio variable-interval contingency were maintained: Responding in one component advanced only its ratio schedule while responding in the other component advanced both ratio schedules. The bias toward the latter component predicted by maximization theory was found.     

Matching and maximizing with variable-time schedules.

Pigeons were offered choices between a variable-time schedule that arranged reinforcers throughout the session and a variable-time schedule that arranged reinforcers only when the pigeon was spending time on it. The subjects could maximize the overall rate of reinforcement in this situation by biasing their time allocation towards the latter schedule. This arrangement provides an alternative to concurrent variable-interval variable-ratio schedules for testing whether animals maximize overall rates or match relative rates, and has the advantage of being free of the asymmetrical response requirements present with those schedules. The results were contrary to those predicted by maximizing: The bias it predicts did not appear.     

Effects of alternative reinforcement sources: A reevaluation

The effects of two alternative sources of food delivery on the key-peck responding of pigeons were examined. Pecking was maintained by a variable-interval 3-min schedule. In the presence of this schedule in different conditions, either a variable-time 3-min schedule delivering food independently of responding or an equivalent schedule that required a minimum 2-s pause between a key peck and food delivery (a differential-reinforcement-of-other-behavior schedule) was added. The differential-reinforcement-of-other-behavior schedule reduced response rates more than did the variable-time schedule in most instances. The delay between a key peck and the next reinforcer consistently was longer under the differential-reinforcement-of-other-behavior schedule than under the variable-time schedule. Response rates and median delay between responses and reinforcers were negatively correlated. These results contradict earlier conclusions about the behavioral effects of alternative reinforcement. They suggest that an interpretation in terms of response–reinforcer contiguity is consistent with the data.     

The role of discriminative stimuli in concurrent performances

Key pecking in pigeons was examined under concurrent and parallel arrangements of two independent and simultaneously available variable-interval schedules. Pecks on the changeover key alternated the schedule of reinforcement for responses on the main key. Under concurrent schedules, discriminative stimuli were paired with the reinforcement schedule arranged in each component and changeover responses also alternated these stimuli. Under parallel schedules, changeover responses alternated the effective reinforcement schedule, but did not change the discriminative stimulus. On concurrent procedures, changeover response rate was inversely related to the difference in reinforcement rate between the two components, whereas on parallel schedules no consistent relationship was found. With both schedules, absolute response and reinforcement rates were positively related, although for a given set of reinforcement frequencies, rates were often higher on the concurrent schedules. On concurrent schedules, relative response rates and relative times were equal to relative reinforcement rates. On parallel schedules these ratios were positively related, but response and time ratios were much smaller than were obtained with comparable concurrent schedules. This inequality was most pronounced when absolute reinforcement frequencies were lowest.     

Preference in concurrent variable-interval fixed-ratio schedules

Five pigeons were trained on concurrent variable-interval fixed-ratio schedules in three experiments. Experiment 1 used two variable-interval schedules and one fixed-ratio schedule, and the ratio requirement was varied. Using the generalized matching law, sensitivity to reinforcement was close to 1.0, but performance was biased toward the variable-interval schedule with the lower reinforcement rate. In Experiment 2, which used one variable-interval and one fixed-ratio schedule, the interval schedule was varied. All birds showed sensitivities to reinforcement of less than 1.0 and of less than the values obtained in Experiment 1. The performance was also biased toward the fixed-ratio schedule. Because the generalized matching law could not account for the differences in the data from Experiments 1 and 2, an extension of this law was suggested and successfully tested in Experiment 3. The proposed dual-sensitivity model was also shown to clarify some previously reported results.     

Performance on variable-interval schedules arranged singly and concurrently

Extensive parametric data were obtained from pigeons responding on variable-interval schedules arranged on three, two, and one response keys. Number of responses on the keys, the time spent responding on the keys, and the number of reinforcements obtained on the keys were measured. Response rates on each key were an increasing function of the reinforcement rate on that key, and an inverse function of the reinforcement rate on the other keys. In terms of preference, both response and time-allocation ratios undermatched ratios of obtained reinforcements, and the degree of undermatching was consistent both within, and between, two- and three-schedule data. When absolute response-rate data were analyzed according to Herrnstein's (1970) quantitative account, obtained values of assumed constants were not consistent either within or between conditions. However, a power-function modification of Herrnstein's account fitted the data well and provided similar exponent values to those obtained for the undermatching of preference ratios.     

Behavior-dependent reinforcer-rate changes in concurrent schedules: A further analysis

Six pigeons were trained on concurrent variable-interval schedules in three different procedures. The first procedure was a standard concurrent schedule, and the relative reinforcer frequency for responding was varied. The second was a schedule in which a relative left-key response rate (over a fixed period of time) exceeding .75 produced, in the next identical time period a higher reinforcer rate on the right key. If this criterion was not exceeded, equal reinforcer rates were arranged on the two keys in this period. This was the dependent procedure. In the third (independent) procedure, the periods of higher right-key reinforcer rates occurred with the same probability as in the second procedure, but occurred independently of behavior. In the second and third procedures, the fixed-time period (window) was varied from 5 s to 60 s, and to 240 s in the second procedure only. Performance on the two keys was similar in the concurrent and independent procedures. The procedure used in the dependent conditions generally affected performance when the windows were shorter than about 30 s. Models of performance that assume that subjects do not discriminate changes in local relative reinforcer rates cannot account for the data. Moreover, existing models are inherently unable to account for the effects of contingencies of reinforcement between responding on one alternative and gaining reinforcers on another that are arranged or that emerge as a result of time allocated to alternative schedules. Undermatching on concurrent variable-interval schedules may result from such emergent contingencies.     

The economics of daily consumption controlling food- and water-reinforced responding

In the first experiment, two rhesus monkeys earned their entire ration of food and water during daily sessions with no provisions to ensure constant daily intakes. Two variable-interval schedules of food presentations were concurrent with one variable-interval schedule of water presentations; the maximum rate of food presentations arranged by one food schedule was varied. As the rate of food presentations was increased, the absolute level of responding on the two food schedules combined decreased, while responding on the water schedule increased. The preference for the variable food schedule compared to the other food schedule approximately matched the proportion of reinforcers obtained from it. The preference for the variable food schedule compared to the water schedule did not match, but greatly decreased, as the proportion of reinforcers from the food schedule increased. When Experiment I was replicated, with provisions to ensure constant daily intakes of food and water (Experiment II), the absolute response rates under the two food schedules combined and under the water schedule no longer changed with increases in the rate of food during the sessions. On the other hand, choice between the two food schedules remained proportional to the distribution of obtained food pellets. These results were interpreted as indicating that behavior to obtain nonsubstitutable commodities, such as food and water, is strongly controlled by the economic conditions of daily consumption, while choice between substitutable commodities is independent of these factors.     

Response rate and changeover performance on concurrent variable-interval schedules

Six pigeons were exposed to variable-interval schedules arranged on one, two, three, and four response keys. The reinforcement rate was also varied across conditions. Numbers of responses, the time spent responding, the number of reinforcements, and the number of changeovers between keys were recorded. Response rates on each key were an increasing function of reinforcement rate on that key and a decreasing function of the reinforcement rate on other keys. Response and time-allocation ratios under-matched ratios of obtained reinforcements. Three sets of equations were developed to express changeover rate as a function of response rate, time allocation, and reinforcement rate respectively. These functions were then applied to a broad range of experiments in the literature in order to test their generality. Further expressions were developed to account for changeover rates reported in experiments where changeover delays were varied.     

THE LOCAL ORGANIZATION OF BEHAVIOR: DISCRIMINATION OF AND MEMORY FOR SIMPLE BEHAVIORAL PATTERNS

A procedure was developed to enable nonverbal organisms to report what they remember of the temporal organization of their recent behavior. A baseline behavior with known temporal structure was established by a concurrent variable-interval variable-interval schedule for two temporal patterns of behavior (two different classes of reinforced interresponse times). The five pigeon subjects emitted these two temporal patterns on a center key and were occasionally given a short-term memory probe for their most-recently-emitted pattern. The probes consisted of symbolic delayed matching-to-sample tests, in which a response on a green side key was reinforced if the most recent pattern belonged to the shorter reinforced class, and a response to a red side key was reinforced if the most recent pattern belonged to the longer reinforced class. All subjects could report with over ninety percent accuracy what their most recently emitted behavioral pattern was when a retention interval separating the pattern from the memory probe was only .1 seconds. The retention interval was then manipulated, and it was found that recall for a pattern was frequently above chance after a delay of as much as eight seconds. Thus, pigeons can remember their most recent interresponse time not only right after it is emitted, but for several seconds thereafter. In other conditions, the patterns themselves were manipulated. It was found that as the patterns became more similar, discrimination became poorer. These results agree with the view that reinforcement tends to organize and integrate the local structure of behavior to the extent to which that structure is remembered.     

Matching-based hedonic scaling in the pigeon

Four slightly hungry pigeons chose between pairs of grains in a Findley concurrent choice procedure. For Condition I, choice involved hemp versus buckwheat; for Condition II, wheat versus buckwheat; and for Condition III, hemp versus wheat. In all conditions, frequency of reinforcement was arranged according to concurrent variable-interval variable-interval schedules. On the assumption that subjects matched their behavior and time distributions to those of reinforcer value, the choice functions obtained in Conditions I and II were transformed to yield estimates of values of hemp and wheat relative to buckwheat. These, in turn, provided predictions about behavior and time allocation in Condition III. In general, the predicted outcomes were close to those actually obtained. The results evidence the effectiveness of matching-based hedonic scales in the prediction of choice between qualitatively different reinforcers.     

Concurrent performances: rate constancies without changeover delays

Pigeons' pecks on two keys were maintained, without changeover delays, by independent variable-interval schedules of food reinforcement. Four regularly cycling 2-min components scheduled reinforcement respectively for both keys, left key only, both keys, and right key only. Initially, reinforcement scheduled for one key alone produced more responding on that key than reinforcement scheduled concurrently for both keys. Continued sessions reduced this difference; response rate on a given key approached constancy, or invariance with respect to the performance on and schedule for the other key. When extinction replaced the reinforcement schedule on either key, responding on that key decreased more during components that scheduled reinforcement for the other key than during those that did not. This demonstration that responses on one key were not supported by reinforcers on the other key suggested that the alternation of concurrent responding and either-key-alone responding prevented concurrent superstitions from developing.     

IS THERE A DECISIVE TEST BETWEEN MATCHING AND MAXIMIZING?

Reinforcers under typical concurrent variable-interval, variable-ratio schedules may be (a) earned and obtained during the variable-interval component, (b) earned and obtained during the variable-ratio component, or (c) earned during the variable-ratio component and obtained during the variable-interval component. Categories a and b, which have no bearing on matching versus maximizing accounts of choice, were set at zero. The rate of Category c reinforcers and the duration of a changeover delay were varied. Simple matching, which predicts exclusive choice of the variable-interval component, and strict maximizing of overall reinforcement rate, which predicts a bias towards the variable-ratio component, were both disconfirmed: Subjects spent approximately 25% of their time in the variable-ratio component, contrary to the matching prediction, but earned only about one third of the reinforcers predicted by strict maximizing. However, maximizing describes the findings functionally in terms of discounting of delayed reinforcers; matching may describe the data in terms of a restructuring of the alternatives. Matching and maximizing are not competing theories about the fundamental nature of choice, but compatible points of view that may reveal environmental function and behavioral structure.