Concurrent fixed-interval variable-ratio schedules and the matching relation

Five rats responded under concurrent fixed-interval variable-ratio schedules of food reinforcement. Fixed-interval values ranged from 50-seconds to 300-seconds and variable-ratio values ranged from 30 to 360; a five-second changeover delay was in effect throughout the experiment. The relations between reinforcement ratios obtained from the two schedules and the ratios of responses and time spent on the schedules were described by Baum's (1974) generalized matching equation. All subjects undermatched both response and time ratios to reinforcement ratios, and all subjects displayed systematic bias in favor of the variable-ratio schedules. Response ratios undermatched reinforcement ratios less than did time ratios, but response ratios produced greater bias than did time ratios for every subject and for the group as a whole. Local rates of responding were generally higher on the variable-ratio than on the fixed-interval schedules. When responding was maintained by both schedules, a period of no responding on either schedule immediately after fixed-interval reinforcement typically was followed by high-rate responding on the variable-ratio schedule. At short fixed-interval values, when a changeover to the fixed-interval schedule was made, responding usually continued until fixed-interval reinforcement was obtained; at longer values, a changeover back to the variable-ratio schedule usually occurred when fixed-interval reinforcement was not forthcoming within a few seconds, and responding then alternated between the two schedules every few seconds until fixed-interval reinforcement finally was obtained.     

Sensitivity to relative reinforcer rate in concurrent schedules: independence from relative and absolute reinforcer duration

Twelve pigeons responded on two keys under concurrent variable-interval (VI) schedules. Over several series of conditions, relative and absolute magnitudes of reinforcement were varied. Within each series, relative rate of reinforcement was varied and sensitivity of behavior ratios to reinforcer-rate ratios was assessed. When responding at both alternatives was maintained by equal-sized small reinforcers, sensitivity to variation in reinforcer-rate ratios was the same as when large reinforcers were used. This result was observed when the overall rate of reinforcement was constant over conditions, and also in another series of concurrent schedules in which one schedule was kept constant at VI ached 120 s. Similarly, reinforcer magnitude did not affect the rate at which response allocation approached asymptote within a condition. When reinforcer magnitudes differred between the two responses and reinforcer-rate ratios were varied, sensitivity of behavior allocation was unaffected although response bias favored the schedule that arranged the larger reinforcers. Analysis of absolute response rates ratio sensitivity to reinforcement occurrred on the two keys showed that this invariance of response despite changes in reinforcement interaction that were observed in absolute response rates on the constant VI 120-s schedule. Response rate on the constant VI 120-s schedule was inversely related to reinforcer rate on the varied key and the strength of this relation depended on the relative magnitude of reinforcers arranged on varied key. Independence of sensitivity to reinforcer-rate ratios from relative and absolute reinforcer magnitude is consistent with the relativity and independence assumtions of the matching law.     

Concurrent second-order schedules of reinforcement

Responses on one key (the main key) of a two-key chamber produced food according to a second-order variable-interval schedule with fixed-interval schedule components. A response on a second key (the changeover key) alternated colors on the main key and provided a second independent second-order variable-interval schedule with fixed-interval components. The fixed-interval component on one variable-interval schedule was held constant at 8 sec, while the fixed interval on the other variable-interval schedule was varied from 0 to 32 sec. Under some conditions, a brief stimulus terminated each fixed interval and generated fixed-interval patterns; in other conditions, the brief stimulus was omitted. Relative response rate and relative time deviated substantially from scheduled relative reinforcement rate and, to a lesser extent, from obtained relative reinforcement rate under both brief-stimulus and no-stimulus conditions. Matching was observed with equal components on both schedules; with unequal components, increasingly greater proportions of time and responses than the matching relation would predict were spent on the variable-interval schedule containing the shorter component. Preference for the shorter fixed interval was typically more extreme under brief-stimulus than under no-stimulus schedules. The results limit the extension of the matching relation typically observed under simple concurrent variable-interval schedules to concurrent second-order variable-interval schedules.     

Changeover ratio effects on concurrent variable-interval performance

Rats' bar-pressing was maintained by concurrent variable-interval schedules of reinforcement. A fixed-ratio of pulls on a chain (the changeover ratio) was required for switching between schedules. The first experiment employed equal variable-interval schedules and symmetrical changeover ratios. Increasing these ratios resulted in a decrease in the rate of switching between schedules and an increase in local response rate. In the second experiment, a range of asymmetrical changeover ratios was used with equal variable-interval schedules, and a preference was found for the schedule associated with the larger switching-into ratio. Both the distributions of responses and time between the two schedules deviated from those expected on the basis of obtained reinforcers. In the third experiment, the switching-out-of ratio was dependent on the amount of time spent in a variable-interval 2-minute schedule; a constant ratio permitted switching out of the alternative variable-interval 1-minute schedule. A strong preference was shown for the variable-interval 2-minute schedule. The fourth experiment used equal variable-interval schedules; one changeover ratio was varied while the second remained constant. The results failed to show systematic differences in local response rates immediately after a changeover.     

Choice, changeover, and travel: A quantitative model

Six pigeons were trained on concurrent variable-interval schedules in which responding on fixed-interval schedules was required to give access to the alternate schedule. Responding on the concurrent schedules was not allowed, after changing over had commenced, until the changeover schedule had been completed. In Parts 1 to 3 of the experiment, the changeover fixed-interval schedules were equal and were 0 s, 10 s, and 20 s, respectively. In each part, the relative frequency of reinforcement obtained on the concurrent schedules was varied over at least five conditions. In Part 4, the concurrent schedules were equal, and one changeover fixed-interval schedule was twice the other. Under these conditions, the absolute sizes of the changeover schedules were varied. Increasing the changeover requirement from 0 s to 10 s (Parts 1 and 2) resulted in increases in the sensitivity of behavior allocation to reinforcers obtained, but no further increase was obtained when the changeover schedules were increased to 20 s (Part 3). In Part 4, performance was biased towards the concurrent schedule that took less time to enter. These results are consistent with a subtractive punishment model of travel in which the degree of punishment is measured by the number of reinforcers apparently lost from a schedule when the subject changes to that schedule. Absolute times spent on the main keys could be accurately described by a previous model of changeover performance.     

The effect of punishment on free-operant choice behavior in humans

During Phase I, three female human subjects pressed a button for monetary reinforcement in five variable-interval schedules specifying different frequencies of reinforcement. On alternate days, responding was also punished (by subtracting money) according to a variable-ratio 34 schedule. In the absence of punishment, response rates conformed to Herrnstein's equation for single variable-interval schedules. Punishment suppressed responding at all frequencies of reinforcement. This was reflected in a change in the values of both constants in Herrnstein's equation: the value of the theoretical maximum response-rate parameter was reduced, and the parameter describing the reinforcement frequency corresponding to the half-maximal response rate was elevated. During Phase II, the same five schedules (A) were in operation (without punishment), but in addition, a concurrent variable-interval schedule (B) of standard reinforcement frequency was introduced. On alternate days, responding in Component B was punished according to a variable-ratio 34 schedule. In the absence of punishment, absolute response rates conformed to equations proposed by Herrnstein to describe performance in concurrent schedules; the ratios of the response rates in the two components and the ratios of the times spent in the two components conformed to the Matching Law. When responding in Component B was punished, response rates in Component B were reduced and those in Component A were elevated, these changes being reflected in distortions of the matching relationship.     

Choice dynamics in concurrent ratio schedules of reinforcement

The generalized matching law predicts performance on concurrent schedules when variable-interval schedules are programmed but is trivially applicable when independent ratio schedules are used. Responding usually is exclusive to the schedule with the lowest response requirement. Determining a method to program concurrent ratio schedules such that matching analyses can be usefully employed would extend the generality of matching research and lead to new avenues of research. In the present experiments, ratio schedules were programmed dependently such that responses to either of the two options progressed the requirement on both schedules. Responding is not exclusive because the probability of reinforcement increases on both schedules as responses are allocated to either schedule. In Experiment 1, performance on concurrent variable-ratio schedules was assessed, and reinforcer ratios were varied across conditions to investigate changes in sensitivity. Additionally, the length of a changeover delay was manipulated. In Experiment 2, performance was compared under concurrently available, dependently programmed variable-ratio and fixed-ratio schedules. Performance was well described by the generalized matching law. Increases in the changeover delay decreased sensitivity, whereas sensitivity was higher when variable-ratio schedules were employed, compared with fixed-ratio schedules. Concurrent ratio schedules can be a viable approach to studying functional differences between ratio and interval schedules.     

More on concurrent interval-ratio schedules: a replication and review.

It has been suggested that the failure to maximize reinforcement on concurrent variable-interval, variable-ratio schedules may be misleading. Inasmuch as response costs are not directly measured, it is possible that subjects are optimally balancing the benefits of reinforcement against the costs of responding. To evaluate this hypothesis, pigeons were tested in a procedure in which interval and ratio schedules had equal response costs. On a concurrent variable time (VT), variable ratio-time (VRT) schedule, the VT schedule runs throughout the session and the VRT schedule is controlled by responses to a changeover key that switches from one schedule to the other. Reinforcement is presented independent of response. This schedule retains the essential features of concurrent VI VR, but eliminates differential response costs for the two alternatives. It therefore also eliminates at least one significant ambiguity about the reinforcement maximizing performance. Pigeons did not maximize rate of reinforcement on this procedure. Instead, their times spent on the alternative schedules matched the relative rates of reinforcement, even when schedule parameters were such that matching earned the lowest possible overall rate of reinforcement. It was further shown that the observed matching was not a procedural artifact arising from the constraints built into the schedule.     

Behavior of humans in variable-interval schedules of reinforcement

During Phase I, human subjects pressed a button for monetary reinforcement in five variable-interval schedules, each of which specified a different frequency of reinforcement. The rate of responding was an increasing, negatively accelerated function of reinforcement frequency; the data conformed closely to Herrnstein's equation. During Phase II, the same five schedules were in operation, but in addition a concurrent variable-interval schedule (B) was introduced, responses on which were always reinforced at the same frequency. Response rate in component A increased while the response rate in B decreased, as a function of the reinforcement frequency in component A. Relative response rates in the two component schedules matched the relative frequencies of reinforcement. Comparing the absolute response rates in component A during Phase I and Phase II it was found that introduction of the concurrent schedule did not affect the value of the theoretical maximum response rate, but did increase the value of the reinforcement frequency needed to obtain any particular submaximal response rate.     

Local response-rate constancy on concurrent variable-interval schedules of reinforcement

Concurrent variable-interval schedules were arranged with a main key that alternated in color and schedule assignment, along with a changeover key on which a small fixed ratio was required to changeover. Acceptable matching was observed with pigeons in two replications, but there was a tendency toward overmatching. Local response rates were found to differ for unequal schedules of a concurrent pair: local response rate was greater for the variable-interval schedule with the smaller average interreinforcement interval, but qualifications based on an interresponse-time analysis were discussed. In a second experiment, two 3-minute variable-interval schedules were arranged concurrently, and the experimental variable was the changeover procedure: either a changeover delay was incurred by each changeover or a small fixed ratio on a changeover key was required to complete a changeover. Changeover delays of 2 and 5 seconds were compared with a fixed-ratio changeover of five responses. The response output on the main key (associated with the variable-interval schedules) was greater when a changeover delay was arranged than when a fixed ratio was required to changeover. A detailed analysis of stripchart records showed that a 2-second delay generated an increased response rate for 3 seconds after a changeover, while the fixed-ratio requirement generated an increased rate during the first second only, followed by a depressed response rate for 2 seconds.     

Concurrent ratio schedules: Fixed vs. variable response requirements

Rats were trained on concurrent fixed-ratio variable-ratio or concurrent fixed-ratio mixed-ratio schedules of food reinforcement. The variable-ratio schedule was composed of an arithmetic sequence of 11 ratios that averaged 50; the mixed-ratio schedule consisted of equiprobable ratios of 1 and 99. Fixed-ratio values, varied over experimental conditions, included 25, 35, 50, 60, and 99. The proportion of responses and time allocated to the variable- or mixed-ratio schedule increased as the size of the fixed ratio increased. For most subjects, higher proportions of responses and time were maintained on the fixed-ratio schedule at fixed-ratio values of 25 and 35; higher proportions of responses and time were maintained on the variable- or mixed-ratio schedule at fixed-ratio values of 50 or higher. On concurrent variable-ratio fixed-ratio schedules, the tendency for responding to be maintained exclusively by one schedule was related to the difference in local reinforcement rates obtained from those schedules. Exclusive responding was approximated when the difference in local reinforcement rates obtained from those schedules was large; responding was more evenly distributed between the schedules as the difference in the rates at which reinforcement was obtained from each decreased.     

Performance in concurrent interval schedules: a systematic replication

Five pigeons were trained on a variety of concurrent interval schedules that arranged reinforcements at either fixed or variable times after the last reinforcement. Two measures were obtained: the number of responses on each schedule, and the time spent responding on each schedule. Ratios of response rates on the two schedules did not equal ratios of reinforcement rates when both schedules were variable nor when one was variable and the other fixed. Ratios of times spent responding approximately equalled ratios of reinforcement rates when both schedules were variable, but did not do so when one was fixed.     

Changeover behavior and preference in concurrent schedules

Pigeons were trained on a multiple schedule of reinforcement in which separate concurrent schedules occurred in each of two components. Key pecking was reinforced with milo. During one component, a variable-interval 40-s schedule was concurrent with a variable-interval 20-s schedule; during the other component, a variable-interval 40-s schedule was concurrent with a variable-interval 80-s schedule. During probe tests, the stimuli correlated with the two variable-interval 40-s schedules were presented simultaneously to assess preference, measured by the relative response rates to the two stimuli. In Experiment 1, the concurrently available variable-interval 20-s schedule operated normally; that is, reinforcer availability was not signaled. Following this baseline training, relative response rate during the probes favored the variable-interval 40-s alternative that had been paired with the lower valued schedule (i.e., with the variable-interval 80-s schedule). In Experiment 2, a signal for reinforcer availability was added to the high-value alternative (i.e., to the variable-interval 20-s schedule), thus reducing the rate of key pecking maintained by that schedule but leaving the reinforcement rate unchanged. Following that baseline training, relative response rates during probes favored the variable-interval 40-s alternative that had been paired with the higher valued schedule. The reversal in the pattern of preference implies that the pattern of changeover behavior established during training, and not reinforcement rate, determined the preference patterns obtained on the probe tests.     

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.     

Behavioral momentum: the effects of the temporal separation of rates of reinforcement.

In Part 1 of the experiment, rats responded under a variable-interval (VI) 30-s schedule and a VI 120-s schedule, with each in effect for a block of consecutive sessions. That is, the two VI schedules were presented in successive conditions. In Part 2 the VI schedules alternated each day, and in Part 3 the schedules alternated within the session as a multiple schedule. For half of the rats in Parts 1 and 2, the VI schedule alternated every few minutes within the session with a stimulus that signaled extinction. For each part, once response rates had stabilized, resistance to change was measured by prefeeding and extinction. When the schedules were examined in successive conditions (Part 1), resistance to extinction was greater under the VI 120-s schedule of reinforcement than under the VI 30-s schedule, but no consistent differences in resistance to prefeeding were observed between the two VI schedules. When the VI schedules alternated each day (Part 2), resistance to extinction was greater under the VI 120-s schedule. However, no consistent differences in resistance to prefeeding were observed between the VI schedules without extinction in Group A, but resistance to prefeeding was greater under the VI 30-s schedule for rats with the added extinction component in Group B. When the VI schedules alternated within the session as a multiple schedule (Part 3), resistance to extinction and resistance to prefeeding were greater under the VI 30-s schedule. The data suggest that different rates of reinforcement, and their accompanying discriminative stimuli, must be compared within the same session (or at least on alternate days) to produce data consistent with the behavioral momentum model.     

A yoked-chamber comparison of concurrent and multiple schedules

Pigeons were exposed to alternative pairs of variable-interval schedules correlated with red and green lights on one key (the food key). In one experimental chamber, responses on a white key (the changeover key) changed the color of the food key and initiated a 2-sec changeover delay. Pigeons in a second chamber obtained food by pecking on a colored key whenever the pigeons in the first (concurrent) chamber had obtained food for a peck on that key color. There was no changeover key in the second (multiple) chamber: changeover responses in the first chamber alternated the schedules and colors in both chambers. The pigeons in both chambers emitted the same proportion of responses on each of the variable-interval schedules, and mastered discrimination reversals at the same rate. The pigeons differed only in their absolute response rates, which were greater under the concurrent schedules. In a second experiment, changes in key color occurred automatically, with different proportions of time allocated to the two variable-interval schedules. Matching of relative response frequency to relative reinforcement frequency was affected by the relative amounts of time in each component, by rate of changeovers, and by manipulations of the variable-interval scheduling.     

Effects on concurrent performances of a stimulus correlated with reinforcer availability

A multiple schedule was arranged in which each component consisted of two, concurrent variable-interval schedules of reinforcement. A changeover-key procedure was used, and the components of the multiple schedule were distinguished (initially) by the color of the changeover key. During one component of the multiple schedule, the availability of a reinforcer arranged by one of the variable-interval schedules was marked by an exteroceptive stimulus, provided that that variable-interval schedule was not at the time assigned to the main key. During the other component of the multiple schedule, no reinforcer-correlated stimuli were ever presented. During the latter component of the multiple schedule, the distribution of responses and time for the concurrent variable-interval schedules suggested control by the distribution of reinforcements. During the former component, most main-key responses were emitted on the key in the presence of which reinforcer-correlated stimuli were presented. Changeover rate in the presence of that key color was depressed. The discriminative control over the changeover was easily established and was reversible.     

Reinforcement of eye movement with concurrent schedules

Human macrosaccadic eye movements to two areas of a four-dial display were conditioned by concurrent variable-interval schedules of signals. Reinforcers (signals) were delivered to the two right-hand dials on one schedule and to the two left-hand dials on another, independent schedule. The use of a changeover delay between crossover eye movements and reinforcement had the effect of changing the pattern of scanning from fixating four dials in succession or in a Z-shaped pattern to scanning vertically the dials on either side with fewer crossovers. In the presence of a changeover delay, subjects matched relative eye-movement rates and relative reinforcement rates on each schedule. Rate of crossover eye movements, with a changeover delay in effect, was also inversely related to the difference in reinforcements arranged by the concurrent schedules. The results suggest that for stimuli whose critical components are arranged spatially, conditioned eye movements play an important part in selective stimulus control.     

Temporal constraint on choice: Sensitivity and bias in multiple schedules

In multiple schedules of reinforcement, ratios of responses in successive components are relatively insensitive to ratios of obtained reinforcers. An analysis is proposed that attributes changes in absolute response rates to concurrent interactions between programmed reinforcement and extraneous reinforcement in other components. The analysis predicts that ratios of responses in successive components vary with reinforcer ratios, qualified by a term describing the reinforcement context, that is, programmed and extraneous reinforcers. Two main predictions from the analysis were confirmed in an experiment in which pigeons' responses were reinforced in the components of a multiple schedule and analog extraneous reinforcement was scheduled for an alternative response in each component. Sensitivity of response and time ratios to reinforcer ratios in the multiple schedules varied as a function of the rate of extraneous reinforcers. Bias towards responding in one component of the multiple schedule varied as an inverse function of the ratios of extraneous reinforcer rate in the two components. The data from this and previous studies of multiple-concurrent performance were accurately predicted by our analysis and supported our contention that the allocation of behavior in multiple-schedule components depends on the relative values of concurrently-available reinforcers within each component.     

Some effects of relative reinforcement rate and changeover delay in response-independent concurrent schedules of reinforcement

Reinforcements were arranged independently of the pigeon's behavior by concurrent variable-interval schedules. The reinforcements arranged by one of the schedules occurred when the chamber was illuminated with amber light, and the reinforcements arranged by the other schedule occurred when the chamber was illuminated with blue light. Both schedules functioned concurrently, but reinforcers were delivered by each only in the presence of the appropriate stimulus condition. A response on a white key, the only key in the chamber, alternated the stimulus condition and the effective schedule. The results of this procedure were similar to those obtained with concurrent response-dependent variable-interval schedules of reinforcement. The proportion of the total session time spent in the presence of a schedule component approximated the proportion of the total number of reinforcements in the component. Changeover rate was a decreasing function of the changeover delay and of the difference between the relative rates of reinforcement for each pair of concurrent schedules.