Unsignalled delay of reinforcement in variable-interval schedules

Three pigeons responded on several tandem variable-interval fixed-time schedules in which the value of the fixed-time component was varied to assess the effects of different unsignalled delays of reinforcement. Actual (obtained) delays between the last key peck in an interval and reinforcement were consistently shorter than the nominal (programmed) delay. When nominal delays were relatively short, response rates were higher during the delay condition than during the corresponding nondelay condition. At longer nominal delay intervals, response rates decreased monotonically with increasing delays. The results were consistent with those obtained from delay-of-reinforcement procedures that impose either a stimulus change (signal) or a no-response requirement during the delay interval.     

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.     

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.     

In search of the feedback function for variable-interval schedules

Finding a theoretically sound feedback function for variable-interval schedules remains an important unsolved problem. It is important because interval schedules model a significant feature of the world: the dependence of reinforcement on factors beyond the organism's control. The problem remains unsolved because no feedback function yet proposed satisfies all the theoretical and empirical requirements. Previous suggestions that succeed in fitting data fail theoretically because they violate a newly recognized theoretical requirement: The slope of the function must approach or equal 1.0 at the origin. A function is presented that satisfies all requirements but lacks any theoretical justification. This function and two suggested by Prelec and Herrnstein (1978) and Nevin and Baum (1980) are evaluated against several sets of data. All three fitted the data well. The success of the two theoretically incorrect functions raises an empirical puzzle: Low rates of reinforcement are coupled with response rates that seem anomalously high. It remains to be discovered what this reflects about the temporal patterning of operant behavior at low reinforcement rates. A theoretically and empirically correct function derived from basic assumptions about operant behavior also remains to be discovered.     

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.     

Key pecking of pigeons under variable-interval schedules of briefly signaled delayed reinforcement: effects of variable-interval value.

Key pecking of 4 pigeons was maintained under a multiple variable-interval 20-s variable-interval 120-s schedule of food reinforcement. When rates of key pecking were stable, a 5-s unsignaled, nonresetting delay to reinforcement separated the first peck after an interval elapsed from reinforcement in both components. Rates of pecking decreased substantially in both components. When rates were stable, the situation was changed such that the peck that began the 5-s delay also changed the color of the keylight for 0.5 s (i.e., the delay was briefly signaled). Rates increased to near-immediate reinforcement levels. In subsequent conditions, delays of 10 and 20 s, still briefly signaled, were tested. Although rates of key pecking during the component with the variable-interval 120-s schedule did not change appreciably across conditions, rates during the variable-interval 20-s component decreased greatly in 1 pigeon at the 10-s delay and decreased in all pigeons at the 20-s delay. In a control condition, the variable-interval 20-s schedule with 20-s delays was changed to a variable-interval 35-s schedule with 5-s delays, thus equating nominal rates of reinforcement. Rates of pecking increased to baseline levels. Rates of pecking, then, depended on the value of the briefly signaled delay relative to the programmed interfood times, rather than on the absolute delay value. These results are discussed in terms of similar findings in the literature on conditioned reinforcement, delayed matching to sample, and classical conditioning.     

Spatiotemporal patterns of behavior produced by variable-interval schedules of reinforcement

The spatiotemporal patterns of behavior exhibited by two pigeons during a variable-interval 15-second schedule of food reinforcement, a variable-interval 5-minute schedule, and then extinction of key pecking were recorded using an apparatus that continuously tracked the position of the bird in the experimental chamber. The variable-interval 15-second schedule produced a close-to-key pattern between reinforcements with two types of regular excursions from the region of the key frequently occurring after reinforcement. Subsequent exposure to the variable-interval 5-minute schedule produced more extended and extremely regular patterns between responses. Reinstatement of the variable-interval 15-second schedule reestablished the close-to-key pattern with regular excursions frequently occurring after reinforcement. During extinction the spatiotemporal patterns that had developed during the variable-interval 5-minute schedule reappeared and gradually dissipated. These patterns may have been a form of superstitious behavior.     

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.     

Conditioned reinforcement versus time to reinforcement in chain schedules.

Pigeons were trained on three-component chain schedules in which the initial component was either a fixed-interval or variable-interval schedule. The middle and terminal components were varied among fixed-interval fixed-interval, variable-interval variable-interval, and an interdependent variable-interval variable-interval schedule in which the sum of the durations of the two variable-interval components was always equal to the sum of the fixed-interval fixed-interval components. At issue was whether the response rate in the initial component was controlled by its time to primary reinforcement or by the temporal parameters of the stimulus correlated with the middle terminal link. The fixed-interval initial-link schedule maintained much lower response rates than the variable-interval initial-link schedule regardless of the schedules in the middle and terminal links. Nevertheless, the intervening schedules played some role: With fixed-interval schedules in the initial links, response rates were consistently highest with independent variable-interval schedules in the middle and terminal links and intermediate with the interdependent variable-interval schedules; these initial-link differences were predicted by the response rates in the middle link of the chain. With variable-interval schedules in the initial links, response rates were lowest with the fixed-interval fixed-interval schedules following the initial link and were not systematically different for the two types of variable-interval variable-interval schedules. The results suggest that time to reinforcement itself accounts for little if any variance in initial-link responding.     

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.     

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.     

Mechanisms underlying the effects of unsignaled delayed reinforcement on key pecking of pigeons under variable-interval schedules.

Three experiments were conducted to test an interpretation of the response-rate-reducing effects of unsignaled nonresetting delays to reinforcement in pigeons. According to this interpretation, rates of key pecking decrease under these conditions because key pecks alternate with hopper-observing behavior. In Experiment 1, 4 pigeons pecked a food key that raised the hopper provided that pecks on a different variable-interval-schedule key met the requirements of a variable-interval 60-s schedule. The stimuli associated with the availability of the hopper (i.e., houselight and keylight off, food key illuminated, feedback following food-key pecks) were gradually removed across phases while the dependent relation between hopper availability and variable-interval-schedule key pecks was maintained. Rates of pecking the variable-interval-schedule key decreased to low levels and rates of food-key pecks increased when variable-interval-schedule key pecks did not produce hopper-correlated stimuli. In Experiment 2, pigeons initially pecked a single key under a variable-interval 60-s schedule. Then the dependent relation between hopper presentation and key pecks was eliminated by arranging a variable-time 60-s schedule. When rates of pecking had decreased to low levels, conditions were changed so that pecks during the final 5 s of each interval changed the keylight color from green to amber. When pecking produced these hopper-correlated stimuli, pecking occurred at high rates, despite the absence of a peck-food dependency. When peck-produced changes in keylight color were uncorrelated with food, rates of pecking fell to low levels. In Experiment 3, details (obtained delays, interresponse-time distributions, eating times) of the transition from high to low response rates produced by the introduction of a 3-s unsignaled delay were tracked from session to session in 3 pigeons that had been initially trained to peck under a conventional variable-interval 60-s schedule. Decreases in response rates soon after the transition to delayed reinforcement were accompanied by decreases in eating times and alterations in interresponse-time distributions. As response rates decreased and became stable, eating times increased and their variability decreased. These findings support an interpretation of the effects of delayed reinforcement that emphasizes the importance of hopper-observing behavior.     

Undermatching on concurrent variable-interval schedules and the power law.

The phenomenon of undermatching on concurrent variable-interval schedules is shown to be derivable by transforming the individual interreinforcement intervals of each variable-interval schedule and averaging the transformed values to produce an "estimate" of the rate of reinforcement the schedules deliver. If the transformation is based on a power function with a fractional exponent, such as is found in many studies of temporal control in animals, matching response rations to the ratios of these estimated rates of reinforcement yields undermatching. If the concurrent variable-interval schedules are arranged such that the individual intervals in each schedule have a constant proportionality (a procedure found in many commonly used variable-interval schedules) the slope of the line relating logarithms of response ratios and of programmed reinforcement ratios is identical to the exponent of the power transformation applied to the individual time intervals in the variable-interval schedules. In other cases this simple relation does not hold but the degree of undermatching is greater the lower the value of the exponent of the power function. This account of undermatching predicts values similar to those typically observed.     

Temporal control by signals of interval duration within variable-interval schedules

Rats' lever pressing produced sucrose reinforcers on a variable-interval schedule where, in different conditions, the duration of a stimulus presented immediately after reinforcement was either correlated or uncorrelated with the duration of the current interreinforcement interval. Under the baseline schedule, in which no stimulus was presented, the minimum interreinforcement interval was 8 s and the mean postreinforcement pause of each subject approximated this value. Response rates increased slowly over the first 10 to 15 s and then remained roughly constant throughout the remainder of the interval. In both the correlated and uncorrelated conditions, the added stimulus resulted in the postreinforcement pauses lengthening to values in excess of the duration of the preceding stimulus. This resulted in a poststimulus pause which was, in most cases, roughly constant irrespective of the duration of the preceding stimulus, or of the reinforcement contingencies prevailing immediately after stimulus offset. Local response-rate patterns in the uncorrelated conditions were similar to those obtained under the baseline schedule in which no stimulus was presented. However, in the correlated condition local response rates increased across the remainder of the interreinforcer interval. Further, the rate of acceleration was inversely related to the duration of the preceding stimulus. These results show that a correlation between stimulus duration and the ensuing time to reinforcement can control behavior—a type of temporal control not previously reported.     

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.     

Interresponse-time sensitivity during discrete-trial and free-operant concurrent variable-interval schedules

Two experiments investigated the sensitivity of pigeons' choice to elapsed time since the last response (i.e., to inter-response time [IRT]) during concurrent variable-interval variable-interval schedules. Experiment 1 used a two-key discrete-trial procedure with variable intertrial intervals. Experiment 2 employed a three-key free-operant procedure. In both experiments choice was found to be a function of the active-schedule IRT, defined as the time since the most recent response. Monte Carlo simulations show how this finding permits the joining of several seemingly incompatible data sets held to both support and contradict a kind of choice strategy, termed momentary maximizing, which attempts to maximize momentary reinforcement probabilities. The studies suggest that only two variables are needed to describe the static molecular structure of concurrent variable-interval choice: active-schedule IRTs and "response states" consisting of the last one or two schedule choices.     

Matching with a trio of concurrent variable-interval schedules of reinforcement

A trio of concurrent variable-interval schedules of reinforcement was arranged according to a changeover-key procedure, including a changeover delay of 1.5 sec. The three schedules provided a combined maximum reinforcement rate of 45 reinforcements per hour. With that restriction, the nine experimental conditions included several combinations of variable-interval schedules, sometimes including extinction. The pigeons matched relative response rate and relative time to relative reinforcement rate. Relative time appeared to match some-what better than relative response rate. Performance adjusted rapidly from one experimental condition to the next, whether the change involved two or all three schedules of the concurrent trio.     

Responding of pigeons under variable-interval schedules of signaled-delayed reinforcement: effects of delay-signal duration.

Two experiments with pigeons examined the relation of the duration of a signal for delay ("delay signal") to rates of key pecking. The first employed a multiple schedule comprised of two components with equal variable-interval 60-s schedules of 27-s delayed food reinforcement. In one component, a short (0.5-s) delay signal, presented immediately following the key peck that began the delay, was increased in duration across phases; in the second component the delay signal initially was equal to the length of the programmed delay (27 s) and was decreased across phases. Response rates prior to delays were an increasing function of delay-signal duration. As the delay signal was decreased in duration, response rates were generally higher than those obtained under identical delay-signal durations as the signal was increased in duration. In Experiment 2 a single variable-interval 60-s schedule of 27-s delayed reinforcement was used. Delay-signal durations were again increased gradually across phases. As in Experiment 1, response rates increased as the delay-signal duration was increased. Following the phase during which the signal lasted the entire delay, shorter delay-signal-duration conditions were introduced abruptly, rather than gradually as in Experiment 1, to determine whether the gradual shortening of the delay signal accounted for the differences observed in response rates under identical delay-signal conditions in Experiment 1. Response rates obtained during the second exposures to the conditions with shorter signals were higher than those observed under identical conditions as the signal duration was increased, as in Experiment 1. In both experiments, rates and patterns of responding during delays varied greatly across subjects and were not systematically related to delay-signal durations. The effects of the delay signal may be related to the signal's role as a discriminative stimulus for adventitiously reinforced intradelay behavior, or the delay signal may have served as a conditioned reinforcer by virtue of the temporal relation between it and presentation of food.     

Performance of rats under concurrent variable-interval schedules of negative reinforcement

The behavior of rats under concurrent variable-interval schedules of negative reinforcement was examined. A single one-minute variable-interval programmer determined the availability of 30-second timeouts from electric shock. These were assigned to one or the other of the two component schedules with a probability of 0 to 1.0. The response requirement for the component schedules was standing to the right or left of the center of the experimental chamber. With a six-second changeover delay, the relative time spent under one component schedule varied directly and linearly with the relative number of timeouts earned under that component schedule. The absolute number of changeovers was highest when a similar number of timeouts was earned under each component schedule, and lowest when all or nearly all timeouts were earned under one component schedule. In general, these relations are similar to those reported with concurrent variable-interval schedules of positive reinforcement.