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.     

Income maximizing in concurrent interval-ratio schedules

Three mice chose between concurrent variable-ratio variable-interval schedules to produce a warm air stream while they were housed in a cold chamber. Across conditions, the duration of the warm air stream was varied between 10 and 80 s and was equal for both schedules. Preference for the VI schedule covaried with reinforcer duration as predicted by maximizing accounts of choice.     

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.     

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.     

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.     

Reporting contingencies of reinforcement in concurrent schedules

Five pigeons were trained on concurrent variable-interval schedules in which two intensities of yellow light served as discriminative stimuli in a switching-key procedure. A conditional discrimination involving a simultaneous choice between red and green keys followed every reinforcer obtained from both alternatives. A response to the red side key was occasionally reinforced if the prior reinforcer had been obtained from the bright alternative, and a response to the green side key was occasionally reinforced if the prior reinforcer had been obtained from the dim alternative. Measures of the discriminability between the concurrent-schedule alternatives were obtained by varying the reinforcer ratio for correct red and correct green responses across conditions in two parts. Part 1 arranged equal rates of reinforcement in the concurrent schedule, and Part 2 provided a 9:1 concurrent-schedule reinforcer ratio. Part 3 arranged a 1:9 reinforcer ratio in the conditional discrimination, and the concurrent-schedule reinforcer ratio was varied across conditions. Varying the conditional discrimination reinforcer ratio did not affect response allocation in the concurrent schedule, but varying the concurrent-schedule reinforcer ratio did affect conditional discrimination performance. These effects were incompatible with a contingency-discriminability model of concurrent-schedule performance (Davison & Jenkins, 1985), which implies a constant discriminability parameter that is independent of the obtained reinforcer ratio. However, a more detailed analysis of conditional discrimination performance showed that the discriminability between the concurrent-schedule alternatives decreased with time since changing over to an alternative. This effect, combined with aspects of the temporal distribution of reinforcers obtained in the concurrent schedules, qualitatively predicted the molar results and identified the conditions that operate whenever contingency discriminability remains constant.     

Human choice in concurrent ratio-interval schedules of reinforcement.

Seven undergraduates participated in a concurrent-choice experiment with monetary reinforcers. Response-independent analogues of variable-interval and variable-ratio schedules were used to assess whether subjects would maximize reinforcement rate. The optimal pattern of behavior, in terms of maximizing reinforcement rate, involved a large bias toward the ratio alternative, with only occasional sampling of the interval schedule. Most experiments with pigeons, however, demonstrate matching of response rates to reinforcement rates, with only slight biases for the ratio schedule. Although subjects in the present experiment allocated more time to the ratio alternative than required by matching, the magnitude of the bias did not approximate that predicted by a maximizing account. After exposure to clock stimuli correlated with the operation of each schedule, 1 subject's behavior did show a substantial level of bias, increasing the total number of reinforcers obtained, and lay at a point between the predictions of matching and maximizing. The other subjects, however, continued to respond less optimally. The present results can be accounted for by a view of matching that incorporates the effects of delayed reinforcement.     

Momentary maximizing in concurrent schedules with a minimum interchangeover interval

Eight pigeons were trained on concurrent variable-interval variable-interval schedules with a minimum interchangeover time programmed as a consequence of changeovers. In Experiment 1 the reinforcement schedules remained constant while the minimum interchangeover time varied from 0 to 200 s. Relative response rates and relative time deviated from relative reinforcement rates toward indifference with long minimum interchangeover times. In Experiment 2 different reinforcement ratios were scheduled in successive experimental conditions with the minimum interchangeover time constant at 0, 2, 10, or 120 s. The exponent of the generalized matching equation was close to 1.0 when the minimum interchangeover time was 0 s (the typical procedure for concurrent schedules without a changeover delay) and decreased as that duration was increased. The data support the momentary maximizing theory and contradict molar maximizing theories and the melioration theory.     

Choice performance in several concurrent key-peck treadle-press reinforcement schedules

Five pigeons were exposed to several concurrent variable-interval food reinforcement schedules. For three subjects, one component of the schedule required a key-pecking response, the other a treadle-pressing response. For the other two subjects, both schedule components required treadle-pressing responses. The relative probability of reinforcement associated with the manipulanda was varied from 0 to 1.0 in 13 experimental conditions for the Key-Treadle subjects and nine conditions for the Treadle-Treadle subjects. The results indicated that the logarithms of relative time spent responding, and the logarithms of relative number of responses emitted on a manipulandum, approximated direct linear functions of logarithms of the relative frequencies of reinforcement associated with that manipulandum. No systematic bias in favor of time spent key pecking over time spent treadle pressing was apparent for the Key-Treadle subjects. All subjects exhibited undermatching, in that the ratios of time and response allocation at the alternatives systematically differed from the ratios of reinforcers obtained from the alternatives in the direction of indifference. Key pecking appeared to have no special link to food beyond treadle pressing or what would be expected on the basis of the reinforcement dependencies alone.     

Contrast and induction in multiple schedules of discrete-trial concurrent reinforcement

Three pigeons were exposed to two-key discrete-trial concurrent schedules of reinforcement. Red and white key colors alternated irregularly and the assignment of reinforcers depended on key color. The red-key schedules were held constant, with the scheduled relative frequency of reinforcement for left-key pecks set at 0.75, while the white-key schedules varied. When the location of white-key reinforcement was changed from one side to the other, while its overall frequency was constant, red-key choices shifted in the same direction as white-key choices, an induction effect. When the overall frequency of white-key reinforcement was changed while its location remained constant, red key choices shifted in a direction opposite to white-key choices, a contrast effect. Both induction and contrast effects were clearer when the overall frequency of red-key reinforcement was reduced. These data demonstrate that the allocation of responding may exhibit schedule interaction effects similar to those commonly reported for response rate.     

Sensitivity of time allocation to an overall reinforcer rate feedback function in concurrent interval schedules

Six pigeons were trained on concurrent variable-interval schedules in which feedback functions arranged that the overall reinforcer rate either (a) was independent of preference, (b) decreased with increasing absolute preference, or (c) increased with increasing absolute preference. In Experiment 1, the reinforcer rate in an interreinforcement interval was determined by the absolute time-allocation ratio in the previous interval. When arranged reinforcer ratios were varied, there was no evidence of control over preference by overall reinforcer rate. In Experiment 2, the feedback function arranged that reinforcer rates were an inverse function of absolute preference, and window durations were fixed times. In Phase 1, using schedules that provided a four-to-one reinforcer ratio, the window duration was decreased from 20 s to 5 s over four conditions. Then, in Phases 2 and 3, the arranged reinforcer ratios were varied. In Phase 2, the reinforcer rate in the current 5-s time window was determined by preference in the previous 5-s window, and in Phase 3, the window durations were 20 s. Again, there was no indication of control by obtained overall reinforcer rate. These data call into question theories that suggest that the process underlying matching is one of maximizing overall reinforcer rates, or that preference in concurrent aperiodic schedules is controlled to any extent by overall reinforcer rate. They also question the notion that concurrent-schedule preference is controlled by molecular maximizing.     

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.     

Hill-climbing by pigeons

Pigeons were exposed to two types of concurrent operant-reinforcement schedules in order to determine what choice rules determine behavior on these schedules. In the first set of experiments, concurrent variable-interval, variable-interval schedules, key-peck responses to either of two alternative schedules produced food reinforcement after a random time interval. The frequency of food-reinforcement availability for the two schedules was varied over different ranges for different birds. In the second series of experiments, concurrent variable-ratio, variable-interval schedules, key-peck responses to one schedule produced food reinforcement after a random time interval, whereas food reinforcement occurred for an alternative schedule only after a random number of responses. Results from both experiments showed that pigeons consistently follow a behavioral strategy in which the alternative schedule chosen at any time is the one which offers the highest momentary reinforcement probability (momentary maximizing). The quality of momentary maximizing was somewhat higher and more consistent when both alternative reinforcement schedules were time-based than when one schedule was time-based and the alternative response-count based. Previous attempts to provide evidence for the existence of momentary maximizing were shown to be based upon faulty assumptions about the behavior implied by momentary maximizing and resultant inappropriate measures of behavior.     

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.     

Timeout from concurrent schedules.

Response-contingent timeouts of equal duration and frequency were added to both alternatives of unequal concurrent schedules of reinforcement. For each of 4 pigeons in Experiment 1, relative response rates generally became less extreme as the frequency of timeout increased. In Experiment 2, relative response rates consistently approached indifference as the duration of timeout was increased. Variation in time allocation was less consistent in both experiments. Absolute response rates did not vary with the timeout contingency in either experiment. In a third experiment, neither measure of choice varied systematically when the duration of a postreinforcement blackout was varied. In contrast to the present results, preference has been shown to vary directly with the parameters of shock delivery in related procedures. The pattern of results in the first two experiments follows that obtained with other manipulations of the overall rate of reinforcement in concurrent schedules. The results of the third experiment suggest that an intertrial interval following reinforcement is not a critical feature of the overall rate of reinforcement.     

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.     

Human choice on concurrent variable-interval variable-ratio schedules.

Each of 5 adult male humans sat in a 4 degrees C room where they could warm themselves by illuminating six heat lamps for 10-second periods according to a concurrent variable-interval variable-ratio schedule. Left-button presses on a response panel switched between the schedules and started a 2-second changeover delay. Right-button presses illuminated the heat lamps if assigned by the associated schedule and if the changeover delay had timed out. Panel lights identified the schedule in effect and each effective right-button press. A discrimination procedure--either a multiple variable-interval variable-ratio schedule or the presentation of each schedule individually on alternate days--preceded exposure to the choice procedure for some subjects. For subjects not exposed to a discrimination procedure prior to exposure to choice, or if such exposure failed to result in higher rates to the ratio than to the interval schedule, relative response rates matched relative reinforcement rates. However, if subjects responded at higher rates to the ratio schedule than to the interval schedule during a prior discrimination procedure, relative rates on a subsequent choice procedure deviated from matching in the direction of reinforcement-rate maximizing. In eight of 11 conditions, choice appeared to be governed by maximizing processes. In all cases, human concurrent ratio-interval performances differed from those of nonhumans in that matching was never obtained with local ratio-interval rate differences.     

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.     

Time allocation in concurrent schedules: the effect of signalled reinforcement

The responses of five pigeons were reinforced on concurrent variable-interval variable-interval reinforcement schedules in which changeover key responses changed the stimulus and reinforcement schedules associated with the food key. While the reinforcement availability in one component remained unchanged throughout the experiment, the reinforcement availability in the other component was, during several conditions, signalled by the onset of an additional discriminative stimulus. During unsignalled conditions, both the relative frequency of responding and the relative time spent in each component approximated the obtained relative reinforcement frequency in each component. The effect of signalling reinforcer availability in one component was to (1) reduce responding in the signalled component to near-zero levels, and (2) increase the relative time in the unsignalled component, without a corresponding increase in the obtained relative reinforcement frequency. The magnitude of the increase in relative time in the unsignalled component decreased as the overall frequency of reinforcement increased. This deviation in the matching relation between relative time and the obtained relative reinforcement frequency was eliminated if the overall reinforcement frequency was increased before the signal was introduced and then, without removing the signal, gradually reduced.