Concurrent random-interval schedules and the matching law

In Experiment I, a group of eight pigeons performed on concurrent random-interval schedules constructed by holding probability equal and varying cycle time to produce ratios of reinforcer densities of 1:1, 3:1, and 5:1 for key pecking. Schedules for a second group of seven were constructed with equal cycle times and unequal probabilities. Both groups deviated from simple matching, but the two forms of the schedules appeared to produce no consistent patterns of deviation. The data were found to be consistent with those obtained in concurrent variable-interval situations. The parameters of the matching equation in the form of Y=k X^a were estimated; the value of k was unity and a was 0.84. In Experiment II, six pigeons were exposed to two conc RI RI schedules in which one component increasingly approximated an FI schedule. The value of k was not 1.0. Concurrent RI RI schedules were shown to represent a continuum from conc FI VI to conc VI VI schedules. The use of the exponential equation in testing “matching laws” suggests that a<1 will continue to be observed, and this will set limits on the form of new laws and the assumed or rational values of the component variables in these laws.     

Choice and segmented interreinforcement intervals

Pigeons were trained on a two-key concurrent schedule, where food reinforcers on one key were arranged by a simple variable-interval schedule and on the other key by a chain variable-interval variable-interval schedule. When the initial link of the chain was in effect, the pigeons tended to respond more on the simple variable-interval schedule, and hence less on the chain, than would be expected from a comparison of both the local and overall rates of reinforcement of the two schedules. When the terminal link of the chain was in effect, the pigeons responded more on the chain than would be expected from a comparison of the rates of reinforcement of the schedules then in effect. Overall responding on the chain was not proportional to overall reinforcement on the chain but rather was a by-product of responding during initial- and terminal-link phases.     

Choice with a fixed requirement for food, and the generality of the matching relation

Pigeons were trained on choice procedures in which responses on each of two keys were reinforced probabilistically, but only after a schedule requirement had been met. Under one arrangement, a fixed-interval choice procedure was used in which responses were not reinforced until the interval was over; then a response on one key would be reinforced, with the effective key changing irregularly from interval to interval. Under a second, fixed-ratio choice procedure, responses on either key counted towards completion of the ratio and then, once the ratio had been completed, a response on the probabilistically selected key would produce food. In one experiment, the schedule requirements were varied for both fixed-interval and fixed-ratio schedules. In the second experiment, relative reinforcement rate was varied. And in a third experiment, the duration of an intertrial interval separating choices was varied. The results for 11 pigeons across all three experiments indicate that there were often large deviations between relative response rates and relative reinforcement rates. Overall performance measures were characterized by a great deal of variability across conditions. More detailed measures of choice across the schedule requirement were also quite variable across conditions. In spite of this variability, performance was consistent across conditions in its efficiency of producing food. The absence of matching of behavior allocation to reinforcement rate indicates an important difference between the present procedures and other choice procedures; that difference raises questions about the specific conditions that lead to matching as an outcome.     

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.     

Melioration, matching, and maximization

Pigeons were studied in an experiment involving two concurrently available response keys. Conditions were such that in the first condition the predictions of melioration (Herrnstein & Vaughan, 1980), minimization of deviation from matching, and maximization were identical: relative time on the right key should have fallen between .125 and .25, which in fact occurred. In the second condition, melioration predicted a shift in relative time on the right to between .75 and .875, which would involve a transient deviation from matching as well as a substantial drop in rate of reinforcement. All three birds eventually shifted their distribution of behavior to within the range predicted by melioration.     

Choice, relative reinforcer duration, and the changeover ratio

Relative reinforcer duration was varied in concurrent schedules with a fixed-ratio four changeover requirement. The schedule in effect after each reinforcer was randomly chosen. For all three pigeons, relative response rates overmatched relative reinforcer durations. Time allocation was less extreme and, on the average, matched relative reinforcer duration. In a subsequent manipulation, the level of preference was shown to depend on the size of the changeover requirement. These results are similar to those from related unequal reinforcement-frequency procedures.     

Choice between response units: The rate constancy model

In a conjoint schedule, reinforcement is available simultaneously on two or more schedules for the same response. The present experiments provided food for key pecking on both a random-interval and a differential-reinforcement-of-low-rate (DRL) schedule. Experiment 1 involved ordinary DRL schedules; Experiment 2 added an external stimulus to indicate when the required interresponse time had elapsed. In both experiments, the potential reinforcer frequency from each component was varied by means of a second-order fixed-ratio schedule, and the DRL time parameter was changed as well. Response rates were described by a model stating that time allocation to each component matches the relative frequency of reinforcement for that component. When spending time in a given component, the subject is assumed to respond at the rate characteristic of baseline performance. This model appeared preferable to the absolute-rate version of the matching law. The model was shown to be applicable to multiple-response concurrent schedules as well as to conjoint schedules, and it described some of the necessary conditions for response matching, undermatching, and bias. In addition, the pigeons did not optimize reinforcer frequency.     

Choice, changing over, and reinforcement delays

In three experiments, pigeons were used to examine the independent effects of two normally confounded delays to reinforcement associated with changing between concurrently available variable-interval schedules of reinforcement. In Experiments 1 and 2, combinations of changeover-delay durations and fixed-interval travel requirements were arranged in a changeover-key procedure. The delay from a changeover-produced stimulus change to a reinforcer was varied while the delay between the last response on one alternative and a reinforcer on the other (the total obtained delay) was held constant. Changeover rates decreased as a negative power function of the total obtained delay. The delay between a changeover-produced stimulus change had a small and inconsistent effect on changeover rates. In Experiment 3, changeover delays and fixed-interval travel requirements were arranged independently. Changeover rates decreased as a negative power function of the total obtained delay despite variations in the delay from a change in stimulus conditions to a reinforcer. Periods of high-rate responding following a changeover, however, were higher near the end of the delay from a change in stimulus conditions to a reinforcer. The results of these experiments suggest that the effects of changeover delays and travel requirements primarily result from changes in the delay between a response at one alternative and a reinforcer at the other, but the pattern of responding immediately after a changeover depends on the delay from a changeover-produced change in stimulus conditions to a reinforcer.     

Stimulus Effects On Behavior Allocation In Three-alternative Choice

Six pigeons were trained on three-alternative concurrent variable-interval schedules that were available through a switching response and were signaled by colored stimuli. The discriminative stimuli for two of the schedules were always 560 nm and 630 nm, but the stimulus signaling the third alternative was varied across conditions over seven levels between these colors. For each third-alternative stimulus condition, the relative frequency of reinforcers was varied over three conditions with 4:1 and 16:1 reinforcer ratios between each pair of alternatives. The distribution of responses between the alternatives was dependent jointly on the third-alternative reinforcer rate and on the disparity between the stimulus signaling the third alternative and those signaling the other alternatives. A generalized matching approach was unable to provide invariant measures of the discriminability between constant stimuli, but a contingency-discriminability approach provided excellent fits and sensible and invariant stimulus discriminability measures.     

Choice and multiple reinforcers

Pigeons chose between equivalent two-component mixed and multiple terminal-link schedules of reinforcement in the concurrent-chains procedure. The pigeons preferred the multiple schedule over the mixed when the components of the compound schedules were differentiated in terms of density of reinforcement, but the pigeons were indifferent when the components were differentiated in terms of number of reinforcers per cycle. Taken together, these results indicate that a local variable, the interval to the first reinforcer, but not a molar variable, the number of reinforcers, was sufficient to differentiate the components and thereby evoke preference.     

Choice in a variable environment: every reinforcer counts

Six pigeons were trained in sessions composed of seven components, each arranged with a different concurrent-schedule reinforcer ratio. These components occurred in an irregular order with equal frequency, separated by 10-s blackouts. No signals differentiated the different reinforcer ratios. Conditions lasted 50 sessions, and data were collected from the last 35 sessions. In Part 1, the arranged overall reinforcer rate was 2.22 reinforcers per minute. Over conditions, number of reinforcers per component was varied from 4 to 12. In Part 2, the overall reinforcer rate was six per minute, with both 4 and 12 reinforcers per component. Within components, log response-allocation ratios adjusted rapidly as more reinforcers were delivered in the component, and the slope of the choice relation (sensitivity) leveled off at moderately high levels after only about eight reinforcers. When the carryover from previous components was taken into account, the number of reinforcers in the components appeared to have no systematic effect on the speed at which behavior changed after a component started. Consequently, sensitivity values at each reinforcer delivery were superimposable. However, adjustment to changing reinforcer ratios was faster, and reached greater sensitivity values, when overall reinforcer rate was higher. Within a component, each successive reinforcer from the same alternative ("confirming") had a smaller effect than the one before, but single reinforcers from the other alternative ("disconfirming") always had a large effect. Choice in the prior component carried over into the next component, and its effects could be discerned even after five or six reinforcement and nonreinforcement is suggested.     

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.     

Optimization versus response-strength accounts of behavior

Pigeons were run in both single-key and concurrent-key experiments in which, over most of the range of response rates, an increase in response rate gave rise to a continuous decrease in reinforcement rate. In spite of the fact that a low response rate would have produced a high reinforcement rate, all birds responded at relatively high rates, thus keeping reinforcement rates substantially below the maximum possible. In the concurrent-key experiment, in addition to responding at relatively high rates, the birds' ratios of responses approximately matched the corresponding ratios of obtained reinforcers. The results are inconsistent with most theories of optimal performance, which assume that organisms behave in ways that either maximize reinforcement value or minimize deviations from a free-behavior point. On the other hand, the results are consistent with the assumption that reinforcement strengthens the tendency to respond.     

Maximizing versus matching on concurrent variable-interval schedules

Maximization and matching predictions were examined for a time-based analogue of the concurrent variable-interval variable-ratio schedule. One alternative was a variable interval whose time base operated relatively independent of the schedule chosen, and the other was a discontinuous variable interval for which timing progressed only when selected. Pigeons switched between schedules by pecking a changeover key. The maximization hypothesis predicts that subjects will show a bias toward the discontinuous variable interval and undermatching; however the obtained results conformed closely to the predictions of the matching law. Finally, a quantitative comparison was made of the bias and sensitivity estimates obtained in published concurrent variable-interval variable-ratio analogue studies. Results indicated that only the ratio-based analogue of the concurrent variable interval variable ratio studied by Green, Rachlin, and Hanson (1983) produced significant bias toward the variable-ratio alternative and undermatching, as predicted by reinforcement maximization.     

Choice in transition: A comparison of melioration and the kinetic model

Transition-state choice behavior of pigeons was examined in two experiments designed to test predictions of melioration and the kinetic model. Both experiments began with an initial training condition during which subjects were maintained on concurrent variable-interval schedules. In Experiment 1, subjects were then exposed to concurrent variable-ratio schedules, whereas in Experiment 2, subjects were then exposed to concurrent extinction. Contrary to the predictions of melioration, but consistent with the kinetic model, acquisition of preference on concurrent variable-ratio schedules followed a negatively accelerated logistic trajectory, and preference remained stable in concurrent extinction. Predictions made by the kinetic model concerning rates of switching between alternatives were also supported.     

Residence time and choice in concurrent foraging schedules

Five pigeons were trained on a concurrent-schedule analogue of the “some patches are empty” procedure. Two concurrently available alternatives were arranged on a single response key and were signaled by red and green keylights. A subject could travel between these alternatives by responding on a second yellow “switching” key. Following a changeover to a patch, there was a probability (p) that a single reinforcer would be available on that alternative for a response after a time determined by the value of λ, a probability of reinforcement per second. The overall scheduling of reinforcers on the two alternatives was arranged nonindependently, and the available alternative was switched after each reinforcer. In Part 1 of the experiment, the probabilities of reinforcement, ρ_red and ρ_green, were equal on the two alternatives, and the arranged arrival rates of reinforcers, λ_red and λ_green, were varied across conditions. In Part 2, the reinforcer arrival times were arranged to be equal, and the reinforcer probabilities were varied across conditions. In Part 3, both parameters were varied. The results replicated those seen in studies that have investigated time allocation in a single patch: Both response and time allocation to an alternative increased with decreasing values of λ and with increasing values of ρ, and residence times were consistently greater than those that would maximize obtained reinforcer rates. Furthermore, both response- and time-allocation ratios undermatched mean reinforcer-arrival time and reinforcer-frequency ratios.     

Concurrent variable-interval variable-ratio schedules can provide only weak evidence for matching

Herrnstein and Heyman (1979) showed that when pigeons' pecking is reinforced on concurrent variable-interval variable-ratio schedules, (1) their behavior ratios match the ratio of the schedules' reinforcer frequencies, and (2) there is more responding on the variable interval. Since maximizing the reinforcement rate would require responding more on the variable ratio, these results were presented as establishing the primacy of matching over maximizing. In the present report, different ratios of behavior were simulated on a computer to see how they would affect reinforcement rates on these concurrent schedules. Over a wide range of experimenter-specified choice ratios, matching obtained — a result suggesting that changes in choice allocation produced changes in reinforcer frequencies that correspond to the matching outcome. Matching also occurred at arbitrarily selected choice ratios when reinforcement rates were algebraically determined by each schedule's reinforcement-feedback function. Additionally, three birds were exposed to concurrent variable-interval variable-ratio schedules contingent on key pecking in which hopper durations were varied in some conditions to produce experimenter-specified choice ratios. Matching generally obtained between choice ratios and reinforcer-frequency ratios at these different choice ratios. By suggesting that reinforcer frequencies track choice on this procedure, instead of vice versa, this outcome questions whether matching-as-outcome was due to matching-as-process in the Herrnstein and Heyman study.     

Choice and transformed interreinforcement intervals

Pigeons chose between two aperiodic, time-based schedules of reinforcement. The arithmetic mean interreinforcement interval of the first schedule was short, but the harmonic mean was long, whereas the arithmetic mean interreinforcement interval of the second schedule was long, but the harmonic mean was short. The pigeons preferred the schedule with the shorter harmonic mean in a concurrent-chains procedure when a terminal link ended after the first scheduled reinforcer had been gained on a terminal-link entry, but reversed their preferences, such that they preferred the schedule with the shorter arithmetic mean, when the terminal links ended after a fixed duration of exposure to the schedule. Moreover, the pigeons preferred the schedule with the shorter arithmetic mean in a two-key concurrent variable-interval variable-interval procedure, as well as in a concurrent variable-time variable-time, changeover-key procedure. The data suggest that an aggregate property of a schedule may not yield valid information about the responding that schedule will maintain as a choice alternative.     

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.