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.     

Effects of Interspersing Briefer Mathematics Problems on Assignment Choice in Students with Learning Disabilities

Within- and between-series designs were combined and used to evaluate the effects of interspersing briefer math problems (i.e., one-digit by one-digit multiplication, 1 × 1) with more time-consuming problems (i.e., two-digit by one-digit multiplication, 2 × 1) on time allocation to one of two concurrent computer-delivered mathematics computation assignments in four high school students with specific learning disabilities in mathematics. The computer presented students with a choice of 2 × 1 problems with 1 × 1 problems interspersed every third problem (i.e., experimental assignment) or 2 × 1 problems without interspersal (i.e., control assignment). Visual and statistical analysis showed that students allocated a greater percentage of their time to the interspersal assignments. These results support previous research on the matching law and problem completion rates and suggest that students' preference for assignments can be improved through the interspersal technique. Discussion focuses on future research and applied educational implications for curricula construction and task sequencing.     

Choice and the rate of punishment in concurrent schedules

Rats' responses on two levers were reinforced according to independent random-interval 1.5-min food schedules. In addition, both lever presses were intermittently punished according to several concurrent random-interval random-interval shock schedules. For the left, the scheduled rate of punishment was kept constant according to a random-interval 6-min schedule. For the right, the rate of punishment varied. As the frequency of punishment for the right lever press increased, its rate decreased. The rate of the left punished lever press increased, however, even though its scheduled reinforcement rate and punishment rate remained unchanged.     

The applied importance of research on the matching law

In this essay, we evaluate the applied implications of two articles related to the matching law and published in the Journal of the Experimental Analysis of Behavior, May 1994. Building on Mace's (1994) criteria for increasing the applied relevance of basic research, we evaluate the applied implications of basic research studies. Research by Elsmore and McBride (1994) and Savastano and Fantino (1994) involve an extension of the behavioral model of choice. Elsmore and McBride used rats as subjects, but arranged a multioperant environment that resembles some of the complex contingencies of human behavior. Savastino and Fantino used human subjects and extended the matching law to ratio and interval contingencies. These experiments contribute to a growing body of knowledge on the matching law and its relevance for human behavior.     

Optimization and the matching law as accounts of instrumental behavior

The interaction between instrumental behavior and environment can be conveniently described at a molar level as a feedback system. Two different possible theories, the matching law and optimization, differ primarily in the reference criterion they suggest for the system. Both offer accounts of most of the known phenomena of performance on concurrent and single variable-interval and variable-ratio schedules. The matching law appears stronger in describing concurrent performances, whereas optimization appears stronger in describing performance on single schedules.     

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.     

An eight-alternative concurrent schedule: foraging in a radial maze.

In two experiments conducted in an eight-arm radial maze, food pellets were delivered when a photocell beam was broken at the end of each arm via a nose poke, according to either fixed-interval or random-interval schedules of reinforcement, with each arm providing a different frequency of reinforcement. The behavior of rats exposed to these procedures was well described by the generalized matching law; that is, the relationships between log behavior ratios and log pellet ratios were approximated by linear functions. The slopes of these log-log functions, an index of sensitivity to reinforcement frequency, were greatest for nose pokes, intermediate for time spent in an arm, and least for arm entries. Similar results were obtained with both fixed-interval and random-interval schedules. Addition of a 10-s changeover delay in both experiments eliminated the slope differentials between nose pokes and time spent in an arm by reducing the slopes of the nose-poke functions. These results suggest that different aspects of foraging may be differentially sensitive to reinforcement frequency. With concurrent fixed-interval schedules, the degree of temporal control exerted by individual fixed-interval schedules was directly related to reinforcement frequency.     

A COLLATERAL EFFECT OF REWARD PREDICTED BY MATCHING THEORY

Matching theory describes a process by which organisms distribute their behavior between two or more concurrent schedules of reinforcement (Herrnstein, 1961). In an attempt to determine the generality of matching theory to applied settings, 2 students receiving special education were provided with academic response alternatives. Using a combined simultaneous treatments design and reversal design, unequal ratio schedules of reinforcement were varied across two academic responses. Findings indicated that both subjects allocated higher rates of responses to the richer schedule of reinforcement, although only one responded exclusively to the richer schedule. The present results lend support to a postulation that positive reinforcement may have undesirable collateral effects that are predicted by matching theory (Balsam & Bondy, 1983).     

EFFECTS OF PROBLEM DIFFICULTY AND REINFORCER QUALITY ON TIME ALLOCATED TO CONCURRENT ARITHMETIC PROBLEMS

Students with learning difficulties participated in two studies that analyzed the effects of problem difficulty and reinforcer quality upon time allocated to two sets of arithmetic problems reinforced according to a concurrent variable-interval 30-s variable-interval 120-s schedule. In Study 1, high- and low-difficulty arithmetic problems were systematically combined with rich and lean concurrent schedules (nickels used as reinforcers) across conditions using a single-subject design. The pairing of the high-difficulty problems with the richer schedule failed to offset time allocated to that alternative. Study 2 investigated the interactive effects of problem difficulty and reinforcer quality (nickels vs. program money) upon time allocation to arithmetic problems maintained by the concurrent schedules of reinforcement. Unlike problem difficulty, the pairing of the lesser quality reinforcer (program money) with the richer schedule reduced the time allocated to that alternative. The magnitude of this effect was greatest when combined with the low-difficulty problems. These studies have important implications for a matching law analysis of asymmetrical reinforcement variables that influence time allocation.     

Response and time allocation in concurrent second-order schedules

Six pigeons were trained on two-key concurrent variable-interval schedules in which the required response was the completion of a fixed number of key pecks. When the required number of pecks was equal on the two keys, response- and time-allocation ratios under-matched obtained reinforcement rate ratios. A similar result was found when the required number of pecks was unequal, except that performance, measured in response terms, was biased to the shorter required number of pecks and was less sensitive to reinforcement-rate changes. No such differences were found in the data on time spent responding. When the variable-interval schedules were kept constant and the required numbers of pecks were systematically varied, response ratios changed inversely with the ratio of the required number of pecks, but time-allocation ratios varied directly with the same independent variable. Thus, on response measures, pigeons “prefer” the schedule with the smaller peck requirement, but on time measures they “prefer” the schedule with the larger peck requirement. This finding is inconsistent with a commonsense notion of choice, which sees response and time-allocation measures as equivalent.