An application of the matching law to evaluate the allocation of two- and three-point shots by college basketball players

We applied the matching equation to evaluate the allocation of two- and three-point shots by male and female college basketball players from a large Division 1 university. The matching law predicts that the proportion of shots taken from three-point range should match the proportional reinforcement rate produced by such shots. Thus, we compared the proportion of three-point shots taken relative to all shots to the proportion of three-point shots scored relative to all shots scored. However, the matching equation was adjusted to account for the greater reinforcer magnitude of the three-point basket (i.e., 1.5 times greater than the two-point basket reinforcer magnitude). For players with substantial playing time, results showed that the overall distribution of two- and three-point shots was predicted by the matching equation. Game-by-game shot distribution was variable, but the cumulative proportion of shots taken from three-point range as the season progressed was predicted almost perfectly on a player-by-player basis for both male and female basketball players.     

Matching, maximizing, and the behavioral unit: concurrent reinforcement of response sequences.

Pigeons pecked two keys in a probability matching situation in which four two-peck sequences were intermittently reinforced: left-left, left-right, right-left and right-right. In Phase 1, relative reinforcement rate was varied with respect to the first response of a sequence: reinforcers were differentially assigned for left-left and left-right sequences as opposed to right-left and right-right sequences. The second response of reinforced sequences occurred equally on the left and right keys across conditions. In Phase II, relative reinforcement rate was varied for sequences that involve an alternation as opposed to those that did not. The relative outputs of the different sequences matched the relative reinforcement rates for the different sequences in both phases. Relative response rates for key pecks did not always match relative reinforcement rates. The intertrial interval separating responses was varied in both phases; increases in the intertrial interval affected the relative frequency of different sequences. The results demonstrate that response sequences acted as functional units influencing choice and thus support a structural account of choice. At the same time, the matching of relative sequence proportion and relative reinforcement rate supports a matching account.     

Matching and conditioned reinforcement rate

Attempts to examine the effects of variations in relative conditioned reinforcement rate on choice have been confounded by changes in rates of primary reinforcement or changes in the value of the conditioned reinforcer. To avoid these problems, this experiment used concurrent observing responses to examine sensitivity of choice to relative conditioned reinforcement rate. In the absence of observing responses, unsignaled periods of food delivery on a variable-interval 90-s schedule alternated with extinction on a center key (i.e., a mixed schedule was in effect). Two concurrently available observing responses produced 15-s access to a stimulus differentially associated with the schedule of food delivery (S+). The relative rate of S+ deliveries arranged by independent variable-interval schedules for the two observing responses varied across conditions. The relation between the ratio of observing responses and the ratio of S+ deliveries was well described by the generalized matching law, despite the absence of changes in the rate of food delivery. In addition, the value of the S+ deliveries likely remained constant across conditions because the ratio of S+ to mixed schedule food deliveries remained constant. Assuming that S+ deliveries serve as conditioned reinforcers, these findings are consistent with the functional similarity between primary and conditioned reinforcers suggested by general choice theories based on the concatenated matching law (e.g., contextual choice and hyperbolic value-added models). These findings are inconsistent with delay reduction theory, which has no terms for the effects of rate of conditioned reinforcement in the absence of changes in rate of primary reinforcement.     

Choosing among natural stimuli

Pigeons worked on concurrent variable-interval, variable-interval schedules with the alternatives signaled by slides either containing trees or not. The schedules were designed to hold both overall and relative rates of reinforcement within narrowly constrained limits, and slides were quasi-randomly ordered each day. Responding to the two alternatives was well described by the generalized matching equation with substantial undermatching. Using an adaptation of the matching law, we estimated that the subjects were correctly classifying 82% to 95% of exemplars. The matching performance transferred to new exemplars of trees and nontrees with only slight generalization decrement. The pigeons appeared to be discriminating among exemplars even when the alternatives provided equal rates of reinforcement and the average relative performances were close to 50%.     

The general matching law describes choice on concurrent variable-interval schedules of wheel-running reinforcement.

Six male Wistar rats were exposed to concurrent variable-interval schedules of wheel-running reinforcement. The reinforcer associated with each alternative was the opportunity to run for 15 s, and the duration of the changeover delay was 1 s. Results suggested that time allocation was more sensitive to relative reinforcement rate than was response allocation. For time allocation, the mean slopes and intercepts were 0.82 and 0.008, respectively. In contrast, for response allocation, mean slopes and intercepts were 0.60 and 0.03, respectively. Correction for low response rates and high rates of changing over, however, increased slopes for response allocation to about equal those for time allocation. The results of the present study suggest that the two-operant form of the matching law can be extended to wheel-running reinforcement. 'I'he effects of a low overall response rate, a short Changeover delay, and long postreinforcement pausing on the assessment of matching in the present study are discussed.     

How to teach a pigeon to maximize overall reinforcement rate

In two experiments deviations from matching earned higher overall reinforcement rates than did matching. In Experiment 1 response proportions were calculated over a 360-response moving average, updated with each response. Response proportions that differed from the nominal reinforcement proportions, by a criterion that was gradually increased, were eligible for reinforcement. Response proportions that did not differ from matching were not eligible for reinforcement. When the deviation requirement was relatively small, the contingency proved to be effective. However, there was a limit as to how far response proportions could be pushed from matching. Consequently, when the deviation requirement was large, overall reinforcement rate decreased and pecking was eventually extinguished. In Experiment 2 a discriminative stimulus was added to the procedure. The houselight was correlated with the relationship between response proportions and the nominal (programmed) reinforcement proportions. When the difference between response and reinforcement proportions met the deviation requirement, the light was white and responses were eligible for reinforcement. When the difference between response and reinforcement proportions failed to exceed the deviation requirement, the light was blue and responses were not eligible for reinforcement. With the addition of the light, it proved to be possible to shape deviations from matching without any apparent limit. Thus, in Experiment 2 overall reinforcement rate predicted choice proportions and relative reinforcement rate did not. In contrast, in previous experiments on the relationship between matching and overall reinforcement maximization, relative reinforcement rate was usually the better predictor of responding. The results show that whether overall or relative reinforcement rate better predicts choice proportions may in part be determined by stimulus conditions.     

Relative and absolute reinforcement frequency as determinants of choice in concurrent variable interval schedules

Rats were trained under concurrent schedules consisting of two equal variable interval component shedules providing sucrose solutions of different concentrations (0.6 M, 0.2 M; 50 μl in each case) as the reinforcers. The mean interreinforcement interval specified by the schedules was varied from 10 to 640 sec. Absolute response rate in each component was an increasing hyperbolic function of reinforcement frequency. Relative response rate and relative time allocation revealed a consistent preference for the more concentrated solution; neither measure of preference was systematically related to reinforcement frequency. The results are consistent with Baum and Rachlin's (1969) extension of the matching law, and with a derivation of the matching law from the hyperbolic relation between absolute response rate and reinforcement frequency in variable interval schedules (Herrnstein, 1970).     

The identities hidden in the matching laws, and their uses

Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help identify the referents for certain empirical or theoretical coefficients. This identity can be applied to both single and compound schedules of reinforcement, absolute and relative measures, and to local, global and overall rates and probabilities. The rate matching equations of Hernstein and Catania appear to have been approximations to, and to have been evolving toward, one form of this algebraic identity. Estimates of the bias and sensitivity terms in the generalized ratio and logarithmic matching models are here held to be averaging artifacts arising from fitting procedures applied to models that violate or conceal the underlying identities.     

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.     

Short-component multiple schedules: effects of relative reinforcement duration

Pigeons were exposed to multiple variable-interval 2-min variable-interval 2-min schedules of food presentation in which relative duration of food presentation was manipulated. When components alternated every 5 sec and were scheduled on separate response keys, relative response rates closely matched relative reinforcement duration in three of four pigeons. On the other hand, relative response rates were insensitive to relative reinforcement duration when components scheduled on a single response key alternated every 5 sec, and when components scheduled on separate response keys alternated every 2 min. Thus, both rapid alternation and spatial separation of components were necessary to produce approximate matching of relative responding to relative reinforcement duration. This finding contrasts with previous findings that only rapid component alternation is necessary for matching when relative rate of reinforcement is manipulated.     

Effects of qualitatively different reinforcers on the parameters of the response-strength equation.

This experiment examined the relationship between two qualitatively different reinforcers and the parameters of a quantitative model of reinforced responding, referred to as the response-strength equation or the Herrnstein equation. A group of rats was first food deprived and later water deprived. An 11.5% sucrose solution served as the reinforcer in the food-deprivation condition, and water was the reinforcer in the water-deprivation condition. Each experimental session consisted of a series of seven variable-interval schedules, providing reinforcement rates that varied between 20 and 1,200 reinforcers per hour. The response rates increased in a negatively accelerating function in a manner consistent with the response-strength equation. This equation has two fitted parameters, k and Re. According to one theory, the k parameter is a measure of motor performance, and Re is indicative of the relative reinforcement efficacy of the background uncontrollable sources of reinforcement in relation to the experimentally arranged reinforcer. In this study, k did not change as a result of the different reinforcers, but Re was significantly larger in the sucrose-reinforcement condition. These results are consistent with the interpretation that k and Re measure two independent and experimentally distinguishable parameters and provide further evidence that absolute response rate is a function of relative reinforcement rate, as implied by the derivation of the response-strength equation based on the matching law.     

The law of effect and avoidance: a quantitative relationship between response rate and shock-frequency reduction

Two experiments were conducted to investigate the quantitative relationship between response rate and reinforcement frequency in single and multiple variable-interval avoidance schedules. Responses cancelled delivery of shocks that were scheduled by variable-interval schedules. When shock-frequency reduction was taken as the measure of reinforcement, the relationship between response rate and reinforcement frequency on single variable-interval avoidance schedules was accurately described by Herrnstein's (1970) equation for responding on single variable-interval schedules of positive reinforcement. On multiple variable-interval avoidance schedules with brief components, asymptotic relative response rate matched relative shock-frequency reduction. The results suggest that many interactions between response rates and shock-frequency reduction in avoidance can be understood within the framework of the generalized matching relation, as applied by Herrnstein (1970) to positive reinforcement.     

Choice, matching, and human behavior: A review of the literature

This review concerns human performance on concurrent schedules of reinforcement. Studies indicate that humans match relative behavior to relative rate of reinforcement. Herrnstein's proportional matching equation describes human performance but most studies do not evaluate the equation at the individual level. Baum's generalized matching equation has received strong support with humans as subjects. This equation permits the investigation of sources of deviation from ideal matching and a few studies have suggested variables which control such deviations in humans. While problems with instructional control are raised, the overall findings support the matching law as a principle of human choice.     

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.     

The assumptions underlying the generalized matching law

Allen (1981) derived the power-function generalization of the matching law from a functional equation involving relative response rates on three concurrently available schedules of reinforcement. This paper defines the conditions (relative homogeneity and independence) under which a more general class of behavioral laws reduces to the power law. The proof also removes two deficiencies of Allen's result (discussed by Houston, 1982), which are, first, that his derivation produces a power law without a bias coefficient, and second, that it holds only for experiments with three or more concurrent schedules.     

The effect of signaled reinforcement availability on concurrent performances in humans

During Phase I, three female human subjects pressed a button for monetary reinforcement in two-component concurrent variable-interval schedules. Five different reinforcement frequencies were used in component A, whereas the reinforcement frequency in component B was held constant. Absolute rates of responding conformed to equations proposed by Herrnstein to describe concurent performances, and the ratios of the response rates and the times spent in the two components conformed to the matching law. During Phase II, the availability of reinforcement in component A was signaled by the illumination of a lamp. This resulted in suppression of response rates in component A and elevation of response rates in component B, these changes being reflected in a distortion of the matching relationship which took the form of a bias in favor of component B.     

Undermatching and overmatching as deviations from the matching law

A model of performance under concurrent variable-interval reinforcement schedules that takes as its starting point the hypothetical “burst” structure of operant responding is presented. Undermatching and overmatching are derived from two separate, and opposing, tendencies. The first is a tendency to allocate a certain proportion of response bursts randomly to a response alternative without regard for the rate of reinforcement it provides, others being allocated according to the simple matching law. This produces undermatching. The second is a tendency to prolong response bursts that have a high probability of initiation relative to those for which initiation probability is lower. This process produces overmatching. A model embodying both tendencies predicts (1) that undermatching will be more common than overmatching, (2) that overmatching, when it occurs, will tend to be of limited extent. Both predictions are consistent with available data. The model thus accounts for undermatching and overmatching deviations from the matching law in terms of additional processes added on to behavior allocation obeying the simple matching relation. Such a model thus enables processes that have been hypothesized to underlie matching, such as some type of reinforcement rate or probability optimization, to remain as explanatory mechanisms even though the simple matching law may not generally be obeyed.     

Effects of methylphenidate on sensitivity to reinforcement in children diagnosed with attention deficit hyperactivity disorder: an application of the matching law

The behavior of children diagnosed with attention deficit hyperactivity disorder (ADHD) has been hypothesized to be the result of decreased sensitivity to consequences compared to typical children. The present study examined sensitivity to reinforcement in 2 boys diagnosed with ADHD using the matching law to provide more precise and quantitative measurement of this construct. This experiment also evaluated the effects of methylphenidate (MPH) on sensitivity to reinforcement of children with ADHD. Subjects completed math problems to earn tokens under four different variable-interval (VI) schedules of reinforcement presented in random order under both medicated and nonmedicated conditions. Results showed that, in the medicated condition, the matching functions for both subjects resulted in higher asymptotic values, indicating an overall elevation of behavior rate under these conditions. The variance accounted for by the matching law was also higher under the medicated conditions, suggesting that their behavior more closely tracked the changing rates of reinforcement while taking MPH compared to placebo. Under medicated conditions, the reinforcing efficacy of response-contingent tokens decreased. Results are discussed with respect to quantifying behavioral changes and the extent to which the drug interacts with prevailing contingencies (i.e., schedule values) to influence behavioral variability.     

Timescale invariance and Weber's law in choice

Pigeons were exposed to concurrent schedules for which reinforcement was alternately available at different times for each of two choices. In Experiment 1 (in which reinforcement times progressed arithmetically), overall, but not relative, response rate was timescale invariant. In Experiment 2 (in which reinforcement times progressed geometrically and were more spaced out), there was temporal control at all reinforcement times, but the amplitude of left-right response alternation decreased as time in the trial increased. These results indicate that the temporal regulation of both overall and relative response rates conforms to Weber's law although relative rate is heavily influenced by processes other than timing. It also adds support to the idea that overall and relative response rate reflects the operation of two independent processes.