Generalized Matching Equation Research Articles

Concurrent performance in a three-alternative choice situation: Response allocation in a Rock/Paper/Scissors game

Adult human subjects engaged in a simulated Rock/Paper/Scissors game against a computer opponent. The computer opponent's responses were determined by programmed probabilities that differed across 10 blocks of 100 trials each. Response allocation in Experiment 1 was well described by a modified version of the generalized matching equation, with undermatching observed in all subjects. To assess the effects of instructions on response allocation, accurate probability-related information on how the computer was programmed to respond was provided to subjects in Experiment 2. Five of 6 subjects played the counter response of the computer's dominant programmed response near-exclusively (e.g., subjects played paper almost exclusively if the probability of rock was high), resulting in minor overmatching, and higher reinforcement rates relative to Experiment 1. On the whole, the study shows that the generalized matching law provides a good description of complex human choice in a gaming context, and illustrates a promising set of laboratory methods and analytic techniques that capture important features of human choice outside the laboratory.

WITHIN‐SESSION TRANSITIONS IN CHOICE: A STRUCTURAL AND QUANTITATIVE ANALYSIS

The present study used within-session transitions between two concurrent schedules to evaluate choice in transition. Eight female Long-Evans rats were trained to respond under concurrent schedules of reinforcement during experimental sessions that lasted 22 hr. The generalized matching equation was used to model steady-state behavior at the end of each session, while transitional behavior that emerged following the change in reinforcement schedules was modeled using a logistic equation. The generalized matching and logistic equations were appropriate models for behavior generated during single-session transitions. A local analysis of behavior on the two response alternatives during acquisition was used to determine the source of preference as revealed in response ratios. The number of "low-response" visits, those containing three to five responses, remained stable. Preference ratios largely reflected a sharp increase in the number of visits with long response bouts on the rich alternative and a decrease in the number of such visits to the leaner alternative.

SENSITIVITY AND BIAS UNDER CONDITIONS OF EQUAL AND UNEQUAL ACADEMIC TASK DIFFICULTY

We conducted an experimental analysis of children's relative problem-completion rates across two workstations under conditions of equal (Experiment 1) and unequal (Experiment 2) problem difficulty. Results were described using the generalized matching equation and were evaluated for degree of schedule versus stimulus control. Experiment 1 involved a symmetrical choice arrangement in which the children could earn points exchangeable for rewards contingent on correct math problem completion. Points were delivered according to signaled variable-interval schedules at each workstation. For 2 children, relative rates of problem completion appeared to have been controlled by the schedule requirements in effect and matched relative rates of reinforcement, with sensitivity values near 1 and bias values near 0. Experiment 2 involved increasing the difficulty of math problems at one of the workstations. Sensitivity values for all 3 participants were near 1, but a substantial increase in bias toward the easier math problems was observed. This bias was possibly associated with responding at the more difficult workstation coming under stimulus control rather than schedule control.

Quantitative assessments of sensitivity to reinforcement contingencies in mental retardation.

Sensitivity to differences in reinforcement contingencies was examined in 6 individuals with mental retardation. A concurrent operants procedure was implemented in the context of a computer game. Participants selected two different types of animated figures displayed concurrently in the left and right portions of a touchscreen monitor. Over a series of conditions, the relative rates or magnitudes of reinforcers following selections of the two options were changed in ratios of 5:1, 3:1, 1:1, 1:3, and 1:5. Sensitivity was quantified by application of the generalized matching equation. Results included individual differences in sensitivity and differential sensitivity to rate and magnitude variation. The results suggest that comprehensive assessments of potential reinforcers may benefit from including tests of delivery parameters.

HUMAN SENSITIVITY TO CONCURRENT SCHEDULES OF REINFORCEMENT: EFFECTS OF OBSERVING SCHEDULE‐CORRELATED STIMULI

The determinants of human sensitivity to concurrent variable‐interval variable‐interval schedules of reinforcement have been difficult to identify, in part because of procedural differences separating published experiments. This experiment investigated vigilance to stimuli correlated with concurrent schedules. Across phases, 3 college students were provided with either no schedule‐correlated stimuli, an observing response that provided brief access to the stimuli, or a contingency that required the subject to identify the stimulus correlated with the source of each obtained reinforcer. Sensitivity, as quantified by the generalized matching equation, was low when no stimuli were available. When the stimuli were response contingent, 1 subject observed them, and her behavior became more sensitive to the distribution of reinforcers across the concurrent schedules. When the procedure required discrimination of the stimulus correlated with each reinforcer, the other 2 subjects also observed the stimuli, and their schedule sensitivity was increased as well. These results implicate procedural differences, rather than inherent behavioral differences, as the source of differences in sensitivity to schedules of reinforcement between humans and nonhumans.

Concurrent schedule performance in domestic goats: persistent undermatching

Performance of nine domestic goats responding under concurrent variable-interval variable-interval schedules of food delivery was examined, with results analyzed in terms of the generalized matching equation. Substantial undermatching of response and time allocation ratios to obtained reinforcement ratios was evident. Post-reinforcement pause time ratios approximately matched obtained reinforcement ratios. Subtracting these times from total time allocation values yielded net time allocation ratios, which undermatched obtained reinforcement ratios to a greater degree than whole-session time allocation ratios. Slopes of regression lines relating behavioral outputs to environmental inputs characteristically were below 0.6, which is similar to previous findings in dairy cows tested under comparable conditions.

CONCURRENT‐SCHEDULE PERFORMANCE IN DAIRY COWS: PERSISTENT UNDERMATCHING

Performance of dairy cows responding under concurrent variable-interval variable-interval schedules of food delivery was examined, with results analyzed in terms of the generalized matching equation. In Experiment 1, bias measures indicated that crushed barley was preferred over meatmeal when these foods were available under the alternative schedules. For whole-session data, substantial undermatching of response and time-allocation ratios to obtained reinforcement ratios was evident. Postreinforcement pause time ratios approximately matched obtained reinforcement rates. Subtracting these times from total time-allocation values yielded net time-allocation ratios that undermatched obtained reinforcement ratios to a greater degree than did whole-session time-allocation ratios. In Experiment 2, substantial undermatching was evident when the same foods (hay for 2 cows, crushed barley for 2 others) were available under the alternative schedules. Food-related activities and other defined behavior not related to food were quantified by direct observation, and were found to occupy a substantial proportion (roughly 40% to 80%) of experimental sessions. Subtracting the time spent in these activities from the time allocated to each component schedule did not reduce the degree of undermatching obtained. Across all conditions in both experiments, slopes of regression lines relating behavioral outputs to environmental inputs characteristically were below 0.6, which agrees with prior findings and suggests that, contrary to suggestions in the literature, undermatching in dairy cows is not the result of using different foods under alternative schedules or differential pausing under those schedules.

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.

DETERMINANTS OF HUMAN PERFORMANCE ON CONCURRENT SCHEDULES

Six experiments, each with 5 human adults, were conducted to investigate the determinants of human performance on multiple concurrent variable-interval schedules. A two-key procedure was employed in which subjects' key presses produced points exchangeable for money. Variables manipulated across experiments were (a) changeover delay (Experiments 2, 4, and 6), (b) ordinal cues related to scheduled reinforcement frequencies (Experiments 3 and 4), and (c) instructions describing the ordinal relations between schedule-correlated stimuli and scheduled reinforcement frequency (Experiments 5 and 6). The performances of only 13 of the 30 subjects could be described by the generalized matching equation and were within a range of values typical of those reported in the animal literature. Eight subjects showed indifference, 9 undermatched, 7 approximated matching, 3 overmatched, and a further 3 responded exclusively to the richer component of the concurrent schedules. These differing modes of responding were closely related to the different types of performance rules reported by subjects in postexperimental questionnaires. The results are in good agreement with those from studies of human performance on single schedules, suggesting that rule-governed behavior, in interaction with contingencies, may be an important determinant of human choice.

Quantitative Analyses of Choice in Rat and Pigeon

Studies of choice in nonhuman animals that presented concurrent variable-interval variable-interval schedules of reinforcement were quantitatively reviewed. Empirical parameters representing bias and sensitivity in the generalized matching equation were calculated across studies and compared. Undermatching was the most common relationship observed. No consistent bias was shown. An analysis of moderating variables also revealed that the laboratory in which an experiment was conducted, and the year of publication were related to the sensitivity parameter. This reanalysis confirms that a proportional relationship between the ratios of response rates and ratios of reinforcement rates is not an accurate description of concurrent VI VI performance.

BEHAVIORAL STEREOTYPY AND THE GENERALIZED MATCHING EQUATION

The development of behavioral stereotypy is a common result of exposure to both response-dependent and response-independent reinforcement procedures. The generalized matching equation and two dynamic versions of that equation, which take into account the time differential between reinforcements and their effect on behavior, predict this outcome of many procedures involving reinforcement. Following from the assumption that distinct response topographies, distinct response sequences, or orientations to distinct stimuli can be treated in the equations as distinct classes of behavior, the equations predict that-at least for matching and undermatching-the behavior class that is most biased relative to other behavior classes of the same type will tend to predominate to the exclusion or near exclusion of those behavior classes.

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%.

Choice, experience, and the generalized matching law.

Five pigeons were exposed to different pairs of concurrent variable-interval, variable-interval schedules on nine experimental conditions of 30 sessions each. For every session, the parameters of the generalized matching equation were computed for the first five, six, seven, eight, and nine experimental conditions. The exponent a, both for response and time distribution, tended to decrease with increases in number of experimental conditions and to increase with number of sessions per condition, but values of k (bias) varied unsystematically. When the subjects were exposed to five new pairs of schedules, with 55 sessions per condition, the findings were confirmed. Data from the literature on the generalized matching law suggest that the variability of exponent values may be explained in part by the use of naive or experienced subjects in different investigations and by the variability in number of experimental conditions and in number of sessions per condition.

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.

CONCURRENT FIXED‐INTERVAL VARIABLE‐RATIO SCHEDULES AND THE MATCHING RELATION

Five rats responded under concurrent fixed-interval variable-ratio schedules of food reinforcement. Fixed-interval values ranged from 50-seconds to 300-seconds and variable-ratio values ranged from 30 to 360; a five-second changeover delay was in effect throughout the experiment. The relations between reinforcement ratios obtained from the two schedules and the ratios of responses and time spent on the schedules were described by Baum's (1974) generalized matching equation. All subjects undermatched both response and time ratios to reinforcement ratios, and all subjects displayed systematic bias in favor of the variable-ratio schedules. Response ratios undermatched reinforcement ratios less than did time ratios, but response ratios produced greater bias than did time ratios for every subject and for the group as a whole. Local rates of responding were generally higher on the variable-ratio than on the fixed-interval schedules. When responding was maintained by both schedules, a period of no responding on either schedule immediately after fixed-interval reinforcement typically was followed by high-rate responding on the variable-ratio schedule. At short fixed-interval values, when a changeover to the fixed-interval schedule was made, responding usually continued until fixed-interval reinforcement was obtained; at longer values, a changeover back to the variable-ratio schedule usually occurred when fixed-interval reinforcement was not forthcoming within a few seconds, and responding then alternated between the two schedules every few seconds until fixed-interval reinforcement finally was obtained.

Behaviour of Humans in Concurrent Schedules Programmed on Spatially Separated Operanda

Six human subjects responded on either of two levers for monetary reinforcement in a continuous choice situation. Responses on one lever (lever A) were reinforced at different frequencies specified by five variable-interval schedules. Reinforcements for responding on the other lever (lever B) were delivered according to a variable-interval schedule of standard reinforcement frequency. Results indicated that absolute response rate on lever A increased, while absolute response rate on lever B decreased as a function of reinforcement frequency for lever A. In terms of preference, the data conformed closely to Baum's (1974) generalized matching equation. Performances departing from perfect matching were obtained in four cases, but these deviations were not systematic either in the direction of overmatching or undermatching.