On the classic and modern theories of matching

Classic matching theory, which is based on Herrnstein's (1961) original matching equation and includes the well-known quantitative law of effect, is almost certainly false. The theory is logically inconsistent with known experimental findings, and experiments have shown that its central constant-k assumption is not tenable. Modern matching theory, which is based on the power function version of the original matching equation, remains tenable, although it has not been discussed or studied extensively. The modern theory is logically consistent with known experimental findings, it predicts the fact and details of the violation of the classic theory's constant-k assumption, and it accurately describes at least some data that are inconsistent with the classic theory.     

The generalized matching law as a description of multiple-schedule responding

The literature was examined to determine how well the generalized matching law (Baum, 1974) describes multiple-schedule responding. In general, it describes the data well, accounting for a median of 91% of the variance. The median size of the undermatching parameter was 0.46; the median bias parameter was 1.00. The size of the undermatching parameter, and the proportion of the variance accounted for by the equation, varied inversely with the number of schedules conducted, with the number of sessions conducted per schedule, and with the time within a component. The undermatching parameter also varied with the operanda used to produce reinforcers and with the reinforcer used. The undermatching parameter did not vary consistently with component duration or with several other variables. Bias was greater when fewer rather than more schedules were conducted, when two rather than one operanda were used, and when White Carneaux rather than homing pigeons served as subjects. These results imply that the generalized matching law may describe both concurrent and multiple-schedule responding, but that the same variables do not always influence the bias and undermatching parameters in the same way for the two types of schedules.     

An analytic comparison of Herrnstein's equations and a multivariate rate equation

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.     

The relation between the generalized matching law and signal-detection theory

The generalized matching law can be applied to a signal-detection matrix to give two equations. The first relates responding in the presence of the stimulus to the reinforcements for the responses, and the second relates responding in the absence of the stimulus to the reinforcements for the responses. Evidence for stimulus discrimination is given by biases that are opposite in sign in the two equations. As the logarithmic ratio and z proportion transformations are similar, the combination of the absolute values of the two logarithmic biases gives a measure equivalent to the signal-detection measures d′ and η. The two equations can also be combined to eliminate the biases caused by the signalling stimuli and to produce a generalized matching-law statement relating overall performance to the obtained reinforcements.     

Signal detection and matching: analyzing choice on concurrent variable-interval schedules.

Pigeons' pecks on a red key and a green key were followed by access to grain according to pairs of concurrent independent variable-interval schedules in a combined signal detection/matching law paradigm. Pecks on the red key were reinforced by the richer variable-interval schedule if a short-duration tone had been presented; pecks on the green key were reinforced by the richer variable-interval schedule if a long-duration tone had been presented. Pecks on the green key given a short-duration tone, or on the red key given a long-duration tone, were reinforced by the leaner variable-interval schedule. The data were analyzed according to both signal detection's and the matching law's separate measures of, first, the discrimination of the choices and, second, the bias to make one response or another. Increasing the difficulty of the tone-duration discrimination decreased both methods' measures of the discrimination of the choices and did not change both methods' measures of the bias to make one response or another. Changing the leaner variable-interval schedule so that it approached the richer variable-interval schedule decreased signal detection's measure of discrimination but left its measure of response bias and the matching law measures unchanged. Data collected only until a subject's first changeover response following presentation of a long or a short tone showed higher values for both methods' measures of discrimination, no change in signal detection's measure of response bias, and lower values for the matching law's measure of response bias. Relationships between the matching law's and signal detection's methods of analyzing choice are discussed. It is concluded that a signal detection analysis is more efficient for examining changes in the difficulty of a discrimination, whereas a matching law analysis is more effective for examining the effects of changes in relative reinforcer frequency.     

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.     

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.     

Matching since Baum (1979).

Data from recent studies employing concurrent variable-interval schedules are reviewed. Subject species employed in different experiments have included rats, pigeons, and humans, and reinforcers have varied from food and shock avoidance to points exchangeable for money. Undermatching (a greater preference for the schedule of the concurrent pair that delivers the lower rate of reinforcement than the Matching Law predicts) has been preponderant in recent studies, irrespective of whether behavior has been measured in terms of response ratios or time allocation, with the possible exception of data produced by human subjects. Little difference in the degree of undermatching exhibited by response and time measures has been found, except in the results from a single laboratory, in which time-allocation measures have tended to undermatch less than response measures. Procedural features, such as type of manipulandum used and changeover delay, seem to have little effect on the degree of undermatching exhibited, but asymmetrical response manipulanda (such as lever and key) for the different concurrent schedules, or other asymmetries in the experimental situation, show up clearly in bias measures, in a manner consistent with previous analyses.     

The symmetrical law of effect and the matching relation in choice behavior

In a concurrent-chains procedure, pigeons chose between outcomes that differed in the rate of response-independent delivery of food and electric shocks. The application of functional measurement techniques confirmed the matching relation—between choice and rate of reinforcement value—for two of three pigeons. Scale values of the outcomes were extracted for the two birds that conformed to matching, and the value of a single occurrence of shock per minute—in terms of negative food units—was estimated. A second experiment with concurrent chains provided a test of these parameter estimates. The close correspondence between predicted and obtained choice behavior found in Experiment II indicated that the estimates of outcome value were indeed reliable. Both experiments together support the contention that the effects on choice behavior of positive and aversive stimuli appear to be equal, though opposite in sign.     

Application of the generalized matching law to chess openings: A gambit analysis

During the opening moves of a chess game, a player (typically White) may offer a number of gambits, which involve sacrificing a chess piece for an opponent for capture to achieve long-term positional advantages. One of the most popular gambits is called the Queen's Gambit and involves White offering a pawn to Black, which will open a lane for White's Queen if accepted by Black. In the present study, the generalized matching law (GML) was applied to chess openings involving the Queen's Gambit using over 71,000 archived chess games. Overall, chess players' opening moves involving the Queen's Gambit exhibited orderly matching as predicted by the GML, and the GML accounted for more variance in players' chess decision making as their relative playing experience increased. This study provides support for the generality of the GML and its application to complex operant behavior outside of laboratory contexts.     

Application of the matching law to pitch selection in professional baseball

This study applied the generalized matching equation (GME) to pitch selection in professional baseball. The GME was fitted to the relation between pitch selection and hitter outcomes for five professional baseball pitchers during the 2014 Major League Baseball season. The GME described pitch selection well. Pitch allocation varied across different game contexts such as inning, count, and number of outs in a manner consistent with the GME. Finally, within games, bias decreased for four of the five pitchers and the sensitivity parameter increased for three of the five pitchers. The results extend the generality of the GME to multialternative natural sporting contexts, and demonstrate the influence of context on behavior in natural environments.     

Multilevel analysis of matching behavior

Multilevel modeling has been considered a promising statistical tool in the field of the experimental analysis of behavior and may serve as a convenient statistical analysis for matching behavior because it structures data in groups (or levels) to account simultaneously for the within‐subject and between‐subject variances. Heretofore, researchers have sometimes pooled data erroneously from different subjects in a single analysis by using average ratios, average response and reinforcer rates, aggregation of subjects, etc. Unfortunately, this leads to loss of information and biased estimations, which can severely undermine generalization of the results. Instead, a multilevel approach is advocated to combine several subjects' matching behavior. A reanalysis of previous data on matching behavior is provided to illustrate the method and point out its advantages. It illustrates that multilevel regression leads to better estimations, is more convenient, and offers more behavioral information. We hope this paper will encourage the use of multilevel modeling in the statistical practices of behavior analysts.     

Concurrent reinforcement schedules for problem behavior and appropriate behavior: experimental applications of the matching law

This study evaluated how children who exhibited functionally equivalent problem and appropriate behavior allocate responding to experimentally arranged reinforcer rates. Relative reinforcer rates were arranged on concurrent variable-interval schedules and effects on relative response rates were interpreted using the generalized matching equation. Results showed that relative rates of responding approximated relative rates of reinforcement. Finally, interventions for problem behavior were evaluated and differential reinforcement of alternative behavior and extinction procedures were implemented to increase appropriate behavior and decrease problem behavior. Practical considerations for the application of the generalized matching equation specific to severe problem behavior are discussed, including difficulties associated with defining a reinforced response, and obtaining steady state responding in clinical settings.     

An application of the matching law to social dynamics

Using a procedure similar to the one described by Conger and Killeen (1974), we evaluated levels of attending for 25 college students who participated in either a 20-min (n = 12) or 30-min (n = 13) discussion on juvenile delinquency. Confederates delivered statements of agreement (e.g., "I agree with that point") according to independent variable-interval schedules. Pooled results were evaluated using three generalized formulations of the matching law, and showed that matching was more likely during the first 5 min of the discussion than during the last 5 min. Individual data for 7 of 9 participants were better described by the generalized response-rate matching equation than by the generalized time-allocation matching equation when response allocation was characterized in terms of frequency rather than duration.     

Matching, undermatching, and overmatching in studies of choice

Almost all of 103 sets of data from 23 different studies of choice conformed closely to the equation: log (B(1)/B(2)) = a log (r(1)/r(2)) + log b, where B(1) and B(2) are either numbers of responses or times spent at Alternatives 1 and 2, r(1) and r(2) are the rates of reinforcement obtained from Alternatives 1 and 2, and a and b are empirical constants. Although the matching relation requires the slope a to equal 1.0, the best-fitting values of a frequently deviated from this. For B(1) and B(2) measured as numbers of responses, a tended to fall short of 1.0 (undermatching). For B(1) and B(2) measured as times, a fell to both sides of 1.0, with the largest mode at about 1.0. Those experiments that produced values of a for both responses and time revealed only a rough correspondence between the two values; a was often noticeably larger for time. Statistical techniques for assessing significance of a deviation of a from 1.0 suggested that values of a between .90 and 1.11 can be considered good approximations to matching. Of the two experimenters who contributed the most data, one generally found undermatching, while the other generally found matching. The difference in results probably arises from differences in procedure. The procedural variations that lead to undermatching appear to be those that produce (a) asymmetrical pausing that favors the poorer alternative; (b) systematic temporal variation in preference that favors the poorer alternative; and (c) patterns of responding that involve changing over between alternatives or brief bouts at the alternatives.     

GROUP FORAGING SENSITIVITY TO PREDICTABLE AND UNPREDICTABLE CHANGES IN FOOD DISTRIBUTION: PAST EXPERIENCE OR PRESENT CIRCUMSTANCES?

The ideal free distribution theory (Fretwell & Lucas, 1970) predicts that the ratio of foragers at two patches will equal the ratio of food resources obtained at the two patches. The theory assumes that foragers have "perfect knowledge" of patch profitability and that patch choice maximizes fitness. How foragers assess patch profitability has been debated extensively. One assessment strategy may be the use of past experience with a patch. Under stable environmental conditions, this strategy enhances fitness. However, in a highly unpredictable environment, past experience may provide inaccurate information about current conditions. Thus, in a nonstable environment, a strategy that allows rapid adjustment to present circumstances may be more beneficial. Evidence for this type of strategy has been found in individual choice. In the present experiments, a flock of pigeons foraged at two patches for food items and demonstrated results similar to those found in individual choice. Experiment 1 utilized predictable and unpredictable sequences of resource ratios presented across days or within a single session. Current foraging decisions depended on past experience, but that dependence diminished when the current foraging environment became more unpredictable. Experiment 2 repeated Experiment I with a different flock of pigeons under more controlled circumstances in an indoor coop and produced similar results.     

Time-allocation matching between punishing situations

In the presence and absence of white noise, response-independent aversive events were delivered to rats according to several variable-time electric-shock schedules. The animals could switch from the noise component to the no-noise component and vice versa by making a single lever-press response. If the schedule in one component was not in operation when the animal was in the other component, the proportion of time allocated to one component equalled or matched the proportion of obtained punishers in the other component. If both schedules were always in operation, minimizing tended to occur: the animals allocated almost all of their time to the component having the lower shock rate. An analysis of these results, in terms of the expected time until an aversive event, is presented.     

Dynamic response-by-response models of matching behavior in rhesus monkeys

We studied the choice behavior of 2 monkeys in a discrete-trial task with reinforcement contingencies similar to those Herrnstein (1961) used when he described the matching law. In each session, the monkeys experienced blocks of discrete trials at different relative-reinforcer frequencies or magnitudes with unsignalled transitions between the blocks. Steady-state data following adjustment to each transition were well characterized by the generalized matching law; response ratios undermatched reinforcer frequency ratios but matched reinforcer magnitude ratios. We modelled response-by-response behavior with linear models that used past reinforcers as well as past choices to predict the monkeys' choices on each trial. We found that more recently obtained reinforcers more strongly influenced choice behavior. Perhaps surprisingly, we also found that the monkeys' actions were influenced by the pattern of their own past choices. It was necessary to incorporate both past reinforcers and past choices in order to accurately capture steady-state behavior as well as the fluctuations during block transitions and the response-by-response patterns of behavior. Our results suggest that simple reinforcement learning models must account for the effects of past choices to accurately characterize behavior in this task, and that models with these properties provide a conceptual tool for studying how both past reinforcers and past choices are integrated by the neural systems that generate behavior.