Matching theory in natural human environments

Grounding applied animal behavior practices in the experimental analysis of behavior

Some of the earliest applications outside the laboratory of principles derived from the experimental analysis of behavior (EAB), such as the pioneering work of Keller and Marian Breland, involved animals. This translational tradition continues to the present as EAB-related behavior principles are applied with increasing frequency to behavior management and training practices with animals in nonlaboratory settings. Such translations, and those populations to which they are applied, benefit from a rigorous experimental analysis of practices that are promulgated in popular outlets. These translations both affirm the generality of those principles and serve as goads for laboratory and field research that can further articulate extant principles, develop new ones, and refine methods of application and assessment. This review considered several areas of basic EAB research and contemporary applied animal behavior (AAB) practices in relation to one another: (1) response establishment and maintenance, (2) response reduction and elimination, (3) chaining and conditioned reinforcement, and (4) discriminative stimulus control. Within each topic, a selection of processes and procedures in both EAB and AAB work were reviewed in relation to one another.

The matching law provides a quantitative description of social time allocation in children with autism

Recent research has developed and evaluated assessments of sociability in which time allocation near or away from an adult who initiates social interactions is used to characterize the participant as social, indifferent, or avoidant of social interaction. Though these qualitative outcomes have been useful, no studies have evaluated methods of obtaining more quantitative measures of sociability. The matching law has been demonstrated to describe a wide range of human behavior and may also be useful in describing social time allocation. We adapted the matching law and assessment of sociability procedures with the aim of providing a more precise, quantitative measure of sociability. We fitted the matching equation to the social time allocation data of 8 children with autism spectrum disorder. The equation was effective in quantifying sociability, accounted for a large proportion of variance in participants' behavior, did so equally well for participants who were social and avoidant, and provided a more sensitive measure relative to those used in previous research. The implications of this methodology, its potential utility, and directions for future research are discussed.

Review of Behavioral Psychology in Transition to Solar Photovoltaics for Low-Income Individuals

The increase in nonrenewable energy (non-RE) has been a growing concern for low-income individuals’ quality of life, health, economy, and environment. At the same time, the use of non-RE is also a great concern for the whole population as we are breathing the same environment. The photovoltaics (PV) solar panel is one solution to decrease low-income individuals’ energy bills and increase the quality of life of all individuals. Knowing the behavioral theory of why low-income individuals do not adopt PV would allow further insights and possible interventions to help low-income individuals install PV. Research has found that low-income individuals are more likely to have financial and knowledge barriers that hinder them from installing PV. Providing a way for low-income individuals to combat these barriers would help them to use PV. This review showed that low-income individuals are likely to benefit from policy programs that incentivize them to use PV. More knowledge about PV can also be aided by policy programs that inform low-income individuals how to save financially and at the same time work their way to install PV. Social groups could also be formed in the same policy programs to help low-income individuals share strategies on saving financially and knowledge about the benefit of installing PV. These social groups can act as a social reinforcement to low-income individuals to install PV. Helping low-income individuals to install PV would help low-income individuals financially and improve the population’s quality of life.

Conditional Regard in the Classroom: A Double-Edged Sword

Teachers’ conditional positive and negative regard are widely endorsed teaching practices aimed to enhance students’ involvement and achievement in school. Previous research has mostly tapped the need frustration and harmful psychological well-being implications of these practices. Yet knowledge of their specific effects on school engagement is scant. This study investigated the association between students’ perceptions of homeroom teachers’ conditional positive and negative regard and their behavioral engagement, while considering the levels at which these practices are conceptualized and operate (a teacher characteristic and a student characteristic). Participants were n = 2533 students from 107 classes in the 7th to 10th grades. Multilevel analysis found conditional positive regard was positively associated with school engagement while conditional negative regard was inversely related. These findings were obtained at both the within- and between-class levels. Based on the findings, we argue conditional regard is a double-edged sword. Consistent with previous research, we suggest conditional negative regard has an undermining effect, and we point to conditional positive regard’s potential to enhance engagement. Lastly, we discuss the importance of the level of analysis and the alignment of theory with measurement.

Resurgence as Choice: Implications for promoting durable behavior change

Resurgence is an increase in a previously suppressed behavior resulting from a worsening in reinforcement conditions for current behavior. Resurgence is often observed following successful treatment of problem behavior with differential reinforcement when reinforcement for an alternative behavior is subsequently omitted or reduced. The efficacy of differential reinforcement has long been conceptualized in terms of quantitative models of choice between concurrent operants (i.e., the matching law). Here, we provide an overview of a novel quantitative model of resurgence called Resurgence as Choice (RaC), which suggests that resurgence results from these same basic choice processes. We review the failures of the only other quantitative model of resurgence (i.e., Behavioral Momentum Theory) and discuss its shortcomings with respect to the limited range of circumstances about which it makes predictions in applied settings. Finally, we describe how RaC overcomes these shortcomings and discuss implications of the model for promoting durable behavior change.

Whether to push or pull? Nicotine reduction and non-combusted alternatives - Two strategies for reducing smoking and improving public health

Combustible cigarettes remain the most harmful and addictive tobacco product, and reducing the prevalence of smoking continues to be a critical public health goal. While nicotine is the constituent primarily responsible for addiction to cigarettes, most of the harm associated with smoking comes from byproducts of tobacco combustion. Recently, two different approaches for reducing the harms of smoking have emerged, both of which focus on breaking the link between the addiction to nicotine and the harms caused by smoking. First, the addictive potential of cigarettes could be minimized by requiring a large reduction in the nicotine content of cigarettes. Evidence for a nicotine reduction policy thus far shows that the use of very low nicotine content cigarettes results in a reduction in the number of cigarettes people smoke per day and a reduction in cigarette dependence. Second, emerging alternative nicotine delivery systems (ANDS) like electronic cigarettes may provide sufficient nicotine to act as substitutes for cigarettes while delivering much lower levels of toxicants. Evidence suggests that the emergence of ANDS has increased the percentage of smokers who are able to quit. The present paper will briefly review the evidence for each of these approaches, and consider what contemporary reinforcement and addiction theories can tell us about their likely success. We argue that the most effective endgame approach is one that pursues both nicotine reduction and alternative nicotine delivery systems as complementary.

Applying the matching law as micro-foundation of social phenomena

Social phenomena are suggested to be explained by the matching law - an empirical regularity of individual behaviour. While a considerable amount of psychological research on this law exists, only a few sociological applications can be found. This paper points to the problems that come with its usage as micro-foundation of social behaviour and provides solutions. In particular, a model of melioration learning enables the derivation of social phenomena from the matching law. The proposed approach is illustrated by the application of the learning model to the volunteer's dilemma. In contrast to game-theoretical solutions, the matching law leads to more intuitive results in case of the asymmetric dilemma. The relationship between the matching law and utility maximisation is discussed by its integration into economic consumer theory.

Melioration Learning in Two-Person Games

Melioration learning is an empirically well-grounded model of reinforcement learning. By means of computer simulations, this paper derives predictions for several repeatedly played two-person games from this model. The results indicate a likely convergence to a pure Nash equilibrium of the game. If no pure equilibrium exists, the relative frequencies of choice may approach the predictions of the mixed Nash equilibrium. Yet in some games, no stable state is reached.

Application of Behavioral Activation Treatment for Depression to an Adolescent With a History of Child Maltreatment

Behavioral Activation Treatment for Depression (BATD) is a promising new intervention that has received preliminary empirical support with adult samples. However, researchers have not yet examined the application of BATD to children and adolescents. The authors describe implementation of BATD with a 17-year-old girl in foster care who endorsed several symptoms of depression and reported a low rate of potentially reinforcing life events. Over the course of treatment, patient adherence to the BATD protocol was associated with an increased rate of reinforcing life events that coincided with decreased symptoms of depression. The authors discuss special considerations (e.g., parental involvement, supplemental activity selection methods) that are unique to the use of BATD with children and adolescents and offer recommendations for researchers and clinicians.

Extra (non-contingent) food does not affect behavioral tolerance to d-amphetamine's rate-decreasing effects

Tolerance refers to the diminished effect of a drug following its repeated administration such that a larger dose is needed to obtain the initial effect. Tolerance to a drug's effects on operant behavior is more likely to develop when initial drug administration results in a loss of reinforcement. It remains unknown how offsetting loss of reinforcement influences the development of tolerance. Providing extra (non-contingent) food pellets was hypothesized to impede the development of tolerance to effects of a repeatedly administered dose of d-amphetamine that reduced the number of food pellets earned by rats. A multiple schedule with two variable-interval (VI) 60-s components resulting in food delivery was used to maintain lever pressing. Extra food pellets were provided in one of those components according to a variable-time (VT) 120-s or a VT 30-s schedule for separate groups of rats (n=6 for each group). Effects of d-amphetamine (0.1-3.0mg/kg) were tested before (acute) and during (chronic) injections (45days) of an individually selected, repeatedly administered dose that reduced the number of food pellets earned by at least 50% compared to when saline was tested acutely. There was a dose-dependent decrease in lever-press rates and food pellets earned. The development of tolerance was quantified by dividing the area-under-the-curve (AUC) of the chronic dose-effect function by the AUC of the acute dose-effect function. Tolerance developed to a similar extent in components with and without extra food pellets for rats in both groups. These results indicate that offsetting reinforcement loss, at least as studied here, did not differentially affect tolerance development.

From antisocial behavior to violence: a model for the amplifying role of coercive joining in adolescent friendships

BACKGROUND

Aggression is one of the more stable characteristics of child and adolescent development, and violent behavior in early adulthood is often foreshadowed by aggressive behavior in childhood and early adolescence. Considerable evidence has linked coercive family interactions to aggressive behavior in childhood, but less research has been conducted on the joint role of family and peer interaction in the escalation of aggression to violence in adulthood.

METHODS

We coded family interactions at age 12-13 and friendship interaction at age 16-17 in a multiethnic sample of youth and families. Violence in young adulthood (age 22-23) was measured using self-report, criminal records, and parent report. We tested the hypothesis that a process of 'coercive joining' in friendship interactions mediated the relationship between coercive family interactions and serious violence.

RESULTS

We found that observed coercive joining in friendships at age 16-17 predicted early-adulthood violent behavior over and above an established tendency toward antisocial behavior. We also found that observed coercive family interactions at age 12 predicted early-adulthood violence, and that coercive joining with friends fully mediated this link.

CONCLUSIONS

These results significantly extend coercion theory by suggesting that coercive joining in the context of peer groups is an additional mechanism by which coercive processes in the family are extended and amplified to violent behavior in early adulthood. Our findings suggest the importance of addressing both individual interpersonal skills and self-organizing peer groups when intervening to prevent violent behavior.

The substitutability of reinforcers

Substitutability is a construct borrowed from microeconomics that describes a continuum of possible interactions among the reinforcers in a given situation. Highly substitutable reinforcers, which occupy one end of the continuum, are readily traded for each other due to their functional similarity. Complementary reinforcers, at the other end of the continuum, tend to be consumed jointly in fairly rigid proportion, and therefore cannot be traded for one another except to achieve that proportion. At the center of the continuum are reinforcers that are independent with respect to each other; consumption of one has no influence on consumption of another. Psychological research and analyses in terms of substitutability employ standard operant conditioning paradigms in which humans and nonhumans choose between alternative reinforcers. The range of reinforcer interactions found in these studies is more readily accommodated and predicted when behavior-analytic models of choice consider issues of substitutability. New insights are gained into such areas as eating and drinking, electrical brain stimulation, temporal separation of choice alternatives, behavior therapy, drug use, and addictions. Moreover, the generalized matching law (Baum, 1974) gains greater explanatory power and comprehensiveness when measures of substitutability are included.

Envisioning cultural practices

Graphic visualization has demonstrated its value for organizing transactional data and modeling complex phenomena in a wide variety of fields, from theoretical physics to medicine. Behavior analysts have historically used a variety of graphic tools not only for presentation but also for analysis and teaching. As they turn increasingly to the analysis and design of cultural practices, the phenomena behavior analysts study are becoming increasingly complicated. Many cultural practices of interest are embedded in extensive webs of interlocking practices and contingencies that can be difficult to grasp comprehensively. Building on contingency diagrams, which have proven to be useful for the analysis of operant behavior, and graphic tools developed for object-oriented systems analysis, this paper suggests graphic tools for capturing the interlocking contingencies that constitute cultures. These diagrams offer a broad-bandwidth technology for analyzing and designing cultural practices.

Two modern developments in matching theory

Matching theory is a mathematical theory of choice behavior, parts of which have been shown to hold in natural human environments and to have important therapeutic applications. Two modern developments in matching theory are discussed in this article. The first is the mathematical description of behavior in asymmetrical choice situations, which are situations where different reinforcers and/or different behaviors are associated with concurrently available response alternatives. Most choice situations in natural human environments are probably asymmetrical. The second development in matching theory is the mathematical description of a tendency toward indifferent responding in all choice situations. Behavior in asymmetrical choice situations and the tendency toward indifferent responding in all choice situations can be described by modifications of the matching equations, which change the equations from lines into power functions. These modern forms have been extraordinarily successful in describing behavior in choice situations, and are the forms most likely to accurately describe human behavior in naturally occurring environments.

Windows on the 21st century

Behavior analysis is a cultural system of which the Association for Behavior Analysis is a component cultural system. As cultural systems, they are composed of interlocking behavioral contingencies that constitute their cultural practices. Critical to the survival of both cultural systems is the frequency of interaction with and the nature of the content of the behavioral contingencies composing those cultural practices. The strengths of behavior analysis as a cultural system include its disciplinary character and its worldwide community of scientists and practitioners; its ability to be integrated into a scientific worldview; its track record in providing effective solutions to problems of importance to society; and the high levels of intellect, competence, and commitment that are characteristic of its participants. Weaknesses of behavior analysis are its status as an academic orphan, its relatively small size and its underdeveloped professional identity, and a lack of sociopolitical sophistication among many of its members. Behavior analysis will need to maximize its strengths and mitigate its weaknesses if it is to take advantage of the many opportunities available for growth in the modern world.

The history of behavior analysis: Some historiography and a bibliography

This article has two main purposes. First, it introduces the discipline of historiography and, second, it provides a selected bibliography on the history of behavior analysis. In introducing the former in the context of the latter, four important methodological considerations involved in the process and product of historiography are described: The sources from which historical materials are drawn (i.e., primary, secondary, and tertiary) and three dimensions along which historiography is conducted and evaluated-internalist vs. externalist, great person vs. Zeitgeist, and presentist vs. historicist. Integrated throughout are four purposes for the historiography of behavior analysis, as well as an overview of the topics covered in the extant literature. The manuscript concludes with a listing of current bibliographic material by publication type and topic.

The matching law: a tutorial for practitioners

The application of the matching law has historically been limited to use as a quantitative measurement tool in the experimental analysis of behavior to describe temporally extended patterns of behavior-environment relations. In recent years, however, applications of the matching law have been translated to clinical settings and populations to gain a better understanding of how naturally-occurring events affect socially important behaviors. This tutorial provides a brief background of the conceptual foundations of matching, an overview of the various matching equations that have been used in research, and a description of how to interpret the data derived from these equations in the context of numerous examples of matching analyses conducted with socially important behavior. An appendix of resources is provided to direct readers to primary sources, as well as useful articles and books on the topic.

Functions of the environment in behavioral evolution

This paper explores some of the ways in which the environment functions with respect to behavior within an explanatory framework analogous to that of evolutionary biology. In both the behavioral and organic domains, the environment functions differently with respect to individual occurrences and evolutionary units. Within the behavioral domain, the problem of accounting for an occurrence of an operant instance differs from that of accounting for the existence of the operant unit of which the instance is a part. Maintaining these distinctions in levels of analysis within the behavioral domain, we focus first on operant units and operant instances as products of evolutionary processes occurring in the behavioral domain and second upon the causal role of the environment with respect to the existence of operant units and the occurrence of operant instances. The environment's function is selective with respect to origin, maintenance, suppression, and extinction of behavioral populations. At the level of operant instances, the environment has instantiating functions-evocative or alterative. Evocative functions are exemplified by discriminative relations, and alterative functions include both conditional and motivative relations. Implications are considered regarding extension of the analogy to more complex behavior-environment relations.

The speciation of behavior analysis

The relationship between the Experimental Analysis of Behavior (EAB) and Applied Behavior Analysis (ABA) has been the subject of several editorials and commentaries in recent years. Various authors have argued that researchers in these two fields (a) have become isolated from each other, (b) face different requirements for survival in their respective fields, and (c) possess different skills to meet those requirements. The present paper provides an allegory for the relationship between EAB and ABA in terms of biological speciation. The conditions that have changed the relationship between EAB and ABA are parallel to those responsible for biological speciation: (a) isolation of some members of a species from the rest of the population, (b) different contingencies of survival for members of the two separate groups, and (c) divergence in the adaptive characteristics displayed by the two groups. When members of two different groups, descendants of common ancestors, no longer are capable of producing viable offspring by interbreeding, the different groups then represent different species. To the extent that members of the EAB group and members of the ABA group interact with each other only trivially, they each represent allegorically different species. Changes in the relationship between EAB and ABA are part of a natural process that takes place in many other sciences, and the course of that process can hardly be reversed by us.

Technology at the zoo: the influence of a touchscreen computer on orangutans and zoo visitors

A computer-controlled touchscreen apparatus (hereafter referred to as "touchscreen") in the orangutan exhibit at Zoo Atlanta provides enrichment to the animals and allows cognitive research to take place on exhibit. This study investigated the impact of the touchscreen on orangutan behavior and visibility, as well as its impact on zoo visitors. Despite previous research suggesting that providing a single computer system may negatively affect orangutan behavior, there was not a significant increase in aggression, stereotypic, or distress-related behaviors following the activation of the on-exhibit touchscreen. We also investigated the possibility that zoo visitors may be negatively affected by technology because it deviates from naturalism. However, we did not find a change in stay time or overall experience rating when the computer was turned on. This research was the first to assess visitor attitudes toward technology at the zoo, and we found that visitors report highly positive attitudes about technology for both animals and visitors. If subjects visited the exhibit when the computer was turned on, they more strongly agreed that orangutans benefit from interacting with computerized enrichment. This study is the first investigation of an on-exhibit touchscreen in group-housed apes; our findings of no negative effects on the animals or zoo visitors and positive attitudes toward technology suggest a significant value of this practice.

Using Microsoft Office Excel 2007 to conduct generalized matching analyses

The generalized matching equation is a robust and empirically supported means of analyzing relations between reinforcement and behavior. Unfortunately, no simple task analysis is available to behavior analysts interested in using the matching equation to evaluate data in clinical or applied settings. This technical article presents a task analysis for the use of Microsoft Excel to analyze and plot the generalized matching equation. Using a data-based case example and a step-by-step guide for completing the analysis, these instructions are intended to promote the use of quantitative analyses by researchers with little to no experience in quantitative analyses or the matching law.

Effects of reinforcer rate and reinforcer quality on time allocation: Extensions of matching theory to educational settings

We examined how 3 special education students allocated their responding across two concurrently available tasks associated with unequal rates and equal versus unequal qualities of reinforcement. The students completed math problems from two alternative sets on concurrent variable-interval (VI) 30-s VI 120-s schedules of reinforcement. During the equal-quality reinforcer condition, high-quality (nickels) and low-quality items ("program money" in the school's token economy) were alternated across sessions as the reinforcer for both sets of problems. During the unequal-quality reinforcer condition, the low-quality reinforcer was used for the set of problems on the VI 30-s schedule, and the high-quality reinforcer was used for the set of problems on the VI 120-s schedule. Equal- and unequal-quality reinforcer conditions were alternated using a reversal design. Results showed that sensitivity to the features of the VI reinforcement schedules developed only after the reinforcement intervals were signaled through countdown timers. Thereafter, when reinforcer quality was equal, the time allocated to concurrent response alternatives was approximately proportional to obtained reinforcement, as predicted by the matching law. However the matching relation was disrupted when, as occurs in most natural choice situations, the quality of the reinforcers differed across the response options.

Limited matching on concurrent-schedule reinforcement of academic behavior

Three adolescent students with special educational needs were given a choice between completing one of two available sets of math problems. Reinforcers (nickels) across these alternatives were arranged systematically in separate experimental phases according to three different concurrent variable-interval schedules (reinforcement ratios of 2:1, 6:1, and 12:1). Time allocated to the two stacks of math problems stood in linear relationship to the reinforcement rate obtained from each stack, although substantial undermatching and bias were observed for all subjects. However, changes in the schedules were not followed by changes in allocation patterns until adjunct procedures (e.g., changeover delays, limited holds, timers, and demonstrations) were introduced. The necessity of adjunct procedures in establishing matching in applied situations is discussed as a limitation to quantitative applications of the matching law in applied behavior analysis.

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.

Basic research needed for stimulating the development of behavioral technologies

The costs of disconnection between the basic and applied sectors of behavior analysis are reviewed, and some solutions to these problems are proposed. Central to these solutions are collaborations between basic and applied behavioral scientists in programmatic research that addresses the behavioral basis and solution of human behavior problems. This kind of collaboration parallels the deliberate interactions between basic and applied researchers that have proven to be so profitable in other scientific fields, such as medicine. Basic research questions of particular relevance to the development of behavioral technologies are posed in the following areas: response allocation, resistance to change, countercontrol, formation and differentiation/discrimination of stimulus and response classes, analysis of low-rate behavior, and rule-governed behavior. Three interrelated strategies to build connections between the basic and applied analysis of behavior are identified: (a) the development of nonhuman animal models of human behavior problems using operations that parallel plausible human circumstances, (b) replication of the modeled relations with human subjects in the operant laboratory, and (c) tests of the generality of the model with actual human problems in natural settings.

The momentum of human behavior in a natural setting

Adults with mental retardation in a group home received popcorn or coffee reinforcers for sorting plastic dinnerware. In Part 1 of the experiment, reinforcers were dispensed according to a variable-interval 60-s schedule for sorting dinnerware of one color and according to a variable-interval 240-s schedule for sorting dinnerware of a different color in successive components of a multiple schedule. Sorting rates were similar in baseline, but when a video program was shown concurrently, sorting of dinnerware was more resistant to distraction when correlated with a higher rate of reinforcement. In Part 2 of the experiment, popcorn or coffee reinforcers were contingent upon sorting both colors of dinnerware according to variable-interval 60-s schedules, but additional reinforcers were given independently of sorting according to a variable-time 30-s schedule during one dinnerware-color component. Baseline sorting rate was lower but resistance to distraction by the video program was greater in the component with additional variable-time reinforcers. These results demonstrate that resistance to distraction depends on the rate of reinforcers obtained in the presence of component stimuli but is independent of baseline response rates and response-reinforcer contingencies. Moreover, these results are similar to those obtained in laboratory studies with pigeons, demonstrating that the determination of resistance to change by stimulus-reinforcer relations is not confined to controlled laboratory settings or unique to the pigeon.

Recent reinforcement-schedule research and applied behavior analysis

Reinforcement schedules are considered in relation to applied behavior analysis by examining several recent laboratory experiments with humans and other animals. The experiments are drawn from three areas of contemporary schedule research: behavioral history effects on schedule performance, the role of instructions in schedule performance of humans, and dynamic schedules of reinforcement. All of the experiments are discussed in relation to the role of behavioral history in current schedule performance. The paper concludes by extracting from the experiments some more general issues concerning reinforcement schedules in applied research and practice.

Application of Herrnstein's hyperbola to time allocation of naturalistic human behavior maintained by naturalistic social reinforcement

Five college students talked to an experimenter about various topics. Time spent looking at the experimenter was reinforced by verbal statements of praise and interest on five variable-interval schedules. Herrnstein's hyperbola provided a good description of the time-allocation data for 4 of the 5 subjects, and accounted for 95% of the variance of the median time-allocation data. The hyperbola provided a significantly better description of the data than a two-parameter ramp function with similar differential properties. Estimates of the asymptote, k, of the hyperbola varied among subjects from about 2 to about 15 seconds of eye contact per minute. These estimates were much smaller than the constant 60 seconds of eye contact per minute required by Herrnstein's matching theory. These results support the conclusion that Herrnstein's hyperbola describes naturalistic human behavior maintained by naturalistic social reinforcement as well as it describes the behavior of humans and nonhumans in typical laboratory preparations. The results also indicate that the hyperbolic form of the time-allocation version of Herrnstein's equation is accurate, but that the constant k requirement of matching theory may not hold.