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.     

Discounting: A practical guide to multilevel analysis of choice data

Multilevel modeling provides the ability to simultaneously evaluate the discounting of individuals and groups by examining choices between smaller sooner and larger later rewards. A multilevel logistic regression approach is advocated in which sensitivity to relative reward magnitude and relative delay are considered as separate contributors to choice. Examples of how to fit choice data using multilevel logistic models are provided to help researchers in the adoption of these methods.     

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.     

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.     

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.     

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.     

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.     

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.     

Incorporating Mobility in Growth Modeling for Multilevel and Longitudinal Item Response Data

Multilevel data often cannot be represented by the strict form of hierarchy typically assumed in multilevel modeling. A common example is the case in which subjects change their group membership in longitudinal studies (e.g., students transfer schools; employees transition between different departments). In this study, cross-classified and multiple membership models for multilevel and longitudinal item response data (CCMM-MLIRD) are developed to incorporate such mobility, focusing on students' school change in large-scale longitudinal studies. Furthermore, we investigate the effect of incorrectly modeling school membership in the analysis of multilevel and longitudinal item response data. Two types of school mobility are described, and corresponding models are specified. Results of the simulation studies suggested that appropriate modeling of the two types of school mobility using the CCMM-MLIRD yielded good recovery of the parameters and improvement over models that did not incorporate mobility properly. In addition, the consequences of incorrectly modeling the school effects on the variance estimates of the random effects and the standard errors of the fixed effects depended upon mobility patterns and model specifications. Two sets of large-scale longitudinal data are analyzed to illustrate applications of the CCMM-MLIRD for each type of school mobility.     

Molar functional relations and clinical behavior analysis: implications for assessment and treatment

The experimental analysis of behavior has identified several molar functional relations that are highly relevant to clinical behavior analysis. These include matching, discounting, momentum, and variability. Matching provides a broader analysis of how multiple sources of reinforcement influence how individuals choose to allocate their time and offers an empirical rationale for reducing problem behavior by increasing adaptive behavior. Discounting highlights the functional relations that affect self-control. Momentum specifies the variables responsible for persistence in challenging situations. Variability characterizes a functional dimension of behavior that is essential for learning and problem solving. These concepts have important implications for clinical practice and research. A selective review of these concepts is presented, and their implications for assessment and treatment are discussed with two goals: to inform basic scientists about the relevance of their work and to invite clinical behavior analysts to broaden the conceptual basis for their work.     

Optimization and the matching law as accounts of instrumental behavior

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

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.     

A Multilevel Contextual Model of Family Conflict and Deviance

This paper presents a multiple-group multivariate hierarchical specification of family problem behaviors across ethnicities using structural equation modeling techniques which explicitly model the individual-level and family-level covariance matrices in familial problem behavior. Analyses were conducted across White and African American ethnic groups. The sample (N = 1,168; 647 White and 521 African Americans) comprised children and their parents from 392 families. In addition to relations between family conflict and deviant behaviors, covariates were included at each level of analysis: neighborhood desirability at the family (between) level, and age and gender at the individual (within) level. At the between level, neighborhood desirability influenced family conflict and family conflict influenced family levels of deviance. At the within level, conflict was significantly related to individual levels of deviance. Discussion focuses on the substantive results as well as the application of multilevel analyses to contextual influences of family problem behavior.     

On the falsifiability of matching theory.

Herrnstein's matching theory requires the parameter, k, which appears in the single-alternative form of the matching equation, to remain invariant with respect to changes in reinforcement parameters like magnitude or immediacy. Recent experiments have disconfirmed matching theory by showing that the invariant-k requirement does not hold. However, the theory can be asserted in a purely algebraic form that does not require an invariant k and that is not disconfirmed by the recent findings. In addition, both the original and the purely algebraic versions of matching theory can be asserted in forms that allow for commonly observed deviations from matching (bias, undermatching, and overmatching). The recent finding of a variable k does not disconfirm these versions of matching theory either. As a consequence, matching remains a viable theory of behavior, the strength of which lies in its general conceptualization of all behavior as choice, and in its unified mathematical treatment of single- and multialternative environments.