Generalized Matching Equation Research Articles

An evaluation of methods for studying the effects of conditioned reinforcement on human choice.

Shahan et al. (2006) found that the relative rate of pigeons' pecking on two observing responses (i.e., responses that only produced an S+ or stimulus correlated with primary reinforcement) was well described by the relative rate of S+ delivery. Researchers have not evaluated the effects of S+ delivery rate in a concurrent observing response procedure with human subjects, so the necessary procedural modifications for studying the effects of conditioned reinforcement on human choice remain unclear. The purpose of the current study was to conduct an additive component analysis of modifications to the procedures of Shahan et al. (2006). We evaluated the additive effects of introducing response cost, a changeover response, and ordinal discriminative stimuli on correspondence with the results of Shahan et al. and the quality of fits of the generalized matching equation. When our procedures were most similar to those of Shahan et al., we observed low rates of observing and indifference between the two observing responses. For the group of subjects with whom all three additive components were included, we obtained the highest level of sensitivity to relative rate of S+ delivery, but the slope and R2 of our fits of the generalized matching equation were still much lower than those obtained by Shahan et al. Potential reasons for these discrepancies, methods of resolving them, and implications for future research are discussed.

Attacking and Scoring by College Volleyball Players: A Matching Analysis

We applied the generalized matching equation to examine the relation between attacks attempted and points scored by the top five attackers on 36 Division 1 National Collegiate Athletic Association women’s volleyball teams during the 2017–2018 season. Twelve of the teams were highest-ranked at season’s end, 12 were mid-ranked, and 12 were lowest-ranked. The equation accounted for at least 70% of the variance in responding for 31 teams. Of them, 16 teams demonstrated undermatching, 4 demonstrated overmatching, and 11 approached strict matching. Eight of these teams were biased towards attempts by a single player and four towards attempts by the other four players combined. Team ranking did not consistently affect performance. These findings are significant in extending the provenance of matching to another sport and to a naturalistic setting involving three or more response options.

Human Choice Predicted by Obtained Reinforcers, Not by Reinforcement Predictors.

Macphail (1985) proposed that “intelligence” should not vary across vertebrate species when contextual variables are accounted for. Focusing on research involving choice behavior, the propensity for choosing an option that produces stimuli that predict the presence or absence of reinforcement but that also results in less food over time can be examined. This choice preference has been found multiple times in pigeons (Stagner and Zentall, 2010; Zentall and Stagner, 2011; Laude et al., 2014) and has been likened to gambling behavior demonstrated by humans (Zentall, 2014, 2016). The present experiments used a similarly structured task to examine adult human preferences for reinforcement predictors and compared findings to choice behavior demonstrated by children (Lalli et al., 2000), monkeys (Smith et al., 2017; Smith and Beran, 2020), dogs (Jackson et al., 2020), rats (Chow et al., 2017; Cunningham and Shahan, 2019; Jackson et al., 2020), and pigeons (Roper and Zentall, 1999; Stagner and Zentall, 2010). In Experiment 1, adult human participants showed no preference for reinforcement predictors. Results from Experiment 2 suggest that not only were reinforcement predictors not preferred, but that perhaps reinforcement predictors had no effect at all on choice behavior. Results from Experiments 1 and 2 were further assessed using a generalized matching equation, the findings from which support that adult human choice behavior in the present research was largely determined by reinforcement history. Overall, the present results obtained from human adult participants are different than those found from pigeons in particular, suggesting that further examination of Macphail (1985) hypothesis is warranted.

Application of the matching law to Mixed Martial Arts

In the contemporary behavior-analytic literature, athletic performance (e.g.,choice of shot or movement) across multiple team sports has been found to correspond with predictions of the generalized matching equation. However, the research in this area has focused primarily on team sports. In the current study the Generalized Matching Equation (GME) was applied to Mixed Martial Arts (MMA) performance by examining strike selection as a function of landing significant strikes among fighters from various weight classes in the Ultimate Fighting Championship. The results suggest that the GME is a good descriptor of strike selection in MMA, an individual sport that is dynamic and fast paced where responding results in immediate feedback from an opponent.

Experimental evaluation of matching via a commercially available basketball video game.

Many recent nonlaboratory-based quantitative analyses of behavior have relied on archival competitive sporting data. However, the ratio-based reinforcement schedules in most athletic competitions and the correlational nature of archival data analyses raise concern over the contributions of such findings to the behavior analytic literature. The current experiment evaluated whether matching in a human operant paradigm would approximate matching observed in nonlaboratory-based sports data. To this end, we used in-game attributes to parametrically manipulate 2- and 3-point shooting in a commercially available basketball video game. The behavior of 6 of 9 participants conformed to the generalized matching equation. These results suggest matching in sporting contexts may be a product of restricted nonindependent concurrent random-ratio schedules. Implications of this experiment, including those in applied behavior analysis and potential influence on gamification, are discussed.

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.

Allocation of speech in conversation.

In a replication and extension of Conger and Killeen's (1974) widely cited demonstration of matching in conversations, we evaluated nine participants' allocation of speech and gaze to two conversational partners. German speakers participated in two 90-min sessions in which confederates uttered approval on independent variable-interval schedules. In one of the sessions, confederates uttered approval contingent upon and contiguous with eye contact whereas in the other session approval was uttered independent of the participant's gaze. Several measures of participants' verbal behavior were taken, including relative duration and rate of speech and gaze. These were compared to confederates' relative rate of approval and relative duration and rate of talk. The generalized matching equation was fitted to the various relations between participants' behavior and confederates' behavior. Conger and Killeen's results were not replicated; participants' response allocation did not show a systematic relation to the confederates' relative rate of approval. The strongest relations were to overall talk, rather than approval. In both conditions, the participant talked more to the confederate who talked less-inverse or antimatching. Participants' gaze showed the same inverse relation to the confederates' talk. Requiring gaze to be directed toward a confederate for delivery of approval made no difference in the results. The absence of a difference combined with prior research suggests that matching or antimatching in conversations is more likely due to induction than to reinforcement.

Choice in transition: Replication and extension to preschool children in a naturalistic setting.

This study replicated previous basic research into the dynamics of choice and extended this analysis to children's behavior in a naturalistic setting. Two preschoolers with disabilities were observed interacting with their teachers at baseline and during an experimental analysis involving four pairs of concurrent variable-interval schedules of adult attention implemented by an experimenter. Each child was exposed to four experimental phases in which the relative reinforcer rates for on- and off-task behavior were 10:1, 1:1, 1:10, and reversed back to 10:1. The 10:1 phase was designed to mimic the same schedules and types of adult attention observed at baseline. We used the generalized matching equation to model steady-state behavior at the end of the transition phases and to evaluate changes in sensitivity at various points throughout the phases. Choice in transition was evaluated by plotting log behavior ratios by session, cumulated time on- and off-task and cumulated attention for on- and off-task behavior by session, and interreinforcer behavior ratios following different sequences of the first four reinforcer deliveries. The generalized matching equation accounted for a large proportion of variance in steady-state responding, sensitivity values increased steadily throughout the phases, patterns of choice in transition were similar to those reported in basic research, and interreinforcer preference generally shifted toward the just-reinforced alternative. These findings are consistent with previous basic research and support the generality of the dynamics of choice to children's on- and off-task behavior reinforced by adult attention.

Human choice under schedules of negative reinforcement

The generalized matching equation provides a good description of response allocation in concurrent schedules of positive reinforcement in nonhumans as well as in humans. The present experiment was conducted to further investigate the allocation of responding under concurrent schedules of negative reinforcement (i.e., timeouts from pressing a force cell) in humans. Each of three participants was exposed to different reinforcement ratios (9:1, 1:1 and 1:9) in the terminal links of a concurrent-chains schedule of negative reinforcement. The allocation of responding under this schedule was well described by the generalized matching equation, for each participant. These results replicate previous findings obtained with nonhumans and humans under concurrent schedules of positive reinforcement. In addition, they extend the results reported by Alessandri and Rivière (2013) showing that human behavior maintained by timeouts from an effortful response is sensitive to changes in relative reinforcement ratios as well as relative delays of reinforcement.

An alternative to the stay/switch equation assessed when using a changeover-delay

An alternative to the generalized matching equation for understanding concurrent performances is the stay/switch model. For the stay/switch model, the important events are the contingencies and behaviors at each alternative. The current experiment compares the descriptions by two stay/switch equations, the original, empirically derived stay/switch equation and a more theoretically derived equation based on ratios of stay to switch responses matching ratios of stay to switch reinforcers. The present experiment compared descriptions by the original stay/switch equation when using and not using a changeover delay. It also compared descriptions by the more theoretical equation with and without a changeover delay. Finally, it compared descriptions of the concurrent performances by these two equations. Rats were trained in 15 conditions on identical concurrent random-interval schedules in each component of a multiple schedule. A COD operated in only one component. There were no consistent differences in the variance accounted for by each equation of concurrent performances whether or not a COD was used. The simpler equation found greater sensitivity to stay than to switch reinforcers. It also found a COD eliminated the influence of switch reinforcers. Because estimates of parameters were more meaningful when using the more theoretical stay/switch equation it is preferred.

Matching Law and Sensitivity to Therapist’s Attention in Children with Autism Spectrum Disorders

A social skills deficit is one of the core symptoms in children with autism spectrum disorders (ASDs) and has been hypothesized to be related to a deficit in social sensitivity. This study set out to investigate the social sensitivity of preschoolers with ASDs in an early behavioral intervention setting, using the generalized matching equation as an analytical tool. We evaluated how 14 children with ASDs, ages 3 to 5, allocated their appropriate social behaviors compared to their inappropriate and nonsocial behaviors, based on the attention of their therapists during early behavioral intervention. The objectives were to (a) measure the social sensitivity of participants using the generalized matching equation and describe the stability of this measure after 8 months of intervention and (b) evaluate the relationship between the parameters of the GME and the participants’ level of functioning as measured by their intellectual functioning and severity of autistic symptoms. Seven of the participants significantly varied their appropriate social behaviors based on changes in the social attention of their therapists at the onset of intervention, and 3 more participants showed this pattern after 8 months of intervention. Changes in behavioral bias and social sensitivity after 8 months of observation were significantly correlated with some measures of functioning in the participants. The usefulness of the matching law as a tool for measuring social sensitivity in this population, and its clinical applications, is discussed.

Obese and lean Zucker rats demonstrate differential sensitivity to rates of food reinforcement in a choice procedure

The obese Zucker rat carries two recessive fa alleles that result in the expression of an obese phenotype. Obese Zuckers have higher food intake than lean controls in free-feed studies in which rats have ready access to a large amount of one type of food. The present study examined differences in obese and lean Zucker rats using concurrent schedules of reinforcement, which more ecologically models food selection using two food choices that have limited, but generally predictable availability. Lever-pressing of ten lean (Fa/Fa or Fa/fa) and ten obese (fa/fa) Zucker rats was placed under three concurrent variable interval variable interval (conc VI VI) schedules of sucrose and carrot reinforcement, in which the programmed reinforcer ratios for 45-mg food pellets were 5:1, 1:1, and 1:5. Allocation of responses to the two food alternatives was characterized using the generalized matching equation, which allows sensitivity to reinforcer rates (a) and bias toward one alternative (log k) to be quantified. All rats showed a bias toward sucrose, though there were no differences between lean and obese Zucker rats. In addition, obese Zucker rats exhibited higher sensitivity to reinforcement rates than lean rats. This efficient pattern of responding was related to overall higher deliveries of food pellets. Effective matching for food, then, may be another behavioral pattern that contributes to an obese phenotype.

An agent simulation study of a unified mechanism on individual foraging behaviour with individual scalar expectancies

In this study, we present a selective overview of group and individual foraging behaviour and propose an agent-based model to unify these methodologies, enabling the exploration of plausible causal mechanisms and novel hypothesis generation. Despite its popular place in behavioural science, little work has been done with the generalized matching equation as it pertains to simulation work. Our model uses psychologically justifiable and theoretically grounded models of memory for time and decision rules. The results of preliminary simulation experiments suggest that the model is sensitive to changes in pay-off density and correctly replicates Ideal Free Distributions.

SWITCH HITTING IN BASEBALL: APPARENT RULE-FOLLOWING, NOT MATCHING

Many studies, including some dealing with shot selection in basketball and play selection in football, demonstrate that the generalized matching equation provides a good description of the allocation of time and effort to alternative responses as a function of the consequences of those alternatives. We examined whether it did so with respect to left- and right-handed at bats (alternative responses) and left- and right-handed total bases earned, runs batted in, and home runs (three consequences) for the outstanding baseball switch-hitters Mickey Mantle, Eddie Murray, and Pete Rose. With all hitters, undermatching, suggesting insensitivity to the consequences of behavior (reinforcement), was evident and there was substantial bias towards left-handed at bats. These players apparently chose handedness based on the rule "bat opposite the pitcher," not on differential consequences obtained in major league games. The present findings are significant in representing a counter-instance of demonstrations of a matching relationship in sports in particular and in human behavior in general and in calling attention to the need for further study of the variables that affect choice.

The Matching Law

This article introduces the quantitative analysis of choice behavior by describing a number of equations developed over the years to describe the relation between the allocation of behavior under concurrent schedules of reinforcement and the consequences received for alternative responses. Direct proportionality between rate of responding and rate of reinforcement was observed in early studies, suggesting that behavioral output matched environmental input in a mathematical sense. This relation is termed “strict matching,” and the equation that describes it is referred to as “the matching law.” Later data showed systematic departures from strict matching, and a generalized version of the matching equation is now used to describe such data. This equation, referred to as “the generalized matching equation,” also describes data that follow strict matching. It has become convention to refer to either of these equations as “the matching law.” Empirical support for the matching law is briefly summarized, as is the applied and practical significance of matching analyses.

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.

The matching relation and situation-specific bias modulation in professional football play selection.

The utility of a quantitative model depends on the extent to which its fitted parameters vary systematically with environmental events of interest. Professional football statistics were analyzed to determine whether play selection (passing versus rushing plays) could be accounted for with the generalized matching equation, and in particular whether variations in play selection across game situations would manifest as changes in the equation's fitted parameters. Statistically significant changes in bias were found for each of five types of game situations; no systematic changes in sensitivity were observed. Further analyses suggested relationships between play selection bias and both turnover probability (which can be described in terms of punishment) and yards-gained variance (which can be described in terms of variable-magnitude reinforcement schedules). The present investigation provides a useful demonstration of association between face-valid, situation-specific effects in a domain of everyday interest, and a theoretically important term of a quantitative model of behavior. Such associations, we argue, are an essential focus in translational extensions of quantitative models.

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.

QUANTIFICATION OF ETHANOL'S ANTIPUNISHMENT EFFECT IN HUMANS USING THE GENERALIZED MATCHING EQUATION

Increases in rates of punished behavior by the administration of anxiolytic drugs (called antipunishment effects) are well established in animals but not humans. The present study examined antipunishment effects of ethanol in humans using a choice procedure. The behavior of 5 participants was placed under six concurrent variable-interval schedules of monetary reinforcement. In three of the six concurrent schedules, punishment, in the form of monetary loss, was superimposed on one alternative. Data were analyzed according to the generalized matching equation which distinguishes between bias (allocation of behavior beyond what matching to relative reinforcer densities would predict) and sensitivity to reinforcement (how well behavior tracks relative reinforcer densities). In addition, participants completed a pencil-tapping test. Under placebo punishment conditions, all participants demonstrated low response rates and a bias against the alternative associated with punishment, despite a resultant loss of available reinforcers. Bias against the punished alternative was dose-dependently reduced in participants shown to be most sensitive to ethanol (0.6, 1.2, and 1.8 g/kg) in measures of overall responding and on the pencil-tapping test. No ethanol-induced change in bias was noted when punishment was not imposed. Sensitivity to reinforcement also decreased for participants shown to be sensitive to ethanol. In addition to extending antipunishment effects to humans, these results also show that antipunishment effects can be quantified via the matching equation.

Coyotes (Canis latrans) and the matching law

Environmental change is accelerating due to anthropogenic influence. Species that have greater behavioral flexibility may be better adapted to exploit new or constantly changing habitats. There are few mammals and even fewer carnivores that better illustrate widespread adaptability and behavioral flexibility in the wake of human disturbance than coyotes (Canis latrans). Yet how such predators successfully track resources, enabling them to survive and extend their range in stochastic environments remains unknown. We tested eight wild-born, captive coyotes individually on an operant two-choice test using concurrent variable interval (VI) schedules. We held the overall rate of reinforcement constant but manipulated the ratio of reinforcement available from the two choices. We analyzed sensitivity of coyotes' tracking of resource change by fitting the generalized matching equation to the data. Results showed all coyotes efficiently tracked changes in reinforcement ratios within the first few sessions of each new condition and matched their relative rate of foraging time to relative rate of resources. We suggest the matching paradigm provides a methodology to explore coyote foraging strategies, and a potential framework to compare behavioral flexibility across species, by measuring the ability to track resource change under variable resource conditions.