Table 2 - uploaded by Derek D Reed
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Fitted Parameters of Equation 1 for Two National Football League Teams That Experienced a Change of Head Coach During the 2004 Regular Season
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A mathematical model of operant choice, the generalized matching law was used to analyze play-calling data from the 2004 National Football League season. In all analyses, the relative ratio of passing to rushing plays was examined as a function of the relative ratio of reinforcement, defined as yards gained, from passing versus rushing. Different a...
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Citations
... The matching law is one quantitative model of behavior that has been demonstrated to provide a good description of many applied phenomena such as problem behavior (Borrero et al., 2010;Borrero & Vollmer, 2002;Kronfli et al., 2021;McDowell, 1982McDowell, , 1988 and response allocation in sports (Alferink et al., 2009;Cero & Falligant, 2019;Cox et al., 2021;Cox et al., 2017;Reed et al., 2006;Romanowich et al., 2007;Rotta et al., 2020;Vollmer & Bourret, 2000). The matching law has also been applied to social behavior such as conversation (Borrero et al., 2007;Conger & Killeen, 1974;McDowell & Caron, 2010;Pierce et al., 1981). ...
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.
... Behavior analysts have previously used basketball data not only because of the social relevance of sports performance, but to describe relations consistent with known principles. For example, one of the most-often-applied analytic methods to athlete behavior (e.g., Cox et al., 2017;Reed et al., 2006), the matching law, has been used by behavior analysts to describe variables influencing the rate of shots made by a given player in a basketball game. Perhaps other principles of behavior could be evaluated, given the wealth of existing data from previously played basketball games. ...
... However, a systematic manipulation of the putative controlling variables of timeout calling would be impractical (although use of virtual reality or video games could obviate this; e.g., Schenk & Reed, 2020). We acknowledge this limitation, but it is an inevitable component of analyses of in-game sports behavior that has previously been addressed (e.g., Reed et al., 2006;Schenk & Reed, 2020). ...
... We acknowledge again that the timeout might serve multiple functions: opportunities for a coach to change strategy and substitute players and a subsequent increase in points scored might function as positive reinforcement for timeout calling in some cases. Reed et al. (2006) asserted that a simplified analysis of one putative controlling variable of sports behavior might disregard many of the factors typically considered to be important to sports success. Behavioral researchers have identified order in the simplest of socially important relations repeatedly and should continue to do so. ...
Using data streams from ESPN.com (n.d.), we identified possible contingencies on timeout calling for 594 timeouts in 117 games for four Division 1 NCAA college basketball teams. We found that the probability of calling a timeout increased as the relative points scored by the opposing team increased. In addition to analyzing all timeouts inclusively, we analyzed timeouts called within the last 2 min of each half separately from those called in the rest of the game. Regardless of the window of analysis, the probability of calling a timeout increased as the relative points scored by the opposing team increased. To the extent that shots made and points scored by the opposing team were aversive to the coach (and players), the reductions that immediately followed the timeout may have functioned as negative reinforcement.
... Extensive empirical support exists for the GML from laboratory and nonlaboratory research (see McDowell, 2013). For example, the GML accounts well for choice across an array of operants such as lever pressing in rodents (e.g., Boomhower & Newland, 2016), key pressing or mouse clicking in undergraduates (e.g., Klapes et al., 2020;Madden & Perone, 1999), play selection and performance in a number of amateur (e.g., Alferink et al., 2009;Romanowich et al., 2007;Rotta et al., 2020) and professional (e.g., Cox et al., 2017;Reed et al., 2006) sports contexts, gambit selection among expert chess players (Cero & Falligant, 2019), conversation allocation among young adults (Borrero et al., 2007), and much more. ...
... Past researchers have examined how well the GML describes pitch allocation in a small sample of professional baseball pitchers (Cox et al., 2017;Falligant et al., 2020). Similar to studies extending the GML to other nonlaboratory contexts (e.g., Alferink et al., 2009;Reed et al., 2006;Romanowich et al., 2007;Rotta et al., 2020), these studies found that: the GML described pitch allocation well (i.e., VACs often greater than 0.80); estimated parameters were similar to laboratory studies on the GML; and estimated parameters changed in ways logically consistent with game contexts. But nonlaboratory settings are much more complex than laboratory settings. ...
... The reinforcers maintaining one organism's behavior may not be the reinforcers maintaining another organism's behavior. But this is often not the communicated finding in past research fitting the GML to behavior in nonlaboratory contexts (e.g., Alferink et al., 2009;Cox et al., 2017;Falligant et al., 2016;Reed et al., 2006;Romanowich et al., 2007;Rotta et al., 2020). The findings of this study raise the question of whether the participants in previous studies applying the GML to nonlaboratory contexts were also uniquely special in the extent the GML described their choices. ...
The generalized matching law (GML) has been used to describe the behavior of individual organisms in operant chambers, artificial environments, and nonlaboratory human settings. Most of these analyses have used a handful of participants to determine how well the GML describes choice in the experimental arrangement or how some experimental manipulation influences estimated matching parameters. Though the GML accounts very well for choice in a variety of contexts, the generality of the GML to all individuals in a population is unknown. That is, no known studies have used the GML to describe the individual behavior of all individuals in a population. This is likely because the data from every individual in the population has not historically been available or because time and computational constraints made population-level analyses prohibitive. In this study, we use open data on baseball pitches to provide an example of how big data methods can be combined with the GML to: (1) scale within-subjects designs to the population level; (2) track individual members of a population over time; (3) easily segment the population into subgroups for further analyses within and between groups; and (4) compare GML fits and estimated parameters to performance. These were accomplished for each of 2,374 individuals in a population using 8,467,473 observations of behavior-environment relationships spanning 11 years. In total, this study is a proof of concept for how behavior analysts can use data-science techniques to extend individual-level quantitative analyses of behavior to the population-level focused on domains of social relevance.
... Approximately 40 years of research from 1961 to 2000 showed that the equations provided excellent descriptions of the behavior of many vertebrate animal species (McDowell, 2013a). An extensive body of research dating through at least 2010 has shown that human behavior is also well-described by these equations (Beardsley & McDowell, 1992;Bradshaw et al., 1976Bradshaw et al., , 1977Bradshaw et al., 1979;Buskist & Miller, 1981;Conger & Killeen, 1974;Dallery et al., 2005;Kollins, Newland, & Critchfield, 1997a, 1997bMcDowell & Wood, 1984, 1985Moffat & Koch, 1973;Ruddle et al., 1981;Ruddle et al., 1982;Takahashi & Iwamoto, 1986), including naturally occurring human behavior (McDowell, 1981(McDowell, , 1988McDowell & Caron, 2010a;Pierce et al., 1981;Plaud, 1992;Reed et al., 2006;Stilling & Critchfield, 2010;Symons et al., 2003;Vollmer & Bourret, 2000). ...
This article provides an overview of highlights from 60 years of basic research on choice that are relevant to the assessment and treatment of clinical problems. The quantitative relations developed in this research provide useful information about a variety of clinical problems including aggressive, antisocial, and delinquent behavior, attention-deficit/hyperactivity disorder (ADHD), bipolar disorder, chronic pain syndrome, intellectual disabilities, pedophilia, and self-injurious behavior. A recent development in this field is an evolutionary theory of behavior dynamics that is used to animate artificial organisms (AOs). The behavior of AOs animated by the theory has been shown to conform to the quantitative relations that have been developed in the choice literature over the years, which means that the theory generates these relations as emergent outcomes, and therefore provides a theoretical basis for them. The theory has also been used to create AOs that exhibit specific psychopathological behavior, the assessment and treatment of which has been studied virtually. This modeling of psychopathological behavior has contributed to our understanding of the nature and treatment of the problems in humans.
... It is important to note that although the matching law equations described above were initially devised to characterize how reward modulates response allocation in pigeons, they have also proven to be highly effective in accounting for responding under symmetrical and asymmetrical choice conditions in subsequent controlled laboratory investigations of rodents, monkeys, and humans (e.g., Belke & Belliveau, 2001;Corrado et al., 2005;Ecott & Critchfield, 2004;Elsmore & McBride, 1994;Kangas et al., 2009;Lau & Glimcher, 2005; reviewed in Davison & McCarthy, 2017). Furthermore, although procedural variables need to be carefully considered when extending the matching framework to human subjects (e.g., Kollins et al., 1997;Pierce & Epling, 1983;Simon & Baum, 2017;Takahashi & Iwamoto, 1986), these equations have been shown to account for how reward can modulate choice and decision making outside of the laboratory, including complex phenomena such as social interactions in adults (e.g., Borrero et al., 2007;Conger & Killeen, 1974;Pierce et al., 1981), alcohol consumption in adults (Oscar-Berman et al., 1980), problem behavior in children (e.g., Borrero et al., 2010), completion of academic tasks in educational settings (e.g., Billington & DiTommaso, 2003;Mace et al., 1996;Neef et al., 1992), and elite athletic performance in professional and collegiate sports (e.g., Cox et al., 2017;Reed et al., 2006;Seniuk et al., 2020;Vollmer & Bourret, 2000). ...
Anhedonia, the loss of pleasure from previously rewarding activities, is a core symptom of several neuropsychiatric conditions, including major depressive disorder (MDD). Despite its transdiagnostic relevance, no effective therapeutics exist to treat anhedonia. This is due, in part, to inconsistent assays across clinical populations and laboratory animals, which hamper treatment development. To bridge this gap, recent work has capitalized on two long-standing research domains dedicated to quantifying responsivity to antecedents and consequences across species: the generalized matching law and signal detection theory. This review traces the integration of these quantitative frameworks, which yielded two empirically derived metrics: response bias (log b) and task discriminability (log d). These metrics serve as primary dependent variables in the Probabilistic Reward Task (PRT). In this computerized task, subjects make visual discriminations and probabilistic contingencies are arranged such that correct responses to one alternative are rewarded more often (rich) than correct responses to the other (lean). Under these conditions, healthy participants consistently develop a response bias in favor of the rich alternative, whereas participants with MDD exhibit blunted biases, which correlate with current and predict future anhedonia. Given the correspondence between anhedonic phenotypes and response bias, the PRT has been reverse-translated for rodents and nonhuman primates. Orderly log b and log d values have been observed across diverse clinical populations and laboratory animals. In addition, pharmacological challenges have produced similar outcomes across species. Taken together, this quantitative framework offers a highly translational approach to assaying reward responsiveness to accelerate treatment development for neuropsychiatric disorders involving anhedonia.
... Fisher & Mazur, 1997). The generalized matching law has been widely adopted in studies of choice, including investigations of non-human subjects in closed settings, as in the laboratory (e.g., Baum, 1979), non-human subjects in open settings, in their natural environment (e.g., Baum, 1974b;Graft, Lea, & Whitworth, 1977), human participants in closed settings, in the laboratory and institutions (e.g., Beardsley & McDowell, 1992;Bradshaw & Szabadi, 1988;Martens & Houk, 1989), and human participants in open settings, in sports events (e.g., Reed, Critchfield, & Martens, 2006;Vollmer & Bourret, 2000). ...
Evaluation of firm performance must consider the effects that its products and services have upon consumers. This can be accomplished when measures of consumer behavior inform marketing strategies. Consumer behavior analysis, a field of research that integrates operant behavioral economics and marketing, has developed several measures of consumer buying patterns based on the identification of the types of reinforcement, informational or utilitarian, that are programmed by different products and brands, and of the scope of consumer behavior setting. The present paper describes research that adopted some of these measures and the main results derived from them. Such studies have shown, for instance, that consumers have brand repertoires that include brands offering similar levels of reinforcement, that they tend to change the quantity they buy as a function of package size, price promotions, and utilitarian and informational reinforcement, that consumer individual differences tend to remain relatively stable across time, and that more open settings increase product search duration, decrease the essential value of brands and increase consumers’ reports related to dominance of shopping environments and approach responses. Moreover, these measures of consumer behavior can be integrated with measures of firm behavior to evaluate firm performance, on the basis of an operant interpretation of firm behavior. This paper explains some of these integrated measures and describes results that have shown, for instance, how increases in spending in marketing activities is related to increases in profitability.
... Fisher & Mazur, 1997). The generalized matching law has been widely adopted in studies of choice, including investigations of non-human subjects in closed settings, as in the laboratory (e.g., Baum, 1979), non-human subjects in open settings, in their natural environment (e.g., Baum, 1974b;Graft et al., 1977), human participants in closed settings, in the laboratory and institutions (e.g., Bradshaw & Szabadi, 1988;Martens & Houk, 1989;Beardsley & McDowell, 1992), and human participants in open settings, in sports events (e.g., Vollmer & Bourret, 2000;Reed et al., 2006). ...
Evaluation of firm performance must consider the effects that its products and
services have upon consumers. This can be accomplished when measures of consumer
behavior inform marketing strategies. Consumer behavior analysis, a field of research
that integrates operant behavioral economics and marketing, has developed several
measures of consumer buying patterns based on the identification of the types of
reinforcement, informational or utilitarian, that are programmed by different products
and brands, and of the scope of consumer behavior setting. The present paper describes
research that adopted some of these measures and the main results derived from them.
Such studies have shown, for instance, that consumers have brand repertoires that
include brands offering similar levels of reinforcement, that they tend to change the
quantity they buy as a function of package size, price promotions, and utilitarian and
informational reinforcement, that consumer individual differences tend to remain
relatively stable across time, and that more open settings increase product search
duration, decrease the essential value of brands and increase consumers’ reports related
to dominance of shopping environments and approach responses. Moreover, these
measures of consumer behavior can be integrated with measures of firm behavior to
evaluate firm performance, on the basis of an operant interpretation of firm behavior.
This paper explains some of these integrated measures and describes results that have
shown, for instance, how increases in spending in marketing activities is related to
increases in profitability.
... For most teams, Equation 2 accounted for a substantial proportion of the variance. This finding is consistent with the results of studies examining basketball (Alferink et al., 2009;Bourret & Vollmer, 2003;Romanowich et al., 2007;Vollmer & Bourret, 2000), baseball (Cox et al., 2017), hockey (Seniuk et al., 2015), and football (Reed et al., 2006;Stilling & Critchfield, 2010). Results from these studies, which examined two response alternatives, demonstrated that the relative allocation of responses between the alternatives in general was ...
... It is also worth noting because all of the players whose data we analyzed are women. Although Bourret and Vollmer (2003) and Vollmer and Bourret (2000) analyzed basketball shooting by both female and male players, most previous analyses of matching in sports involved only males (Cox et al., 2017;Poling, Edwards, et al., 2011;Reed et al., 2006;Romanowich et al., 2007;Seniuk et al., 2015;Stilling & Critchfield, 2010). As Li, Wallace, Ehrhardt, and Poling (2017) explained, there are good reasons to include both females and males in all types of behavior-analytic research. ...
... If so, training setters to attend carefully to the outcome of sets to various attackers, and apportion the number of sets accordingly, might be a useful coaching technique of some applied value given the importance of success in volleyball to its players. Reed et al. (2006) previously reported that the national rankings of football teams did not influence any of the values obtained when the GME was used to analyze the relation between number of plays called and number of yards earned. Our results were similar, in that for the volleyball teams we analyzed slopes and y-intercepts for the regression lines describing the data did not differ systematically as a function of team ranking. ...
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.
... First examined in basic laboratory contexts, researchers have increasingly applied the GML to naturalistic, socially significant human behavior (Borrero & Vollmer, 2002). The generality of the GML has also been assessed in describing complex operant behavior in competitive contexts such as basketball (e.g., Alferink et al., 2009), baseball (e.g., Poling et al., 2011), football (e.g., Reed et al., 2006), hockey (Seniuk et al., 2015), martial arts (Seniuk et al., 2019), chess (Cero & Falligant, 2019), volleyball (Rotta et al., 2020), and videogames (Schenk & Reed, 2019). However, in many naturally occurring contexts, there are more than two concurrently available response alternatives/schedules of reinforcement. ...
... In addition to skill-level differences, other factors may modulate sensitivity, bias, and variance accounted for by the GML. Two specific examples are: (1) game-specific situations and strategies (e.g., Reed et al., 2006) and (2) rule changes which alter the difficulty or probability of obtaining reinforcement (Falligant et al., 2016;Romanowich et al., 2007). Specifically, sensitivity and bias may be altered as a function of one (or both) of those factors. ...
Cox et al. (2017) successfully applied the multialternative version of the generalized matching law (GML) to pitch selection among a sample of Major League Baseball (MLB) pitchers. The purpose of the present study was to replicate and extend these findings by fitting the multialternative GML to pitch data among a sample of MLB pitchers with varying levels of success in the major leagues. We also examined how matching parameters changed as a function of novel antecedent game contexts such as the infield shift, game location, and number of times the pitcher faced the batters in the batting order. These results replicate the findings from Cox et al. and suggest the multialternative GML is a robust descriptor of pitch selection among MLB pitchers. Together, these findings further extend the generality of the multialternative GML to naturalistic, non-laboratory environments.
... Matching analyses have been applied to shot selection and points earned in basketball (Alferink, Critchfield, Hitt, & Higgins, 2009;Bourret & Vollmer, 2003;Romanowich, Bourret, & Vollmer, 2007;Vollmer & Bourret, 2000), pitch selection and hitter outcome in baseball (Cox, Sosine, & Dallery, 2017;Falligant, Cero, Kranak, & Kurtz, in press), switch hitting and hitter outcome in baseball (Poling et al., 2011), shot selection and hits of the net (as well as points scored) in hockey (Seniuk, Williams, Reed, & Wright, 2015), and play selection and yards earned in football (Reed, Critchfield, & Martens, 2006;Stilling & Critchfield, 2010). ...
... For the National Football League (Reed et al., 2006), the GME accounted for 76% of the variance in play selection (run vs. pass) at the league level. In eight other leagues, it accounted for 57% to 95% of the variance. ...
... These findings suggest, albeit weakly, that the performance of teams with better records might exhibit stronger matching than teams with poorer records. Reed et al. (2006) failed to detect such a relation with football teams, however, and further data are needed to confirm it in volleyball teams. ...
We investigated the applicability of the generalized matching equation to 6 Division I women’s volleyball teams. Data were collected on team allocation of 2 kinds of offensive attempts, spikes and tips, and on the number of points immediately earned by those responses. The equation accounted for at least 50% of the variance in performance for 3 of the 6 teams, including the 2 top-ranked teams, but for less of the variance for the remaining teams. Overmatching was evident for 5 teams and undermatching for 1 team. Four teams showed a bias for spiking and 2 for tipping. Variables other than points immediately earned by spikes and tips that are likely to influence performance are considered, including rules and trained responses to particular defensive formations.
