Describing Naturally Occurring Schedules: Analysis of Feedback Functions for Shooting During Basketball Games

A number of recent studies have applied the matching law in analyzing behavior occurring in the context of sports including twoand three-point shot allocation by basketball players. This research contributes to our understanding of matching in that it suggests that the matching law may describe responding in naturally occurring human environments. However, because there is no experimental manipulation, the schedule in effect is unknown and it is unclear if obtained matching is a behavioral phenomenon (i.e., behavior conforming to relative reinforcement rates) or merely a relation between responding and reinforcement forced by ratio-like properties of the schedule. In the Study 1, both linear and hyperbolic feedback functions described the naturally occurring basketball shooting data; however data fell along a line which suggests responding was on a ratio schedule. In Study 2, an experiment was conducted in which the rate of basketball shooting was explicitly manipulated to generate a wider range of shooting rates than those observed in Study 1. For five of seven participants both equations described the data equally well and data fell along a linear path. For two participants the hyperbolic function more accurately described the data. Implications of these results for future descriptive analyses are discussed. Describing Naturally Occurring Schedules 3 Describing Naturally Occurring Schedules: Analysis of Feedback Functions for Shooting During Basketball Games The Matching Law states that the relative rate of responding on two or more response alternatives will conform to match the relative rate of reinforcement available for responding on those alternatives. For example, if a variable interval (VI) 2-min schedule is arranged for a response key on the left and a VI 4 min is arranged for a response key on the right in an operant chamber, the matching law predicts that the organism will respond twice as often on the left key (VI 2 min) than the right key (VI 4 min). Baum (1974) proposed an equation which quantifies this relation and also describes bias and sensitivity to relative reinforcement rates. This equation is called the generalized matching law and is expressed quantitatively as: log log log (1) Where B1 and B2 are the response rates on two alternatives and r1 and r2 are the reinforcement rates for those alternatives. The variable s represents the slope of the line and refers to sensitivity to changes in reinforcement rate. When s is equal to 1, for each unit change in relative reinforcement rate there is an equal unit change in relative response rate. Most commonly, s is less than 1 (e.g., behavior is less sensitive to relative reinforcement rate than strict matching would predict) which is referred to as undermatching (Baum, 1979). The variable b represents bias which is an increase in response allocation toward one or the other alternative that is not due to different reinforcement schedules. Log b is the y intercept. If there is no bias, log b equals zero and response bias will shift the line up or down, changing the y intercept. The matching law has been well studied in the experimental laboratory with a variety of species, responses, and reinforcers. There is a growing body of applied literature as well. Researchers have applied matching theory to socially significant settings and behavior in two Describing Naturally Occurring Schedules 4 ways to empirically validate its usefulness and generality. The first is by using the methods described in the experimental literature with socially significant behavior. In these studies, a reinforcer is identified and explicitly controlled by the experimenter to determine if response allocation matches reinforcer allocation. This method has been applied to social dynamics (Borrero, Crisolo, Tu, Rieland, Ross, Francisco, & Yamamoto, 2007), time spent working on math problems (e.g., Mace, Neef, Shade, & Mauro, 1994), and sensitivity to reinforcement in children with Attention Deficit Hyperactivity Disorder when taking methylphenidate compared to a placebo (Murray & Kollins, 2000). In all of these studies, the matching law accounted for much of the variance in response allocation. This suggests that in situations with two response alternatives, complex behavior, such as those involved in social dynamics, will conform to the relative rate of reinforcement obtained. The second way in which applied researchers have examined the generality of the matching law is in describing naturally-occurring response and reinforcer allocations. For example, Borrero and Vollmer (2002) described allocation of problem behavior and an alternative appropriate response according to the matching law. In this study, the maintaining reinforcer for problem behavior was identified via a functional analysis. After identifying the reinforcer, experimenters observed the participants interacting with care providers. The experimenters collected data on appropriate and inappropriate behavior and the occurrence of consequent events including those identified to be maintaining inappropriate behavior. Borrero and Vollmer (2002) found that allocation of inappropriate and appropriate responding was well described by the matching law when the identified reinforcer was used. When the nonreinforcers were used in the analysis there was no or limited matching. This study, unlike those Describing Naturally Occurring Schedules 5 described previously, demonstrates the occurrence of matching with socially significant behavior in a natural setting and without explicitly controlling the reinforcement rate. Several studies have used sports as a context in which to evaluate the generality of matching in non-experimentally controlled environments. Examination of data from professional and collegiate sports, and basketball in particular, is convenient because the statistics are readily available, the responses (e.g., shooting) and reinforcers (e.g., points) are clearly defined, the reinforcer value stays consistent throughout the game, and the opportunity for twoor three-point shots at anytime approximates a concurrent choice arrangement (Vollmer & Bourret, 2000). An example of a line of research in this vein is a series of studies examining allocation of twoand three-point shot taking in professional and collegiate basketball. In this research, the points gained by successful shots were used as the assumed reinforcer. Vollmer and Bourret (2000) examined allocation of twoand three-point shots by collegiate male and female basketball players from a Division I University. The authors found, when accounting for the difference in amount of points earned for twoand three-point shots, the matching equation predicted the relative response ratio of shots taken. Romanowich, Bourret, and Vollmer (2007) replicated and extending these findings by examining three periods of time in professional basketball history in which the three-point line was moved closer and then moved back creating a naturally occurring reversal design. This type of manipulation might be expected to affect the reinforcement rate for three-point shots. When the authors compared the response ratios across the three time periods, they found that moving the line closer increased the relative rate of three-point shots compared to the first time period. When the line was moved further away again, the ratio of attempted three-point shots decreased again but not to the level observed during the first time period. For all three time periods, the Describing Naturally Occurring Schedules 6 matching equation accounted for more than 95% of the variance in the observed data. This study provides further evidence suggesting matching between relative shooting rates and relative reinforcement rates in terms of the fit of the matching equation within each time period and the shift in responding and reinforcement rate across time periods is congruent with what an analysis of responding as a function of reinforcement rate would predict. The previous studies analyzing basketball statistics only examined data from very good teams. Alferink, Critchfield, Hitt, and Higgins (2009) evaluated response allocation of twoand three-point basketball shots for a variety of collegiate basketball teams to determine if better teams or players would better match relative reinforcement rate of twoand three-point shots better than worse teams or players. The authors found that the matching equation accounted for more variance for better players and teams than worse players and teams. In addition, sensitivity to changes in relative reinforcement rate was higher (closer to 1) for better teams and players. In Study 3, Alferink and colleagues (2009) examined data for a single player across seasons. The authors found that the matching equation still accounted for much of the variability in the data for a single player. The benefit of these descriptive analyses is that matching was found to occur in complex and uncontrolled natural environments. These data increase the robustness of the evidence supporting the matching law and suggest that its generality extends into these naturally occurring contexts. One problem with this research is that the schedule of reinforcement arranged for the responses is unknown unlike more controlled studies. In experimental research on the matching law, variable-interval (VI) schedules are typically used because the rate of behavior can vary widely without significantly affecting reinforcement rate. This is in comparison to variable-ratio (VR) schedules in which the rate of reinforcement is directly tied to the rate of behavior. Describing Naturally Occurring Schedules 7 For example, if a pigeon pecks 50 times per minute on a VR 50 reinforcement schedule, then the pigeon will earn one reinforcer per minute. If the pigeon pecks 100 times per minute the pigeon will earn about two reinforcers per minute. However, on a VI 1-m schedule, if the pigeon pecks either 50 or 100 tim