JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2006, 39, 281–297 NUMBER 3(FALL 2006) THE GENERALIZED MATCHING LAW IN ELITE SPORT COMPETITION: FOOTBALL PLAY CALLING AS OPERANT CHOICE DEREK D. REED SYRACUSE UNIVERSITY THOMAS S. CRITCHFIELD ILLINOIS STATE UNIVERSITY AND BRIAN K. MARTENS SYRACUSE UNIVERSITY 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 analyses focused on season-aggregate data for the league as a whole, game-by-game data for the league as a whole, and game-by-game data for individual teams. In all analyses except those for a few individual teams, the generalized matching law accounted for a majority of variance in play calling. The typical play-calling pattern reflected undermatching (suggesting imperfect sensitivity of play calling to yardage- gained reinforcers) and a bias for calling rushing plays. Bias was found to be a function of both the relative risk of turnovers and the relative variability in yards gained associated with passing versus rushing plays. The external validity of the matching analyses was supported by significant correlations between parameters of the generalized matching law and team success on offense and season winning percentage. These results illustrate the broad applicability of the generalized matching law to problems outside of the laboratory. DESCRIPTORS: choice, generalized matching law, football, sport, play calling _______________________________________________________________________________ The matching law (Herrnstein, 1961) states ratory (e.g., Baum, 1979), and has also provided that individuals tend to divide their time and insights into a variety of socially important effort between two or more simultaneously problems such as employee absenteeism (Red- available behavior options proportional to the mon & Lockwood, 1986), teen pregnancy reinforcement that is contingent on each. A small (Bulow & Meller, 1998), and classroom behavior family of mathematical models based on this (Billington & DiTommaso, 2003). Collectively, maxim accounts well for performance under these analyses are noteworthy for suggesting that concurrent reinforcement schedules in the labo- principles of operant choice operate robustly, even amid the complexity of everyday environments. We are grateful to Roland Breech of twominutewarning. Few everyday environments are as complex com for providing raw data that served as the basis for and multiply determined as those in which elite Figure 4; Anthony S. Martens for helping us characterize sport competition occurs. Among the many selected NFL offenses during the 2004 season; and Erin S. Bullett, Florence D. DiGennaro, and David D. Reed for variables believed to influence sport perfor- their insight and assistance throughout the study. mance are coaching strategies and coach Address correspondence to Derek D. Reed, Department personalities; rules of a sport and the extent to of Psychology, Syracuse University, Syracuse, New York 13244. which they are enforced by officials during doi: 10.1901/jaba.2006.146-05 a given contest; the location in which compe- 281 282 DEREK D. REED et al. tition occurs and the behavior of spectators; matching law accounted for about 90% or weather (for outdoor sports) and the condition more of shot-selection variance across players on of the playing surface; the skills of individuals both teams. Players showed near-perfect re- who participate on the team; and the mental inforcement sensitivity (slope ,1), and a bias and physical well-being of those players at the for three-point shots was evident (intercept time of competition. Any lawful principle or ,0.12) (i.e., three-point shots were taken more functional relation found to cut through all of often than frequency of shot making predicted, these variables to reliably predict sport perfor- presumably because of their higher point value). mance would be noteworthy indeed. In the present study, we used the generalized Vollmer and Bourret (2000) provisionally matching law to evaluate choice in a different identified the matching relation as one such sport, American-rules professional football (as predictor when they showed that the generalized contrasted with rugby football, Australian-rules matching law (Baum, 1974, 1979) provided football, or soccer; see McCorduck, 1998). In a good account of shot selection in two college applying Equation 1, we assumed that the basketball teams. The generalized matching law relative rate of calling pass versus rush plays predicts that, given two behaviors, B1 and B2, would vary as a function of relative reinforce- and rates of reinforcement contingent on them, ment, which we operationalized as average yards R1 and R2, respectively, relative effort investment gained for passing and rushing plays. in the behaviors varies linearly with relative Football was chosen for examination for reinforcement rates. This relation is expressed as several reasons. First, play calling can be B1 R1 conceptualized as individual behavior. Typically log ~ a log z log b, ð1Þ B2 R2 in American-rules professional football (hereaf- in which the behavior and reinforcement ratios ter referred to simply as football), each team has are logarithmically transformed to yield an easy- one individual, usually an offensive coordinator to-evaluate linear function in which, if behavior working in conjunction with a head coach, who matches reinforcement perfectly, the slope 5 1 decides what kind of play, passing or rushing and the intercept 5 0. The two fitted (running), will take place on each down or parameters of the equation describe obtained opportunity (McCorduck, 1998). Offensive estimates of slope and intercept, which usually coordinators are typically highly skilled (and deviate from the ideal values. The slope of the highly paid) professionals with considerable function, a, is viewed as a measure of sensitivity football experience and whose play-calling to reinforcement differentials, that is, of how behavior leads to high-stakes outcomes (e.g., much behavior ratios change given one unit of team success, continued employment). Second, change in reinforcement ratio. The intercept, in calling plays, coaches routinely consider the log b, indicates bias, or systematic preference success (i.e., reinforcement) of previously not accounted for by reinforcement rates. attempted plays (Edwards, 2002), suggesting In Vollmer and Bourret’s (2000) analysis of a general sensitivity of play calling to re- players on two college basketball teams, behav- inforcement. Third, individual differences in iors were two-point and three-point shots play-calling patterns may be anticipated. As attempted, and reinforcers were succesful shots a function of coaching staff, player skill, and made of the same two types. Graphic analyses other factors, teams often adopt specific suggested that the generalized matching law offensive styles for which they become known provided a good fit to the shot-selection data, over time. For example, during 2004 the and based on raw data in Table 1 of the report, Indianapolis Colts were regarded as a passing it can be estimated that the generalized team, with Peyton Manning, who threw for OPERANT CHOICE IN FOOTBALL 283 a league record 49 touchdowns for the season, Unless otherwise noted below, data were re- as quarterback (Clayton, 2005). By contrast, the trieved from http://www.nfl.com. The primary Atlanta Falcons were regarded as a rushing team data consisted of the number of passing and with Michael Vick, a strong runner but rushing plays executed and net yards gained inconsistent passer, as quarterback (Winkel- from those plays for each of the 32 NFL teams john, 2005). We wondered whether such during each game of the 16-game 2004 regular differences would be apparent in the fitted season. Several aspects of this data set should be parameters of the generalized matching law. noted. First, plays are categorized as rushing or Fourth, there are situation-specific patterns in passing based on what actually occurred rather play calling. For example, during 2004, Na- than what was called, and these may differ in tional Football League (NFL) teams rushed on the case of ‘‘audibles’’ (i.e., when a quarterback 52% of first-down plays but on only 24% of calls a different play than originally planned). third-down plays (data provided by Roland Second, ‘‘sacks’’ (i.e., when a quarterback is Beech of twominutewarning.com). It is reason- tackled behind the line of scrimmage) are able to suggest that yardage gains associated treated as failed rushing plays even though the with rushing and passing vary across game quarterback’s original intention may have been situations, and thus situational patterns might to pass the ball. Because yards lost as well as the manifest as changes in the generalized matching intention of the quarterback to pass or not law’s fitted parameters. Finally, rushing and during each play that resulted in a sack are not passing statistics for every team in the NFL are specified, we could not apply a correction for readily available from their Web site and other this feature of the data. Third, although it is sources, making possible analyses based on the possible for a player to receive a pass and generalized matching law. subsequently fumble the ball, yards gained from