The Generalized Matching Law in Elite Sport Competition: Football Play Calling as Operant Choice

Abstract

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.

Postreinforcement Pause in Grocery Shopping: Comparing Interpurchase Times Across Products and Consumers

“…This analogy resembles recent investigations of complex phenomena that occur in natural settings, such as playing-calling in football (Reed, Critchfield, & Martens, 2006) and scalloping patterns in congressional bill approval (Critchfield, Haley, Sabo, Colbert, & Macropoulis, 2003). Table 1 presents information concerning the number of consumers, the number of purchases, the average number of purchases, and the total amount spent per consumer for each product category.…”

What are we doing when we translate from quantitative models?

“…Although earlier we stressed the capacity of quantitative models to generate unique predictions, many of the published studies in which quantitative analysis was applied to everyday problems have a distinct exploratory feel to them. For example, Reed et al (2006) summarized their application of the generalized matching law (Figure 2) to football play selection by noting that, ''The present findings are noteworthy in that a fairly simple quantitative model predicts behavior under the complex circumstances of elite sport competition'' (p. 293). This conclusion illustrates a feature that is shared by many quantitative translational investigations to date, namely an interest in whether or not a model's core relation is relevant outside the laboratory.…”

“…When the problems at the two stations were of equal difficulty, little bias was evident, but when problems were more difficult at one station, a bias for the other station emerged ( Figure 6C). Stilling and Critchfield (in press; see also Reed et al, 2006) found that bias in football play selection varied as a function of the down on which a play was executed ( Figure 6D); as part of the same investigation, Stilling (2008) found that bias varied with other situational factors such as the game score, the location on the field from which a play originated, and the amount of competition time remaining when a play began.…”

“…Each team must decide how much it will rely on running versus passing, and some teams are regarded as running teams and others are viewed as passing teams. Reed, Critchfield, and Martens (2006) highlighted the similarities among teams by using the generalized matching law (Figure 2), with frequencies of the two types used in the behavior ratio and yards gained from those plays used in the reinforcement ratio. They found that the 32 National Football League teams, although differing in preference for running versus passing plays, fell along a single linear function that accounted for about 75% of the variance in play selection.…”

Serial correlation in National Football League play calling and its effects on outcomes

“…We also control for various observables, such as whether the game was played in excessively cold conditions, 18 excessively windy conditions, 19 wet 1 7 For more on the optimality of the pass-rush mix, see Alamar (2006Alamar ( , 2010, Reed, Critch…eld, and Martens (2006), Rockerbie (2008), Kovash andLevitt (2009), andStilling andCritch…eld (2010).…”