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.

Figure 1. Some potential outcomes of applying Equation 1 to football play-calling data. See text for explanation.

Figure 2. Top: results of applying the generalized matching law (Equation 1) to 2004 regular-season play calling by NFL teams.

Each data point represents one team. Heavy diagonal lines indicate perfect matching. Bottom three panels: slope, intercept, and variance accounted for by Equation 1 when applied to the 1972 through 2004 NFL regular-season data. Trend lines were determined using least squares linear regression. See text for further explanation.

Figure 3. Results of applying the generalized matching law (Equation 1) to season-aggregate play calling of teams in nine leagues.

Heavy diagonal lines indicate perfect matching.

Figure 4. Fitted parameters of the generalized matching law (Equation 1) when applied to play calling in the NFL on first-, second-, and third-down situations during the 2004 regular season.

In the analysis of each down, each of 32 teams served as an individual observation. Raw data were provided by twominutewarning.com.

Figure 5. Results of applying the generalized matching law (Equation 1) to the game-by-game data of all NFL teams during three phases of the 2004 season.

Figure 6. Illustrative instances of applying the generalized matching law (Equation 1) to the game-by-game data of specific teams.

See text for explanation.