Evaluating NBA end-of-game decision-making

2.1Win probability model

The win probability model in this paper, restricted to the last three minutes of the game, uses time remaining, score differential, possession, and point spread. The data consist of NBA regular season play-by-play descriptions from 2011 to 2015 gathered from stats.nba.com (Luo, 2017). The point spread data come from sportsdatabase.com. The win probability model is built using the statistical modeling technique of logistic regression. Given any game situation, the model takes the state of the game and generates a win probability describing the chances of winning for both teams.

Eq. 1 shows the general logistic regression function of the win probability model,

where P (W) is the win probability for the team with possession, β_i is the coefficient of the ith term, S is the score differential relative to the team with possession, tˆ is the normalized time remaining in the game, λ is an exponent found using the validation set, and L is the point spread relative to the team with possession. The time is normalized to equal 1 at the beginning of regulation and 0 at the end of regulation. The data are split into an 80% training set, 10% validation set, and 10% test set with data assigned randomly from the 2011-2012 through 2014-2015 NBA seasons.

The features in the model consist of nonlinear functions of score differential and time remaining in the game. The motivation behind using these nonlinear functions is the intuition that these variables combine nonlinearly to affect the probability of winning. Many features of this nature were tried with the validation set guiding the choice of the final model via maximum likelihood.

The model was then run on an independent test set. On these test data with a win probability cutoff of 0.5 for class assignment, the model has a precision of 92.3% and a recall of 90.9%, with an overall accuracy of 92.5%. These values indicate that the logistic regression model provides a reasonable estimate of in-game win probability.

Figure 1 provides an illustration of how the logistic regression model maps to in-game win probability given different game states. The figure contains the win probability plots of a team with a given score differential and how the win probability changes within the last three minutes. For each time point, the win probability corresponds to a team starting possession at that moment with the circumstances shown in the figure. This plot reveals several features of the win probability model. As the score differential gets more positive, the resulting increase in win probability gets smaller, which agrees with intuition as the win probability asymptotically approaches one. Smaller score differentials have extreme win probability behavior as the time remaining approaches zero.

Fig.1

Win probabilities for different score differentials S and game time remaining for a team with possession and a point spread of zero.

2.2End-of-game tactics metric

The win probability model serves as the foundation for ETM because, at the beginning of each possession, both teams have an initial win probability and a set of choices to make. At a tactical level, these choices include whether to shoot a two-point or three-point field goal for the offensive team and whether to intentionally foul for the defensive team, which are the only decisions evaluated in this study. The timing of these decisions also factor into the decision-making process. The possibility of a turnover or free throws on a shooting foul exists, but these are not explicit choices made by teams. The power of ETM lies in the ability to calculate the optimal decision and judge a team’s actual decision relative to the optimal one.

The Chapman-Kolmogorov equations allow for the calculation of the win probability of a team after a given decision is made. In this context, the equation states that the probability of winning a game after making a decision k is the sum of the probability of all of the possible outcomes of that decision multiplied by the probability of winning after those outcomes. In equation form, this becomes,

where P (W) _k is the probability of winning after making decision k, P_j is the probability of score differential j occurring after making decision k, P (W) _j is the probability of winning if score differential j occurs after making decision k, and J is the number of possible score differentials of the decision k. The decisions include k ∈ (2 - pointshot, 3 - pointshot) for the offensive team and the decisions include k ∈ (Intentionallyfoul, Nofoul) for the defensive team. The win probability term of Eq. 2 comes directly from the win probability model. The P_j term comes from evaluating how all of the different outcomes of a decision can result in score differential j. For example, for the score differential to remain the same after a decision to shoot a two-point field goal, this can occur if the team misses the field goal attempt, turns the ball over, or misses both free throw attempts followinga shooting foul (for simplicity, this model ignores getting fouled during a three-point field goal attempt). The probabilities of all of these outcomes come from team statistics, including two-point field goal percentage, three-point field goal percentage, turnover percentage, free throw percentage, rebound percentage, and foul percentage. These statistics vary by team and this study uses team statistics rather than aggregated league averages. Eq. 2 evaluates the transition of the current state of the game to all other possible future states of the game, after one possession.

With the ability to evaluate the decision-making of teams both offensively and defensively, ETM can now be defined as,

4.1“Fouling Up-3”

The “Fouling Up-3” tactic is used by teams leading by three points in the final seconds of a game in order to force two free throw attempts instead of a three-point field goal attempt. The tactic has been heavily debated, but analyses and studies on the topic tend to be heuristic in nature (Annis, 2006). While these studies provide value, statistical modeling efforts like ETM can provide more insight.

Using ETM and win probability to examine this tactic is quite straightforward. Consider a team on defense leading by three points. This team has the option to intentionally foul or play conventional defense. Fig. 4 shows the win probability for each option of a defensive team winning by three points and the possession beginning at a given time.

Fig.4

Win probability for a team winning by three points without possession for two tactics. The win probability of the intentional foul tactic exceeds the win probability of playing conventional defense beginning at nine seconds remaining.

Notice how the defensive win probability changes throughout the last 36 seconds of the game in Fig. 4. With 36 seconds remaining, the conventional defensive tactic win probability is 3.5 percentage points higher than the intentionally fouling tactic. However, the intentionally fouling tactic becomes the more optimal tactic around nine seconds remaining. With one second remaining, intentionally fouling is 9.2 percentage points higher than theconventional defensive tactic. This observation validates the “Fouling Up-3” tactic with fewer than nine seconds remaining, which leaves the trailing team with little time to intentionally foul themselves and regain possession. Note, this exercise assumes league average percentages that factor into Eq. 2. These solutions change when a team may be better at three-point field goals or worse at free throws, for example. Also, this exercise assumes that, if a team chooses to play conventional defense, the offense will choose the optimal tactic. Lastly, while “Fouling Up-3” is found to be optimal under certain circumstances, executing this tactic is difficult. The risk of a successful field goal and a foul exists, along with fouling while the offense is attempting a three-point field goal.

4.2Intentional foul timing

While intentionally fouling with the lead is controversial, intentionally fouling while trailing late is a nearly universally accepted tactic. However, both the timing and score differential under which this tactic becomes the optimal choice vary significantly from game to game. The win probability of a trailing team can be used to find the time at which intentionally fouling becomes the optimal tactic given a score differential. For the purposes of this exercise, the shooting weight applied to shooting percentages is set to one.

Fig. 5 shows the win probability difference between intentionally fouling and playing conventional defense for possessions beginning at a given time. The trends intuitively show that for a smaller score differential, the intentionally fouling tactic becomes optimal later in the game. For a team trailing by one through four points, intentionally fouling becomes the optimal tactic with 36, 48, 61, and 74 seconds remaining, respectively. Again, these curves will shift depending on team or player statistics used in Eq. 2. However, Fig. 5 can provide a guideline to maximize win probability when trailing late in games. This exercise also assumes a point spread of zero.

Fig.5

Win probability for different score differentials S for a team without possession. For each score differential, the win probability after deciding to intentionally foul or play conventional defense is shown.