Bayesian model for highly applied decision making in American football

A task

Choose to play the punt or play the 4th attempt in the “4 and j yards” in situation i yards on the field.

Events

From the statement of the problem, it follows that we must consider two possessions (one of our own and the next after the current one, which the opponent takes). During these two possessions, 4 events can occur, fully describing significant and possible results (several events can occur):

  • A: Our team scores a touchdown after two possessions
  • B: our team will miss a touchdown after two possessions (including a return touchdown to our touch zone)
  • С: our team will miss a touchdown after two possessions
  • D: Our team scores a touchdown after two possessions (pick-six)

Decision

General idea

Thus, the task is reduced to comparing four probabilities:

  • P (A): Probability to score a direct touchdown when choosing to play 4th attempt,
  • P (B): Probability of missing a touchdown when choosing to play 4th attempt,
  • P (С): Probability of missing a touchdown when choosing a punt,
  • P (D): Probability to score a touchdown when a punt is selected.

P (A) — P (B)

?

P (C)— P (D)

The events that affect the probabilities on the left side of the inequality are: entered and missed touchdowns on the basis of two possessions, as well as the first down scored on the basis of the 4th attempt.

4th attempt play

The probability of entering a touchdown on the decision to play the 4th try, or P (A), depends on whether the 4th try is successful. And it also depends on how successfully our team implements the situation of the first down, on a specific part of the field, to a touchdown. These probabilities fully describe all possible outcomes, and most conveniently, they can be taken from the accumulated (for your own team) statistics:

  • P (X): the statistical probability of passing j yards in one attempt,
  • P (A | X): The statistical probability of getting a touchdown from the i-th yard of the field (from situation 1–10).

P (A | X) = (P (X | A) * P (A)) / P (X)

where P (X | A) is the probability of realizing the 4th attempt, provided that our team scores a touchdown, which, according to common sense, is equal to one. Thus, our desired prior probability is:

P (A) = P (A | X) * P (X)

As a result, we consider P (A) to be a simple multiplication of the probability of passing j yards by the probability of scoring from the i-th yard from situation 1–10. We take both probabilities from statistics.

  • P (Y): the probabilities of missing a touchdown when unsuccessful play of the 4th attempt (from the place of its drawing, from situation 1–10 to attack the opponent). Moreover, the probability of an unsuccessful play of the 4th attempt is 1 — X.
  • P (Z): Probabilities of missing a touchdown in case of a successful 4th attempt, for example, when changing possession on the following drives.

P (Y) = P (Y | (1-X)) * P (1-X)

and in case of a successful drawing of the 4th attempt:

P (Z) = P (Z | X) * P (X)

For simplicity, let’s take for P (Z | X) the simple statistical probability of missing a touchdown after kickoff. Simplifying a little more, we can reduce the probability of missing after kickoff to the probability of missing from 30 + k yards, that is, from the place where we, on average, move the opponent on the kickoff.

P (B) = P (Y | (1-X)) * P (1-X) + P (Z | X) * P (X)

And the total damage (after all, we are already in a vulnerable situation when playing the 4th attempt and we consider in which case the damage will be less) from the decision to play the 4th attempt:

P (A | X) * P (X) — P (Y | (1-X)) * P (1-X) — P (Z | X) * P (X)

The meaning of the expression reduces to calculating the difference in probable touchdowns for two possessions.

Punt

When playing a punt, we actually give up our attempt (of the two we are considering) and the calculation comes down to the likely damage when our team plays in defense.

  • k: the average number of yards that our team pushes the opponent away by playing the punt, with the returned yards,
  • P (C): The statistical probability of missing a touchdown from i + k yards on the field (from situation 1–10).

Outcome

To make a decision about whether to play the 4th attempt or to play the punt, one must compare the possible damage (taking into account the possible benefits) from the first decision and the possible damage from the second. Moreover, we can take all the data from the accumulated statistics. In addition, the attentive reader will notice that P (Y | (1-X)), P (Z | X) and P(C) are the same thing, only they are taken for different i or positions on the field.

P (A | X) * P (X) — P (Y | (1-X)) * P (1-X) — P (Z | X) * P (X)

?

P (C)

Thus, having the statistics of the plays for your own team and indicating the current position of the team on the field before the 4th attempt (the number of yards to the first down and the position of the scrimmage line), you can evaluate the chances and choose the most effective outcome.

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