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May 2, 2005

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A Model for the Two-Minute Drill

In this article we introduce a model for strategy and win probabilities when the team with possession trails by one score in the final two minutes of the game. By restricting the analysis to a single possession in the closing minutes, we are able to model the game at a level of detail that is computationally infeasible in the footballcommentary.com Dynamic Programming Model, but is essential for usable probability estimates late in the game. The model operates at the level of individual plays, and includes as state variables the line of scrimmage, the time remaining, the down, the yards to go for a first down, the number of timeouts remaining, and the score. The decision to spike the ball or call timeout, as well as the decision to attempt a field goal, are made optimally within the model by backward induction. The defense adapts to the situation, for example by focusing on long passes and conceding a short completion when it's optimal to do so. In fact, on each play, the strategies of the offense and defense comprise a Nash equilibrium.

The model is a dynamic program, solved by backward induction, and not a simulation. With a simulation one would not be able to determine and incorporate optimal strategy for field goal attempts, timeout usage, or spikes. Nor would it be possible to have the pass strategy of the offense and the pass coverage strategy of the defense be in equilibrium.

Overview of the Model

This section lists all the aspects of a football game that are contained in the model. Any features of a game that are not described in this section, such as penalties, are not included in the model.

Win probabilities in the model are conditional on the line of scrimmage, the time remaining, the down, the yards to go for a first down, the number of timeouts remaining for the offense, and the number of points by which the offense trails.

On any play, the offense has the option of attempting a field goal. The probability that the field goal is good is a function of the length of the attempt. As described in a later section, the decision to attempt a field goal is made optimally by backward induction. (Obviously, if the offense trails by more than three points, it will never be optimal to attempt a field goal.)

The model assumes that every play other than a field-goal attempt is a pass play. In reality, except when the time pressure is extreme, the equilibrium strategy for the offense will be a randomized strategy, in which they will run with some small probability. However, if running plays were incorporated into the model, they would have an effect similar to short passes.

On any pass play, there is a possibility of a sack. Both the probability of a sack and the probability distribution for the resulting loss of yardage have been estimated from data. If there is a sack the down advances, and the number of yards for a first down increases by the loss on the play. Since we deal only with the final 2:00, the clock continues to run following a sack (in accordance with the Exception to Supplemental Note 3 following Rule 4-3-7).

If there is not a sack, the quarterback will throw a pass of some chosen distance. The probabilities that the pass is complete, incomplete, or intercepted depend on the length of the pass, and on the pass coverage. In the model, the pass coverage selected by the defense depends on the situation. The defense is able to improve its chances against long passes, at the expense of short routes, or vice versa. The offense's strategy is a randomized strategy, in which they choose a probability for each of the distances that the pass might be thrown. The probabilities depend on the situation. Intuition suggests (for example) that if the offense is deep in its own end with very little time remaining, the probabilities for long passes will be large, while the probability of a short pass will be small or even zero. However, the strategies of the offense and defense are not exogenous in the model. Instead, in each situation, the model solves for the Nash equilibrium strategies, meaning that the randomization probabilities selected by the offense are optimal relative to the defense's coverage, and the defense's coverage is optimal relative to the randomization probabilities selected by the offense. (We explained Nash equilibrium more fully in a previous article.)

If the pass is intercepted, the offense's chances vanish. If the pass is incomplete the clock stops and the down advances, whereas the line of scrimmage and yards to go for a first down are unchanged. If the pass is complete but the gain is short of a first down, the line of scrimmage, down, and yards to go are updated. If the play makes a first down, the line of scrimmage updates and it becomes 1st and 10.

In the model there is a probability that the ball-carrier will go out of bounds following a completed pass. If he does, the clock stops until the next snap (in accordance with Exception 1 following Rule 4-3-2). If the ball-carrier stays inbounds, the offense can call timeout (if they have one) to stop the clock, or spike the ball. Following a spike, of course, the down advances. As described in the next section, the decision to call timeout or spike the ball is made optimally by backward induction.

The model analyzes a single possession. It implicitly assumes that if the trailing team fails to score, they won't get another opportunity. The best chance for a counterexample would be if the trailing team takes possession with 2:00 and three timeouts remaining. Under the scenario in which they then throw three consecutive incomplete passes, punt, and force a three-and-out, they get the ball back with somewhere around 1:06 left and no timeouts. Of course, this scenario is not very likely, and even if it occurred, the trailing team's probability of scoring on their second opportunity wouldn't be high. So, ignoring the possibility of a second opportunity is probably not a huge deficiency, although in applications we should be mindful of cases in which it's material.

Similarly, the model doesn't account for the possibility that if the trailing team scores to tie the game or take the lead, their opponents have a chance for a subsequent score. This deficiency of the model is probably not material when the offense trails by three or more points, because if they score a touchdown they will likely consume most of the clock doing so, and if they score a field goal to tie the game their opponents will probably elect to play for overtime.

Results of the Model

We will postpone until the next section a more detailed description of the model. In this section we will give some sample applications. We begin by revisiting Philadelphia's decision to attempt an onside kick with 1:48 remaining in Super Bowl XXXIX, trailing by three points. The model allows us to refine our original analysis.

Following a successful onside kick, the Eagles take over at around their own 42-yard line, with about 1:48 on the clock and two timeouts. According to the model, their probability of winning is then 0.306. If the attempt fails but the Eagles force the Patriots to go three-and-out, the Eagles can expect to get the ball back at about their own 10-yard line, with around 0:45 left and no timeouts. Their probability of winning at that point is just 0.035.

Alternatively, the Eagles can kick deep. In this case, if they force a three-and-out, they can expect to get the ball back at about their own 30-yard line, with 0:45 and no timeouts. According to the model they then have a 0.109 probability of winning.

Now, let p denote the probability of recovering a fully anticipated onside kick, and let q denote Philadelphia's probability of forcing New England to go three-and-out in a situation in which the Eagles know the Patriots will be running. Then if the Eagles kick deep, their probability of winning is 0.109q, whereas if they attempt the onside kick their probability of winning is

0.306 p + 0.035 (1 − p) q.

If for example p=0.2 and q=0.5, then kicking onside is significantly superior to kicking deep.

We can also use the model to evaluate a team's use of timeouts and spikes to stop the clock. As an example we will examine the Washington-Philadelphia game in Week 5 of the 2003 season. The Redskins took over at their own 43-yard line with 1:13 left and 1 timeout, trailing 27-19.

There were two plays in that possession for which the clock continued to run after the whistle. In the first case, Washington picked up a first down at the Philadelphia 45-yard line, and the play was whistled dead at 0:57. The Redskins chose to let the clock run; the model says it would have been slightly better to call timeout.

The next play was whistled dead with 0:32 remaining, with Washington just having picked up another first down at the Philadelphia 32-yard line. Once again the Redskins let the clock run. In this case the model says that calling time would have been substantially better, with spiking the ball a close second.

We have produced a Table that summarizes some of the results of the model. One notable result that emerges from the Table is that the value of a timeout is typically small. Remember, though, that this is only a statement about the clock-management value of a timeout on offense at the end of the game. It doesn't say anything about the cost of a wasted timeout earlier in the half, because in most cases, the team that trails at the end should have used its timeouts earlier on defense, where they would have saved 40 seconds each. As the Table shows, an extra 0:40 can give a substantial increase in the probability of winning. More generally, one of the potential applications of this model is as a component of a more complete analysis of the clock-management value of timeouts.

Description of the Dynamic Program

This section contains a more detailed description of the model. We assume that the reader has some familiarity with dynamic programming. Those who do not may first want to read the description of the footballcommentary.com Dynamic Programming Model. For a precise description of the model, the reader should consult the MATLAB® source code dyn2min.m and sumfunc.m.

The state variables are the line of scrimmage, the time remaining, the down, the distance to go for a first down, the number of timeouts remaining for the offense, and the number of points by which the offense trails. However, since the point differential is constant throughout the possession, it enters the calculations only through the boundary conditions.

There are two situations in which the model computes the offense's probability of winning the game: when the ball is snapped, and when the whistle blows to end a play.

Let S(yards,t,down,togo,timeouts) denote the offense's probability of winning at the moment the ball is snapped, as a function of the distance of the line of scrimmage from the defending team's goal line, the time in seconds remaining in the game, the down, the distance to go for a first down, and the number of timeouts that the offense still has. S(yards,t,down,togo,timeouts) is also the offense's probability of winning when the whistle blows to end a play, if the clock stops at the whistle because the pass was incomplete or the ball-carrier went out of bounds.

To facilitate the computations, the unit of distance is five yards. Therefore, for example, yards=1 denotes the defending team's 5-yard line, and yards=17 denotes the offense's 15-yard line. Similarly, togo=2 means two units of distance (i.e. 10 yards) to go for a first down.

Let W(yards,t,down,togo,timeouts) denote the offense's probability of winning when the whistle blows, if the clock continues to run.

Boundary Conditions

Let h(points) denote the offense's probability of winning the game if they score a touchdown, as a function of the number of points by which they trailed prior to scoring. If points < 6, then h(points)=1. Also,

     h(6) = p1pt + (1 − p1pt)pot,

where p1pt is the probability of a successful kicked extra point, and pot is the probability of winning in overtime (presumably 0.5). Finally, h(7)=p1ptpot and h(8)=p2ptpot, where p2pt is the probability of a successful two-point conversion.

There are various boundary conditions for the functions S and W. First, if the offense reaches the opponent's goal line, their probability of winning is h(points). Therefore S and W equal h(points) when yards=0, regardless of the values of the other arguments of those functions. If time expires before the offense scores, they lose. So, except when yards=0, S and W equal 0 when t=0.

In addition, if yards=20 it is a safety, so that S and W equal 0 when yards=20 regardless of the values of the other arguments of those functions. Finally, the offense loses if they turn the ball over on downs. We handle this in the model by incorporating a "fifth down;" if down=5, then S and W equal 0.

When the offense trails by more than three points, the probability of winning the game is proportional to h(points). Therefore, for those cases, one can simply use 1 as the boundary value for S and W when yards=0, and apply the factor h(points) to the output win probabilities at the end. This is the way the calculations are actually done in the computer code.

Backward Induction

We compute the win probabilities S and W for other values of the arguments by backward induction. Suppose we have computed those probabilities for all states in which the time remaining is less than t seconds. We will describe how to compute the probabilities when there are t seconds remaining.

First consider S(yards,t,down,togo,timeouts). When the ball is snapped, there is a probability ps that the quarterback will be sacked. We approximate the actual distribution of losses from sacks by assuming that a sack loses either 0, 5, or 10 yards with probabilities ps0, ps5, and ps10 respectively. Therefore, if there is a sack, the probability of winning when the whistle blows is

     sack = ps0 W(yards, t − tplay, down+1, togo, timeouts)
         + ps5 W(yards+1, t − tplay, down+1, togo+1, timeouts)
         + ps10 W(yards+2, t − tplay, down+1, togo+2, timeouts)

where tplay is the number of seconds consumed by the play, up until the whistle blows.

If there is not a sack, the quarterback will throw the ball down the field a distance that corresponds to an expected gain of i distance units on the play if the pass is complete. We require i ≤ 10, corresponding to a maximum gain of 50 yards. (As we explain later, every i gives the same probability of winning in equilibrium, except for certain "dominated" distances.) If the pass is complete but does not result in a first down (i.e., if i < togo), the offense's probability of winning is

     complete = pob S(yards − i, t − tplay, down+1, togo − i, timeouts)
             + (1 − pob) W(yards − i, t − tplay, down+1, togo − i, timeouts),

where pob is the probability that the ball carrier goes out of bounds. If the play does result in a first down, the offense's probability of winning is

     complete = pob S(yards − i, t − tplay, 1, 2, timeouts)
             + (1 − pob) W(yards − i, t − tplay, 1, 2, timeouts).

If the pass is incomplete, the offense's probability of winning is

     incomplete = S(yards, t − tplay, down+1, togo, timeouts).

Finally, if the pass is intercepted, the offense's win probability is 0. In the model we assume that the probability of an interception is β times the probability that the pass is incomplete.

Putting these pieces together, we find that if the offense chooses to run a play rather than attempt a field goal, their probability of winning at the moment the ball is snapped is

     passplay = ps sack + (1 − ps) (pCi complete + pIi incomplete)

where pCi is the probability that the pass is complete, and pIi is the probability that the pass is incomplete. (The probability of an interception enters this equation implicitly, because pCi and pIi sum to less than one. It doesn't enter explicitly because after an interception the offense's win probability is zero.)

Instead of running a play, the offense can attempt a field goal. The probability that the field goal is good depends on the line of scrimmage; denote it by g(yards). Then the probability that the offense wins the game if they attempt a field goal is ψ g(yards), where ψ equals 1, pot, or 0 depending on whether the offense trails by less than 3 points, exactly 3 points, or more than 3 points. Naturally, the offense will attempt a field goal rather than run a play if and only if it gives a higher probability of winning. Therefore,

     S(yards,t,down,togo,timeouts) = max( passplay, ψ g(yards) ).

The formula for W is much simpler. If the clock continues to run when the whistle blows, the offense can simply proceed with its next play, which consumes tnext seconds; or they can spike the ball, which uses tspike seconds; or, if they have a timeout, they can call time, which uses tto seconds. The probabilities of winning corresponding to these decisions are

     nextplay = S(yards, t − tnext, down, togo, timeouts),

     spike = S(yards, t − tspike, down + 1, togo, timeouts),   and

     calltime = δ S(yards, t − tto, down, togo, timeouts − 1)

where δ equals 1 if the offense has a timeout and equals 0 otherwise. Since the offense makes decisions optimally in the model, we have

     W(yards,t,down,togo,timeouts) = max(nextplay, spike, calltime).

Equilibrium

The probability pCi that a pass thrown for an expected gain of i units (if it's complete) will actually be complete is not exogenous in the model. Instead, it depends on the defense's coverage. The defense adapts to the situation, for example by shifting coverage to protect against deep routes while conceding short passes. Formally, the defense devotes xi of effort to stopping a pass that would give a gain of i distance units. The xi are non-negative and subject to the constraint that their sum cannot exceed X. The probability that a pass intended to gain i units is complete is pi*/(1 + αi xi), where the pi* and αi are fixed parameters. The defense's strategy at a given state is its choices for the xi. By increasing some xi while (necessarily) lowering others, the defense affects the completion probabilities for passes of those distances. The offense's strategy at a given state consists of its randomization probabilities. We let πi denote the probability that the offense attempts a pass for a gain of i distance units. We assume that the strategies of the offense and defense in any particular state comprise a Nash equilibrium, in which the πi are optimal relative to the xi, and the xi are optimal relative to the πi. Typically there will be some set of pass distances that are dominated, meaning they are suboptimal for the offense even if the corresponding xi are zero. In equilibrium the defense's coverage is such that the offense is indifferent among the remaining pass distances; i.e., passes of those distances yield identical probabilities of winning the game. In a mathematical appendix we show how that win probability, and the probability pCi of completing a pass for a gain of i units, are determined in equilibrium.

Table 1
Illustrative Nash Equilibrium
Distance
(in yards)
Win prob.
if complete
x Prob. of
completion
Win
probability
π
5 0.11 0 0.88 0.103 0
10 0.14 0 0.76 0.127 0
15 0.17 0.57 0.52 0.131 0.153
20 0.20 0.91 0.37 0.131 0.147
25 0.24 1.11 0.27 0.131 0.140
30 0.28 1.06 0.21 0.131 0.129
35 0.33 1.02 0.17 0.131 0.119
40 0.38 0.99 0.14 0.131 0.111
45 0.44 0.95 0.11 0.131 0.104
50 0.51 0.89 0.10 0.131 0.097

Table 1 illustrates the Nash equilibrium in one specific case. The situation is that the offense has 1st and 10 at its own 25-yard line, with 1:00 and two timeouts remaining, and trails by 4 points. The first column gives the length of the pass play, meaning the expected yardage gained if the pass is complete.

The second column gives the offense's probability of winning the game if the pass is complete. The third column gives x, a measure of the defense's coverage for passes of each distance. Notice that in the equilibrium, the defense essentially concedes 5- or 10-yard pass plays, concentrating more on longer routes. Column 4 gives the probability of a completion that results from the defense's coverage.

Column 5 gives the offense's probability of winning the game at the moment the pass is thrown, before it's known whether the pass is complete. This value depends on the probability of winning if the pass is complete and on the probability of a completion, both of which are shown in the Table, but also on the probabilities for an incomplete pass or an interception, and on the probability of winning after an incompletion. Notice that the offense's probability of winning is the same for all passes longer than 10 yards. This isn't a coincidence, it's a property of the equilibrium. All pass distances yield the same win probability in equilibrium, except for distances that are dominated from the standpoint of the offense — and hence conceded by the defense.

The final column in the Table gives the offense's randomization probabilities in equilibrium. Each entry is the probability that, in this specific situation, the offense will throw a pass with expected gain (if complete) equal to the yardage of the corresponding row. Notice that, consistent with column 3, the offense assigns zero probability to passes that, if complete, would be expected to gain 5 or 10 yards.

Parameter Estimation

There are 50 parameters in the model. However, many of them are known almost exactly, such as the probability of making an 80-yard field goal or the time required to call timeout after the whistle blows. Many others can be estimated in a straightforward way from observation and data, such as the probability of making a 40-yard field goal, or the time required to spike the ball after the whistle blows, or the probability of a sack.

This leaves 21 parameters: the pi* and αi for i = 1,…,10, and X. These are part of a stylized model of pass coverage and have no natural observable analogues. Since the pi* and αi should vary smoothly in i, we imposed restrictions on them that reduce the total number of free parameters to six. Let φ denote this vector of parameters. We estimated it from data.

During the 2002, 2003, and 2004 NFL seasons there were 182 times in which a team took possession with 2:00 or less remaining, trailing by one score. For each of these, we recorded the line of scrimmage, time remaining, and number of timeouts at the start of the possession, along with the outcome of the possession.

We will define variables yj and qj(φ) for observation j, where j = 1,…,182. Suppose first that for observation j, the offense trails by fewer than three points. Let yj equal 1 if that possession ends in a touchdown, and 0 if it ends with neither a touchdown nor a field-goal attempt. If that possession ends in a field-goal attempt, let yj be the probability of making a field goal from that distance. Let qj(φ) be the probability (from the model) that the offense wins the game, given the state variables at the start of the possession.

If the offense trails by exactly three points at the start of the possession, yj and qj(φ) are defined the same way, except that if the possession culminates in a field-goal attempt, yj is half the probability of making a field goal from that distance (because a field goal leads to overtime).

Finally, if the offense trails by more than three points, let yj equal 1 if the possession ends in a touchdown and 0 otherwise, and let qj(φ) be the model probability of a touchdown, given the state variables at the start of the possession.

One can estimate φ by maximizing

     F(φ) = ∑j   yj log qj(φ) + (1 − yj) log(1 − qj(φ)).

(Notice that if, contrary to fact, we observed only who won the game, then for every j we could let yj equal 1 if the offense wins the game and 0 otherwise, and let qj(φ) be the model probability of winning the game. F(φ) would then the logarithm of the likelihood function, and the φ that maximizes F(φ) would be the maximum-likelihood estimator. But since we have information about how the game was decided, the estimator we are using is more efficient. Intuitively, the winner of the game will sometimes depend on the outcome of a field-goal attempt of a particular distance, or the outcome of a two-point try, or the outcome of overtime. But none of these contains information about φ.)


Copyright © 2005 by William S. Krasker