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June 14, 2005

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The Clock-Management Value of Timeouts

In this article we present a model for estimating the clock-management value of a timeout—the value of the option to stop the game clock in order to preserve time. We give examples to show how the model can be used to make and evaluate decisions.

The proper way to use timeouts for clock management is fairly clear: You use them toward the end of the game, on defense, if you're behind. So it's natural to ask why one needs to know the clock-management value of a timeout. The answer is that teams sometimes use timeouts for other reasons, such as to avoid a delay-of-game penalty, to discuss strategy, to challenge an official's ruling, or to try to "ice" the opponent's field-goal kicker. Often one can estimate the benefit of those actions, expressed as the increase in the probability of winning the game. However, the benefit has to be weighed against the cost of using up a timeout. The model presented here provides a way to estimate the most important component of that cost.

Overview of the Model

This section contains an overview of the main features of the model. We defer a more detailed description until the final section. The model is a dynamic program. Readers who are not familiar with dynamic programming might want to consult the description of the footballcommentary.com Dynamic Programming Model (FCDPM). Actually, those who are familiar with the FCDPM might wonder why we haven't simply generalized it to include state variables for the number of timeouts remaining for each team, rather than build a separate model. The reason is computational time. Since for each team there are 4 possibilities for the number of remaining timeouts (0, 1, 2, or 3), the number of states would increase by a factor of 16. The number of calculations required at each state would also be much larger. Consequently, the run time for the model would increase by a factor of about 100. This is out of the question, because the FCDPM is already near the limit of tolerable computational time. So, in order to incorporate timeouts, some state variable in the FCDPM has to be removed. Since the score and the time remaining in the game are obviously essential if the goal is to value timeouts, we have eliminated the state variable for field position. Strictly speaking, this implies that the results apply only for "typical" field position. However, we will show with an example how it is often possible to handle other cases.

The model covers only the second half of the game. Certainly, timeouts have clock-management value in the first half as well. However, the model presented here relies heavily on our detailed analysis of the two-minute drill. Currently we have no analogous model for the closing minutes of the first half.

We evaluate a team's probability of winning the game when they have gained possession, or when they have scored a touchdown but have not yet attempted the try. In either case, the state is characterized by the time left in the game, the score, the number of timeouts remaining for each team, and which team has the ball.

Our primary interest is in determining how a team's probability of winning the game varies according to how many timeouts they have, holding the other state variables constant. The difference in win probability from having one more (or one fewer) timeout is what we call the value of a timeout.

In the final 2:00 of the game, the win probabilities are derived from our model for the two-minute drill, which already incorporates timeouts. If the team with possession trails by 8 points or less, we take their win probability directly from that model. If the team with possession leads by 8 points or less, we assume that they run the ball—forcing the defense to use its remaining timeouts—and have a specified probability of making a first down. Each play uses time, and the offense can run an additional 0:40 off the clock between plays once the defense is out of timeouts. If the offense makes a first down, we assume they win the game. If instead there is a three-and-out (and there is still time on the clock), the trailing team gains possession with no timeouts, and as before we take their probability of winning from our model for the two-minute drill. In the last 2:00, we assume that if a team trails by more than one score, their win probability is zero.

If the game is tied in the last 2:00, we assume that the scoring probability for the team with possession equals the scoring probability for a team that trails by one point in the model for the two-minute drill. If they don't score, we assume the game goes to overtime, in which the teams are equally likely to win. (This formulation presumably overstates the win probability for the team with possession, because teams play more conservatively when tied than when trailing. But alterations to these assumptions had minimal effect on the value of a timeout.)

With more than 2:00 left, we solve for the win probabilities by backward induction. At any state in which a team has just gained possession, we allow that team to choose from four different offensive tempos. For each tempo there are specified probabilities of scoring a touchdown or a field goal, and conditional distributions for the amount of time used on the possession, conditional on whether the drive culminates in a touchdown, a field goal, or no score.

The first tempo is the regular tempo. The second is an extreme slowdown in which we assume, essentially, that the offense runs the ball on every play. For the slowdown tempo, the specified probabilities of scoring a touchdown or field goal are much smaller than under the regular tempo. Against either of these tempos, if the defense has timeouts, they have the option of using one or more of them to stop the clock, saving up to 40 seconds for each timeout used. Against the slowdown tempo, if the defense does not use timeouts, then even if they force a three-and-out, at least 2:10 runs off the game clock before they get possession. The number of timeouts used by the defense is the number that maximizes their probability of winning the game.

The third tempo is a hurry-up. In this case the conditional distributions for the time of the possession are such that the possession takes no more than 3:00. The probabilities for scoring a touchdown or field goal are smaller than under the regular tempo, and, if less than 3:00 remains in the game, depend on the time remaining. If the offense has timeouts, they have the option of using one or more of them, saving 10 seconds for each timeout used. The number of timeouts used by the offense is the number that maximizes their probability of winning the game.

The fourth tempo is actually identical to the hurry-up, except that the probability of a touchdown is higher, and the probability of a field goal is zero. This choice is often optimal when the offense trails by more than a field goal late in the game.

At each state, the offense chooses the tempo that gives them the highest probability of winning the game.

At states in which a team has just scored a touchdown but has not yet attempted the try, that team either kicks the extra point or attempts a two-point conversion, depending on which yields the higher probability of winning the game.

Results of the Model

The table below contains a sample of the output from the model. The table entries are the difference in the probability of winning, for the team with possession, if they have two timeouts remaining rather than one. The entries are computed under the assumption that the team with possession has 1st and 10 at its own 30-yard line, and that the defense has two timeouts remaining. For example, suppose that the offense trails by 6 points at the beginning of the 4th quarter (15:00 remaining). Then according to the model, their probability of winning is 0.0109 higher if they have two timeouts remaining rather than one. Since the probability of winning is approximately linear in the number of timeouts remaining, the table entries can simply be regarded as the incremental value of an additional timeout.

                                TIME REMAINING
          5:00      10:00     15:00     20:00     25:00     30:00
        ----------------------------------------------------------
    8   | 0.0003    0.0010    0.0023    0.0031    0.0037    0.0039
    7   | 0.0007    0.0015    0.0024    0.0032    0.0036    0.0039
    6   | 0.0019    0.0056    0.0059    0.0058    0.0057    0.0055
    3   | 0.0020    0.0055    0.0066    0.0067    0.0066    0.0064
    1   | 0.0024    0.0070    0.0081    0.0079    0.0076    0.0072
L   0   | 0.0057    0.0072    0.0075    0.0073    0.0071    0.0068
E  -1   | 0.0353    0.0174    0.0131    0.0107    0.0093    0.0084
A  -3   | 0.0266    0.0139    0.0109    0.0094    0.0084    0.0078
D  -6   | 0.0238    0.0128    0.0109    0.0098    0.0090    0.0083
   -7   | 0.0140    0.0096    0.0088    0.0083    0.0077    0.0073
   -8   | 0.0170    0.0117    0.0100    0.0091    0.0084    0.0078
  -10   | 0.0120    0.0095    0.0084    0.0077    0.0071    0.0067
  -13   | 0.0079    0.0065    0.0061    0.0061    0.0060    0.0058
  -14   | 0.0041    0.0041    0.0048    0.0049    0.0050    0.0049
Graph of Win Probabilities With 5:00 Left

The graph on the right illustrates a different slice of the output from the model. The depicted situation is that the team with possession has 1st and 10 at its own 30-yard line with 5:00 remaining. The defense has two timeouts. The vertical axis is the win probability for the team with possession. The horizontal axes are the number of points by which they lead (−10 to 10), and the number of timeouts they have left (0 to 3). The win probability is of course an increasing function of the lead; and since a timeout is an option, the win probability is an increasing function of the number of timeouts as well. Win probability increases sharply as the lead increases from −1 to 1. Notice that the win probability is approximately linear in the number of timeouts remaining, and that the value of a timeout is visibly larger when the lead is −1 than when the game is tied.

If you trail by one score with 15:00 left, the clock-management value of a timeout is in the vicinity of 0.01. To put this in perspective, kicking an extra point instead of going for two when trailing by 5 points with 15:00 left reduces your win probability by about 0.013; and punting instead of going for it on 4th and 1 at your own 40-yard line, trailing by 3 points with 15:00 left, reduces your win probability by about 0.035.

Of course, for a team that trails near the end of the game, the clock-management value of a timeout can be substantially larger in certain circumstances, particularly if the opponents have the ball. In the final 2:00, one can work out the value using the tables associated with our model for the two-minute drill.

Applications of the Model

Our first example is from the game between Green Bay and Minnesota in Week 16 of the 2004 season. With 7:35 left in the 4th quarter, the Packers faced 3rd and 10 at their own 20-yard line, trailing 31-24. With the play clock winding down, Green Bay quarterback Brett Favre realized that he would not be able to run a play before the play clock expired. By failing to get off a play in the 40 seconds allowed, the Packers reduced their probability of winning the game; but since Green Bay has the option of using a timeout, the size of the reduction is the smaller of the value of a timeout and the value of 5 yards.

According to the footballcommentary.com Dynamic Programming Model, if the Packers make a first down, their win probability is 0.145. If they punt from their 20-yard line, their win probability is 0.077. If they punt from their 15-yard line, their win probability is 0.073. According to Aaron Schatz at Football Outsiders, the likelihood of making a first down is about 31% on 3rd and 10, and 22% on 3rd and 15. So Green Bay's win probability is 0.31 × 0.145 + (1 − 0.31) × 0.077 = 0.098 if they avoid the delay-of-game penalty, and 0.22 × 0.145 + (1 − 0.22) × 0.073 = 0.088 if they incur the penalty. The value of 5 yards is therefore about 0.01. The clock-management value of the timeout is also around 0.01, but since timeouts have uses other than clock management, the full value of the timeout is somewhat higher. The conclusion is that Green Bay should have let the play clock expire, and that by failing to get off a play, the Packers knocked 0.01 off their win probability.

Our next example is from the game between Cleveland and Baltimore in Week 9 of the 2004 season. The Ravens scored a touchdown with 7:03 left in the 4th quarter to take an 18-13 lead prior to the try, and naturally lined up to attempt a two-point conversion. The Browns then called their second timeout to talk things over.

The model explicitly computes win probabilities just before a try. According to the model, Cleveland's win probability at that point is 0.013 lower if they have just one timeout remaining rather than two. This is an estimate of the cost of calling the timeout—and is actually an underestimate, since it measures only the clock-management value. Cleveland's hoped-for benefit is a reduction in Baltimore's probability of success on the two-point conversion. However, calculations using the footballcommentary.com Dynamic Programming Model indicate that a 0.13 reduction in that success probability is required to produce a compensating 0.013 increase in Cleveland's win probability. It's questionable whether anything Cleveland could come up with during the timeout could make that large a difference.

Our final example, from the game between Kansas City and Tennessee in Week 14 of the 2004 season, shows how the model can sometimes be used even if the field position is very far from typical. Trailing 21-14 with 12:01 left in the 3rd quarter, Kansas City apparently scored on a 13-yard pass play. Tennessee challenged the ruling on the field, claiming that the ball carrier was down before the ball broke the plane of the goal line. Both teams had all of their timeouts.

If Tennessee doesn't challenge, the score is 21-20 pending the extra point. According to the model, Tennessee's win probability is 0.535. If they challenge but the ruling is upheld, the situation is the same except that they have only two timeouts. In this case their win probability is 0.528 (and hence the value of the timeout is 0.007). If the ruling is reversed, Kansas City has 1st and goal at the 6-inch line, and Tennessee's win probability will be either 0.535 or 0.691, depending on whether Kansas City ultimately scores a touchdown or is forced to settle for a field goal. Let p denote the probability that the ruling is reversed, and let q denote the probability that Kansas City ultimately scores a touchdown on the drive if the ruling is reversed. Then it is correct for Tennessee to challenge if

     (1 − p)0.528 + p[q0.535 + (1 − q)0.691] > 0.535.

Certainly q is very high, but even if it is 0.95, p must only exceed 0.48 to justify Tennessee's challenge. Based on the replay that was shown before they made their decision, the likelihood of a reversal was high. So it appears that Tennessee made the right decision—although this analysis ignores the value of timeouts for reasons other than clock management, and the direct cost of using up a challenge.

Description of the Model

This section contains a more complete description of the model. Those who are interested in the full details can peruse the MATLAB® source code dynto.m, which uses the routines lognorparams.m and lognordensity.m.

Denote the two teams by 1 and 2. Let R(t,d,T1,T2,g) be the probability that team 1 wins if team g gains possession with t time periods remaining, when team 1 leads by d points, team 1 has T1 timeouts, and team 2 has T2 timeouts. Let S(t,d,T1,T2,g) be the probability that team 1 wins if team g has just scored a touchdown and is about to attempt the try. Each time period in the model is ten seconds, and hence a timeout used on defense can save 4 time periods.

In the final 2:00 of the game, the values for R(t,d,T1,T2,g) are taken almost directly from our model for the two-minute drill. For a concrete example of how this is done, suppose team 1 gets possession with t time periods remaining, leading by d=8 points. The model assumes team 1 runs the ball three times, forcing team 2 to use its timeouts in the process. If team 1 makes a first down, their win probability is 1. If there is a three-and-out, team 2 gets the ball with t − 14 + 4×T2 time periods left (if this is positive) and no timeouts. Since the model for the two-minute drill incorporates field position, we assume team 2 gets the ball at their own 30-yard line. Let p3out denote the probability of a three-and-out, let p2pt denote the probability of a successful two-point conversion, and suppose that each team has probability 0.5 of winning in overtime. Then

     R(t,8,T1,T2,1) = 1 − p3out Q p2pt 0.5

where Q is the probability—from the model for the two-minute drill— that a team scores a touchdown after taking possession at its own 30-yard line, trailing by 8 points with t − 14 + 4×T2 time periods remaining, and with no timeouts.

With over 2:00 left, we compute the values of R(t,d,T1,T2,g) by backward induction. For each tempo, we specify the probability that the offense scores a touchdown or a field goal. For tempos 1 and 2, these scoring probabilities are independent of t (except that no score occurs if the game ends before the drive is complete). For the hurry-up tempos (3 and 4), the scoring probabilities are increasing linear functions of t if less than 3:00 (18 time periods) is left. Conditional on the tempo and on whether the possession results in a touchdown, field goal, or no score, there is a probability distribution for the number k of time periods consumed on the possession. (Strictly speaking, k is the number of time periods consumed if neither team uses a timeout). For the hurry-up tempos, k is bounded by min{t,18}. For each tempo, we use Bayes' rule to solve for the conditional probability of a touchdown or a field goal, conditional on the time consumed by the possession. Let pTD(k,i) denote the probability that the possession ends with a touchdown if the offense uses tempo i and the possession uses k periods; and similarly for pFG(k,i).

Again assume that team 1 has possession, and suppose first that the tempo is either i=1 (regular) or i=2 (slowdown). Then

     f(k,i,j) = pTD(k,i)   S( min{t − k + 4j, t−3}, d+6, T1, T2−j, 1)
         + pFG(k,i)   R( min{t − k + 4j, t−3}, d+3, T1, T2−j, 2)
         + (1 − pTD(k,i) − pFG(k,i))   R( min{t − k + 4j, t−3}, d, T1, T2−j, 2)

is team 1's win probability if team 2 uses j timeouts when the possession would otherwise have consumed k periods. Since team 2 will choose j so as to minimize team 1's probability of winning,

     F(k,i) = minj f(k,i,j)

is team 1's win probability if it selects tempo i and the possession uses k periods (in the absence of a timeout). Finally, letting p(k,i) denote the probability that a drive under tempo i uses k time periods,

     V(i) = ∑k p(k,i) F(k,i)

is team 1's win probability if they select tempo i. If the tempo is i=3 (hurry-up) or i=4 (hurry-up without field goals), the calculations are similar, except that it is team 1 that considers using timeouts rather than team 2, and they do so in order to maximize their win probability rather than minimize it. (And of course if i=4, pFG(k,i) = 0.) Hence if i equals 3 or 4,

     f(k,i,j) = pTD(k,i)   S( min{t − k + j, t−2}, d+6, T1−j, T2, 1)
         + pFG(k,i)   R( min{t − k + j, t−2}, d+3, T1−j, T2, 2)
         + (1 − pTD(k,i) − pFG(k,i))   R( min{t − k + j, t−2}, d, T1−j, T2, 2)

is team 1's win probability if they use j timeouts when the possession would otherwise have consumed k periods, and

     F(k,i) = maxj f(k,i,j)

is their win probability if they select tempo i and the possession uses k periods (in the absence of a timeout). As before,

     V(i) = ∑k p(k,i) F(k,i)

is team 1's win probability if they select tempo i. Finally, since team 1 chooses the tempo that gives the highest win probability,

     R(t,d,T1,T2,1) = maxi V(i).

The computation of the win probability prior to a try is trivial by comparison. We have

     S(t,d,T1,T2,1) = max{ p2pt R( t−1, d+2, T1, T2, 2) + (1−p2pt) R( t−1, d, T1, T2, 2) ,
         p1pt R( t−1, d+1, T1, T2, 2) + (1−p1pt) R( t−1, d, T1, T2, 2) }.


Copyright © 2005 by William S. Krasker