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Models for Sudden-Death Overtime

In this article we present models that can be used to make and analyze decisions during sudden-death overtime. We describe separate models for regular-season overtime, which lasts at most one period, and playoff overtime, which continues until a team scores. We give an example of the application of each model.

Playoff overtime is relatively straightforward to model. For regular-season overtime, the main complication is the possibility of a tie. We will see that the value a team places on a tie, relative to a win or a loss, is a necessary input to the model, and can determine a team's optimal strategy. For example, in Week 17 of the 2004 season, St. Louis could make the playoffs with a win, but would be eliminated with either a loss or a tie. We will examine how those circumstances should have influenced St. Louis's decision to punt with 8:19 remaining in the overtime session.

Overtime in Regular-Season Games

Model Overview

Our model for regular-season overtime is an adaptation of the footballcommentary.com Dynamic Programming Model (FCDPM). Overtime allows several simplifications. There is no need to model kickoffs or extra points. Since the score is tied throughout, the state variables are just the time remaining, the field position, and which team has the ball. The only decision that is explicit in the model is whether to use a regular tempo or a hurry-up tempo, which is modeled exactly as in the FCDPM. However, the model can also be used for fourth-down decisions, and we anticipate that those will be the most common applications.

The FCDPM doesn't include overtime explicitly, and it ignores the possibility that a game will end in a tie by assigning each team a 0.5 probability of winning if the score is tied at the end of regulation. This assumption conveniently eliminates any ambiguity about the correct decision criterion: when there is a decision to be made, and winning and losing are the only possible outcomes for the game, every team prefers the choice that gives the higher probability of winning.

But once a game is several minutes into overtime, the possibility of a tie is no longer small enough to be ignored. The value of a tie can depend on a team's position in the standings, particularly late in the season. In extreme cases a tie might be almost as good as a win, or (as in the example cited in the introduction) virtually equivalent to a loss. From this it follows that a decision that is correct for one team might be incorrect for another—even without invoking differences in the teams' abilities.

To see how the value of a tie affects strategy, suppose the game is well into the overtime period, and you have to decide whether to punt or go for it on 4th down. Suppose that the probabilities for a win, loss, or tie associated with these choices are as shown in the table on the left. Going for it gives a higher probability of winning, but also a higher probability of losing. There is no universally correct decision. If a tie is like a loss, you go for it, whereas if a tie is like a win, you punt.

The proper way to assign a numerical value to a tie is with a device familiar to people who have studied expected utility. Denote the two teams by 1 and 2. For some probability v₁, team 1 will be indifferent between (a) a tie, and (b) winning with probability v₁ and losing with probability 1−v₁. This probability v₁ represents the value of a tie to team 1. Similarly, for some probability v₂, team 2 will be indifferent between (a) a tie, and (b) winning with probability v₂ and losing with probability 1−v₂. This v₂ is the value of a tie to team 2. In the dynamic program, if a tie occurs, we assign win probability v₁ to team 1 and win probability v₂ to team 2. Backward induction in the dynamic program then proceeds as in the FCDPM. Of course, the "win" probabilities computed at the various states are not really the probability of winning, because they incorporate a fraction of the probability of a tie. We will therefore call them pseudo-win probabilities. However, we show in a mathematical appendix that maximizing pseudo-win probability is the correct decision criterion.

In the NFL's computation of win percentage, a tie counts as half a win and half a loss. Therefore, a coin flip to determine the winner would have the same effect as a tie on a team's expected win percentage. It follows that if the number of games remaining in the season were very large, the value of a tie (as defined above) would be about 0.5. Of course, the season has only 16 games rather than a very large number, so in applications of the model we have to think about the correct value for a tie.

The next section contains a more formal description of the model for regular-season overtime. Many readers may prefer to skip directly to the example.

Model Description

Let g=1 denote team 1, and let g=2 denote team 2. Suppose that with t time periods remaining in overtime, team g has 1st and 10 at field position y. (Field position is modeled exactly as in the FCDPM.) Then the state is represented by (t,y,g). Let R1(t,y,g) denote team 1's pseudo-win probability, and let R2(t,y,g) denote team 2's pseudo-win probability. We will describe how to compute these pseudo-win probabilities by backward induction, assuming we have already computed the teams' pseudo-win probabilities at every state (t',y',g') with t' < t.

If the team with possession chooses the regular tempo, let p_s(t,y) denote the probability that they score on the possession, let p₀(t,y) denote the probability that the overtime period ends during the possession, and let p(k,y',t,y) denote the probability that after k time periods, the opponents gain possession at field position y'. If the team with possession chooses the hurry-up tempo, let q_s(t,y) denote the probability that they score on the possession, let q₀(t,y) denote the probability that the overtime period ends during the possession, and let q(k,y',t,y) denote the probability that after k time periods, the opponents gain possession at field position y'.

If team 1 has possession and chooses the regular tempo, their pseudo-win probability at state (t,y,1) is

     regular = p_s(t,y) + p₀(t,y) v₁ + ∑_k,y' p(k,y',t,y) R1(t−k,y',2),

whereas if they choose the hurry-up tempo, their pseudo-win probability is

     hurry = q_s(t,y) + q₀(t,y) v₁ + ∑_k,y' q(k,y',t,y) R1(t−k,y',2).

Of course, team 1 will choose the tempo that yields the higher pseudo-win probability, and hence

     R1(t,y,1) = max{regular, hurry}.

Notice that if it's optimal for team 1 to use the regular tempo, then

     R2(t,y,1) = p₀(t,y) v₂ + ∑_k,y' p(k,y',t,y) R2(t−k,y',2)

whereas if it's optimal for team 1 to use the hurry-up tempo, then

     R2(t,y,1) = q₀(t,y) v₂ + ∑_k,y' q(k,y',t,y) R2(t−k,y',2).

Analogous formulas apply when team 2 has possession.

Those who want more details about the model can consult the MATLAB^® source code dynprogot.m, which uses the routines lognorparams.m and lognordensity.m.

Example

As an illustration, consider the 2004 Week 17 game between the Jets and St. Louis, which we mentioned in the introduction. St. Louis could make the playoffs with a win, but would be eliminated with a loss or tie. By the time the overtime began, the Jets were locked into the #5 playoff seed. We will refer to the Rams as team 1, and the Jets as team 2.

The decision we will examine came with 8:19 left in overtime, with the Rams facing 4th and 2 at their own 31-yard line. Not surprisingly, they decided to punt. We will analyze this decision two ways. First, we will naively set both v₁ and v₂ equal to 0.5. In this case, if the Rams punt, their pseudo-win probability is 0.424. If they go for the first down, their pseudo-win probability is either 0.6 or 0.197, depending on whether or not they succeed. Since the probability of success is about 0.54, their pseudo-win probability if they go for it is 0.54 × 0.6 + (1 − 0.54) × 0.197 = 0.415. Under these assumptions, then, punting is preferred.

Of course, in St. Louis's actual situation, a tie is nearly equal to a loss (although not exactly equal; they still prefer a tie to a loss). So we will repeat the analysis, keeping v₂= 0.5 but setting v₁= 0.05. With this change, St Louis's pseudo-win probability if they punt is 0.313. If they go for the first down, their pseudo-win probability is either 0.493 or 0.145, depending on whether or not they succeed, and hence their pseudo-win probability if they go for it is 0.54 × 0.493 + (1 − 0.54) × 0.145 = 0.333. Therefore, with a more appropriate assumption about the value of a tie, going for it is preferred.

Overtime in Playoff Games

Model Description

There are no ties in playoff games. The teams change goals at the end of each 15-minute overtime session, but play continues until a team scores.

We could use the model for regular-season overtime, described above, by letting t be very large. Instead, we will take a simpler approach that explicitly uses the fact that in playoff overtime, win probabilities are independent of time.

Suppose there are n field positions. As in the FCDPM, we label these from the perspective of the team that has possession. (Hence field position i when team 1 has the ball is the same spot on the field as field position n − i + 1 when team 2 has the ball.) Let V_i denote team 1's win probability if they gain possession at field position i, and let W_i be team 1's win probability if team 2 gains possession at field position i. Let s_i be the probability that a possession that begins at field position i culminates in a score, and let c_ij be the probability that a possession that begins at field position i culminates with the opponents taking possession at field position j. The probabilities s_i and c_ij are estimated from data. Taking those probabilities as given, we want to solve for the win probabilities V_i. Notice that for a given i,

     V_i = s_i + ∑_j c_ij W_j.

But since W_j = 1 − V_j and s_i + ∑_j c_ij = 1, we have

     V_i = 1 − ∑_j c_ij V_j.

The equations of this form for i=1,…,n comprise n equations for the n unknowns V₁,…,V_n, and have a unique solution.

Since the win probabilities depend only on field position, they are easy to display. In the graph on the left, the horizontal axis is the yard line at which a team has 1st and 10 in playoff overtime. The vertical axis is their estimated probability of winning the game.

Example

As an example, we will examine San Diego's decision to punt on 4th and a foot, at their own 35-yard line, during the overtime of their 2004 wild-card game against the Jets. If the Chargers punt, they can expect the Jets to begin their possession at their own 25-yard line. At that point the Jets have about a 0.56 probability of winning, so that San Diego's win probability is 0.44. If the Chargers go for it and succeed, they have a first down at around their own 37-yard line, where their win probability is 0.63. Finally, if the Chargers go for it and fail, the Jets take over near San Diego's 34 yard line; the Jets then have a 0.81 win probability, so that San Diego's probability of winning is 0.19.

Let α denote San Diego's probability of picking up the first down. Then the Chargers should go for it if 0.63α + 0.19(1−α) > 0.44. It's easy to check that this is the case provided α exceeds 0.57. Since the actual probability of gaining a foot is larger than 0.57, San Diego should have gone for it. In fact, if the probability of making the first down is 0.7, then San Diego's win probability if they go for it is 0.50, which is substantially better than the 0.44 they obtain by punting.

Copyright © 2005 by William S. Krasker