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November 17, 2005

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Model-Independent Results Early in the Game

On the opening drive of the game between New Orleans and Arizona in Week 4 of the 2004 season, New Orleans reached 4th-and-goal at the Arizona 1-foot line, and had to decide whether to go for the touchdown or settle for a field goal. We analyzed this decision using the footballcommentary.com Dynamic Programming Model (FCDPM), and found that going for it is by far the better choice.

In this article we will explain why no model is necessary to analyze New Orleans's decision and certain other decisions that arise early in the game. The reason is that near the start of the game, a team's win probability is a smooth function of their lead, and hence approximately linear over a small range of leads. The smoothness, in turn, arises because there are a large number of possessions still to come. Thus, the considerable uncertainty about how the rest of the game will unfold—a common excuse for kicking—is actually the reason why it's right to go for it.

Even near the start of the game, most decisions require a model, though not necessarily one as elaborate as the FCDPM. In the final section we will determine what conditions have to hold in order to be able to analyze decisions using an "expected points" calculation augmented by an adjustment for field position.

Win Probability vs. Lead

We begin by examining graphs of win probability versus lead at different stages in the game. Figure 1 shows the graph at the end of the 4th quarter. If your lead is positive, your win probability is 1. If your lead is negative (i.e. you trail), your win probability is 0. If the game is tied it goes to overtime. Provided that the teams are roughly evenly matched, the win probability should then be about ½, which is the value shown in the graph.

Figure 1
Scatter Plot
Win probability as a function of
lead, at the end of the 4th quarter.
 
Figure 2
Scatter Plot
Win probability as a function of
lead, at the end of the 3rd quarter.
 
Figure 3
Scatter Plot
Win probability as a function of
lead, near the start of the game.
 
Figure 4
Scatter Plot
Win probability as a function of
lead, near the start of the game.

The graph in Figure 1 is not smooth. By this we mean that the incremental value of a point changes abruptly. For example, going from a lead of 0 to a lead of 1 increases the win probability by ½, but going from a lead of 1 to a lead of 2 leaves win probability unchanged.

Figure 2 graphs win probability as a function of lead at the end of the 3rd quarter. The graph was generated using the footballcommentary.com Dynamic Programming Model (FCDPM) assuming that the team has 1st-and-10 at their own 20-yard line. However, we are concerned here not with the specific numbers, but rather with the overall shape of the graph, which would be similar even the Model's parameters were altered. The graph is considerably smoother than in Figure 1, simply because there is a likelihood of additional scoring before the end of regulation time.

Finally, Figure 3 graphs win probability versus lead near the start of the game, again using the FCDPM. As we will explain later, the graph is very smooth because the remainder of the game will have a large number of possessions, each of which may or may not result in a score. The incremental value of a point changes very little as the lead changes, particularly over the central part of the graph, where the lead isn't too large in absolute value. Figure 4 shows in more detail the central part of Figure 3; over this small portion, the graph is very close to linear.

Implications of Linearity

Before explaining why the graphs in Figures 3 and 4 have the observed form, we will turn to the implication of having win probability be a linear function of lead (over the relevant range). Holding the other state variables constant, we can write

(1)
win probability = α + β L

for some constants α and β, where L is lead. Suppose a team with a current lead of L has two options, one of which scores Xi points with probability pi (for i=1,…,m), and the other scores Yj points with probability qj (for j=1,…,k). Then under the first option, the team's win probability is

(2)
i pi(α + β (L+Xi)) = α + β L + β ∑i piXi

since the pi sum to 1. Similarly, under the second option, the team's win probability is

(3)
α + β L + β ∑j qjYj.

The choice between the two options depends entirely on whether the expected points i piXi for the first option exceed the expected points j qjYj for the second option. Neither the lead L nor the parameters α and β enter into the decision. To summarize: If win probability is a linear function of the lead (holding the other state variables constant), then a decision that affects the score but not the other state variables can be made by maximizing expected points.

Strictly speaking, of course, the only decision that affects the score while leaving the other state variables constant is the choice between a kicked extra point and a two-point try. In this case, the arguments above explain why it is generally correct to kick the extra point early in the game. The kick gives one point virtually for certain. A attempted two-point try gives 2 points with probability p and 0 points with probability 1−p, and hence 2p expected points. Generally p is less than ½, so we can conclude that the kicked extra point is preferred early in the game—without reference to any model.

Of course, it's possible that for some teams against some opponents, the probability of a successful two-point conversion exceeds ½. When that's the case, it's actually correct to go for two in the early part of the game.

With a slightly more elaborate argument, we can also show that near the beginning of the game, a team facing 4th-and-goal very near the opponent's goal line should go for the touchdown. From the 1-yard line, for example, a team will have around a 0.6 probability of scoring. Hence going for it gives 0.6 × 7 = 4.2 expected points, compared to 3 expected points from an attempted field goal. It would therefore be right to go for it if all other things were equal, for example if the rules required the team to kick off to the opponents regardless of the outcome of the 4th-down play. Of course, other things are not equal; if the team goes for the touchdown and fails, the opponents are left with very poor field position. But this only increases the desirability of going for it. In short, as long as the probability of scoring a touchdown exceeds 3/7, one can conclude without reference to any model that it's right to go for it in the early stages of the game.

This point deserves some emphasis. Even among people who accept that it's right to go for the touchdown, it's sometimes thought that the reason has to do with leaving the opponents backed up near their own goal line if you are stopped. But when you are very close to the opponent's goal line—say a foot away—the most important reason to go for it is more mundane: 7 points are much better than 3, and you have a high probability of getting the 7. Indeed, using the FCDPM, we estimate that if the probability of scoring the touchdown is 0.65, only 20% of the benefit of going for it comes from field position. The rest can be regarded as coming from expected points. Again, this result applies only in the early part of the game.

An Explanation for the Smoothness

Denote the teams as Team 1 and Team 2. The graphs in Figures 1 through 4 all have the properties of a distribution function, which is of course what they are: they plot the probability that, during the remainder of the game, Team 2's points minus Team 1's points will be less than Team 1's current lead. The smoothness of the graph in Figure 3 results from the large number of remaining possessions, and the Central Limit Theorem.

Formally, suppose it's early in the game, and let L be the number of points by which Team 1 leads. Suppose there will be n more possessions in the game, starting with a possession for Team 1, and let Si be the number of points scored on the ith of those possessions. Of course, n is a random variable. Then Team 1 wins in regulation if

(4)
i (−1)i Si < L,

where the summation runs from 1 to n. Now, the Si are not strictly independent. For example, a team that trails by 4 points in the closing seconds would never attempt a field goal. But the dependencies, and the disparities in scale, among the Si are small enough that the conditional distribution for the left side of equation (4), given n, should be approximately normally distributed by the Central Limit Theorem. The marginal distribution for the left side of equation (4) should therefore resemble a mixture of normal distributions. This suggests that the graph in Figure 3 should approximate the distribution function for a mixture of normal distributions. But the exact shape of the distribution doesn't matter; all that we require is that it be smooth.

Simplified Models

One interesting question is whether, early in the game, one can dispense with a full model like the footballcommentary.com Dynamic Programming Model (FCDPM) and use a simpler analysis based on expected points and an adjustment for field position. This was the approach introduced by Carter and Machol1 and refined by Romer. In a slightly simplified form, in which a change of state occurs at the end of a possession rather than whenever a first down is made, those models can be described as follows. 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 Vi denote the "point value" of gaining possession at field position i, which is assumed to be independent of time. Let pi be the expected points scored on a possession that begins at field position i. Finally, let cij be the probability that, following a possession that begins at field position i, the opponents begin their next possession at field position j. The expected points pi and transition probabilities cij can be estimated from data. Now, Vi equals the expected points scored on the possession, minus the weighted average of the "point values" of the field positions at which the opponents could take over on their subsequent possession:

(5)
Vi = pi − ∑j cij Vj.

Taking the pi and cij as given, the equations of this form for i=1,…,n comprise n equations for the n unknowns V1,…,Vn.

Equation (5) is superficially very similar to the equation for win probability in playoff sudden-death overtime, which we derived in a previous article. However, the V's in equation (5) are not win probabilities. In addition, in playoff sudden-death overtime, the situation really is independent of both time and the score. During regulation time, however, a stationary solution of the sort described by equation (5) can't really be correct. Nevertheless, under certain assumptions that are implicit in the Carter-Machol-Romer approach, their model can be used for the early part of the game.

Following the notation in the FCDPM, let R(t,L,y,g) denote Team 1's win probability at time t if Team 1 leads by L points, and Team g has 1st-and-10 at their field position y. Now, early in the game the win probabilities should depend only slightly on t, and will be nearly linear in L, so we can rewrite the win probability as

(6)
α(y,g) + β(y,g) L

for some functions α and β. If we make the additional assumption that β is actually a constant function, we can rescale by dividing through by β without affecting any decisions. This gives

(7)
α(y,g)/β + L.

Since the units for β are win probability per point, the units in expression (7) are points. The term α(y,g)/β can be thought of as representing the effect of field position. Any decision that a team makes is associated with a joint distribution for (L,y,g), and the win probability corresponding to the decision is proportional to

(8)
E[α(y,g)/β + L] = E[α(y,g)/β] + E[L].

Therefore, under the assumptions just described, the objective function associated with a decision reduces to expected points, plus the expectation of the field-position adjustment. This may be a good approximation early in the game, although for the bulk of the game one needs a more complete model like the FCDPM.


1 Virgil Carter and Robert E. Machol, "Operations Research on Football," Operations Research, Vol. 19, No. 2 (Mar.–Apr., 1971) 541–544.

Copyright © 2005 by William S. Krasker