Optimal Decision Making

The purpose of our strategy discussions is to determine the optimal course of action for a particular set of game conditions.  Mathematics aside, this is a thorny question in part because we must decide what constitutes an optimum.  (In other words, before deciding which strategy is best, we must agree on the definition of the word "best.")  In our other essays, we chose to define the optimum as the strategy which maximizes the team's chance of eventually winning the game.  However, other criteria are possible (and in many cases easier to analyze).  One such alternative is for a coach to maximize the expected point differential (defined as the difference between his team's score and his opponent's score).  How do these criteria compare?

Initially, maximizing expected point differential and probability of winning appear to be two ways of saying the same thing.  Positive point differentials result in wins; negative point differentials result in losses.  How different can they be?

Example 1: Trailing by four points with 0:01 remaining in the fourth quarter and the ball on his opponent's 20 yard-line, every football coach would call a play designed to score a touchdown because it gives his team the best chance of winning.  Kicking the field goal would maximize his team's expected point differential, but guarantees a loss.

Example 2: Suppose our hypothetical coach faces the same decision at the end of the first half, rather than at the end of the game.  Because he has the entire second half to come back, he would likely attempt the field goal because it's more likely to yield points than trying for the endzone.

Marginal Value of a Point Scored

The two examples illustrate that in some cases maximizing expected point differential and probability of winning are compatible while other times they are at odds.  The critical difference between the two scenarios was the time remaining in the game.  Maximizing expected point differential implicitly assumes that the value of an additional point is constant.

The first example suggests that late in the game, the value of an additional three points is not constant; instead the value of a field goal depends on the current point differential.  Trailing by four points, an additional three is worthless, but had the margin been only two, a field goal would win the game and the coach's decision would certainly be different.  In the second example, however, the point differential weighs less on the coach's decision because the coach knows his team has many possessions remaining to regain the lead.

The graph summarizes these observations.  For point differentials between -10 and +10, the probability of winning is given for a range of times remaining.  (Probabilities were approximated by generating 10,000 simulated game conclusions for each scenario using the SportsQuant football simulator.)  Early in the game (represented by "cold" colors like purples and blues), the plots of probability of winning vs. margin are approximately linear.  This implies that early in the game the value of an additional point (the slope) is constant and therefore doesn't depend on the point differential.  Late in the game (represented by "warm" colors like reds and oranges) the value of an additional point is distorted because there is little time remaining for a potential lead change.  Therefore, at the end of the game, additional points are worth a great deal (i.e. have a steep slope) when the current point differential is near zero (and the lead can change on a score) but are relatively worthless (i.e. have slope near zero) otherwise.

When the relationship between probability of winning and point differential is linear (essentially the first three quarters, represented by purples and blues), the value of an additional point is constant with respect to point differential.  This means that when there is enough time left, the difference between a four-point lead and a three-point lead is the same as between five-point lead and a four-point lead -- something which is clearly not true late in the game.  However, even when the value of an additional point is constant with respect to point differential, it is not constant with respect to time.  This is seen in the increasing slopes of the lines as the time remaining decreases.  In other words, other things being equal, the value of an additional point late in a close game is worth more than an additional point earlier in one.

Conclusions

Early in the game, the value (measured in increased probability of winning the game) of an additional point is constant.  (To paraphrase the Declaration of Independence: All points are created equal.)  At the end of the game, however, additional points are worthless except when the current point differential is near zero, in which case they are extremely valuable.  (To paraphrase George Orwell's "Animal Farm": All points are created equal, but some are more equal than others.)

The plot shows that even as late as the end of the third quarter, the value of a point is approximately constant with respect to point differential.  This suggests that maximizing expected point differential and maximizing probability of winning are equivalent goals for most of the game.  Furthermore, since calculating expected points scored is easier than calculating expected probability of winning, this realization simplifies analysis.  However, late in the game -- particularly in a close game -- the time remaining must be considered and winning, not points, becomes the relevant criterion.

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Last modified: 01/05/2008