Opportunity cost is an essential concept in sports. It is defined as "the evaluation placed on the most highly valued of the rejected alternatives or opportunities." This is of great practical significance : a player must select the best of multiple options in order to have the greatest chance of success, and in doing so, must reject or negate options that may be nearly as viable. Essentially, each decision in a game is a gamble, and opportunity cost represents the ante.

In football more than in any other sport, opportunity cost is of heightened practical significance. Each side selects a play a priori, and then executes it faithfully after the snap. Moreover there are dozens or, potentially, hundreds of plays from which to select. As such, the pre-snap decision is of paramount importance, making the opportunity cost critical to consider.

In football, the opportunity cost would represent the number of yards which could be attained on any play if the correct play was chosen and if the teams were of equal quality - essentially what the offense can expect to get and what the defense can expect to give up. It represents a low-risk, low-reward situation in which the offense attains a small goal that is not able to be defended, but is consequently insufficient to attain victory. The offense must sacrifice this insufficient but quantifiable ante in order to achieve their greater goal.

As such, the opportunity cost essentially represents undefended territory.

The further benefit to modeling football plays as gambles against an ante is that they are consequently divided into failures or successes with this ante as a threshold. A play stopped for no gain has a negative contribution toward victory. As such, it should have a negative valuation. If we model plays simply as their raw yardage, then plays resulting in zero-yard gains are de facto evaluated as having zero contribution to victory. This is false.

Prior attempts have been made to include opportunity cost in decision analysis. This line of thinking is evident in the use of the Sharpe ratio in evaluating football players' respective performance. The Sharpe ratio was developed as a part of modern portfolio theory, in order to evaluate different investments' relative return when related to their total volatility, and against a threshold of minimum acceptable return :

Where:

Rp = Portfolio Return

Rf = Risk free rate

σp = Standard deviation of portfolio returns

Applications of the Sharpe ratio typically take two yards as the risk-free return. This is logical, given that a quarterback dive, fullback dive, or receiver slant (or some other play) should be able to achieve a two-yard gain in most situations, but also given that an unvarying two yards per play will not result in attaining a first down, and as such will not result in victory.

However, this static value does not fully describe the pre-snap decision. Opportunity cost in football is dynamic, dependent both on down and on distance. On 3rd and 20, the defense is focused on defending long gains, and as such the undefended territory is greater. On 2nd and 2, the defense is focused on defending short gains and the undefended territory is effectively zero.

Other statistics do not measure opportunity cost directly, but do account for it at least indirectly. The best-described such statistic is success rate, developed by Football Outsiders. A play is successful if it gains 50% of remaining yardage on 1st down, 70% on second down, and 100% on 3rd or 4th. These values define a threshold of success, which implies a threshold of minumum acceptable return. Moreover, they describe a dynamic relationship between down-and-distance and success. However, that threshold is not specifically defined.

I suggest a dynamic opportunity cost, based on the football application of the Sharpe ratio and the work of Football Outsiders in estimating down-dependent success thresholds :

1. Given : success rates are assumed to be 50% of remaining yardage on 1st down, 70% on second down, and 100% on 3rd or 4th.
2. On 1st and 10, success is defined as 5 yards.
3. Given : in football applications of the Sharpe ratio, the static minumum acceptable return is 2 yards.
4. 1st and 10 is probably the most applicable down for an ante of 2. 2 is 40% of 5
5. On 1st and 10, opportunity cost is 2, or, 40% of the remaining yardage.
6. Therefore, the linear estimation of opportunity cost as 40% of the yards needed for success on the given down.
7. Assumed : opportunity cost does not decay in a purely linear manner.
8. Assumed : opportunity cost increases most rapidly at low values for yards remaining, and increases most slowly at higher values.
9. Therefore, I choose the equation : DOC = a * x ^ (1 / m) - b.
10. Brute force solving yields m = 78.46872, with varying values of a and b for each down-and-distance.
11. Given : opportunity cost must decay to zero.
12. The remaining yardage at which this occurs is likely 2 yards, when the offense's and defense's options are restricted.

This yields the following set of equations, where x is distance to go for a first down :

• 1st down : maximum of { 0 , 95.67 * x ^ (1 / 78.47) - 96.51 }
• 2nd down : maximum of { 0 , 133.93 * x ^ (1 / 78.47) - 135.12 }
• 3rd or 4th down : maximum of { 0 , 191.33 * x ^ (1 / 78.47) - 193.0 }

Graphed :

Even though football statistics are reported in integer values, I permit an ante that is greater than 0 and less than 1. In some respect this is more true to the action on the field, wherein ball position is actually a continuous variable and not segmented in 1-yard increments.

The advantage of using a dynamic opportunity cost here is that instead of measuring the team's success rate as distinct from the yards gained any particular play, the degree to which a play was successful is embedded into the yard total for that play. Integrating these measurements produces a unified statistic and a more universal method of comparison