The Price of a Point

Copyright 2003 by Paul Niquette.  All rights reserved.

irst a few observations about the game of football relevant to this puzzle:
  • Missing a one-point conversions is an extremely rare event, which seems consistent with its one-point value.
  • Two-point conversions are also rare and probably impose twice the risk if not difficulty.
  • Safeties result from happenstance more than stratagems, so two points seems reasonable.
  • Most scoring opportunities in football produce either 7 or 3 points.  Do these make sense?
Our observations lead us to expect that at every scoring opportunity, a team will most likely score either seven points (a touchdown followed by a successful conversion) or three points (a field goal). Many unknowns will influence those decisions as the teams interact with each other.
Time out.  Whereas touchdowns are often scored in a sequence of offensive plays following a kickoff return, touchdowns can also result from a turn-over (pass-interception or fumble) anywhere on the field.  Field goals, on the other hand, can be scored only by the team in possession of the ball -- the offensive team -- and only from some limited distance from the goal line.  Let us embody these observations into assumptions...
Assumption #1 That scoring a touchdown requires an offensive advance of some typical distance XTD to the goal line from the point of taking over possession of the football at XTP in the offensive team's own territory. 
It is evident that  XTD = 100 - XTP .

Assumption #2 That scoring a field goal requires an offensive advance from the point of taking over possession of the football at XTP to within some distance XFG  from the goal line. 

The requisite drive for a field goal would be given by 100 - XTP - XFG .

For insights into these two assumptions, we might plug in some numbers -- "foozling" is the technical term.  Let XTP = 20 yards.  To make a touchdown, after taking possession (either from a kick-off or turn-over), a team needs to drive some 80 yards (XTD = 100 - 20).  The team puts seven points on the scoreboard, with the conversion.  For a field goal, we might assign a value of XFG = 30 yards, which requires an advance of only 50 yards -- but scores only three points.

Taking simple ratios, we see that for a touch down, a team must move the ball 80 / 7 = 11.4 yards-per-point, and for a field goal, the team must move the ball 50 / 3 = 16.7 yards-per-point.  Thus, the "price" for each field goal point in yardage is a lot higher than that for a touchdown.  To the sophisticated puzzle solver this does not make sense.  Thus, the solution is...

he two scoring options ought to be brought into balance, don't you think?  Football is kind of a metaphor for the "territorial imperative."  At four points, say, the typical field goal would cost 12 yards of offensive advance per point, about the same territory as a touchdown.  Just a suggestion.

Whereas touchdown points are easier to get than field-goal points, there is another possible outcome in every offensive drive -- the much dreaded fourth down, particularly with long yardage.  When that arises outside of field goal range, a punt is the most common play, which relinquishes control of the ball.  A fourth down inside field goal range produces a likely three-point score.  Using the present scoring units of football, it should be possible to make reasonable wagers on which type of score will be made at the next opportunity during a game.

If a sophisticated solver gets curious, he or she might test the sensitivity of his or her wagering decisions to various numerical values for those distances (XTD and XFG).  For now, let's just apply the ones we already have postulated.  Thus, it is reasonable to predict that, out of every three successful scoring opportunities, a team will make two touchdowns and one field goal. This prediction merely appropriates our foozling in formulas that balance scoring preferences with realities of the fourth-down.

We are also recognizing another reality -- that our analysis here does not warrant more than a digit of precision, as follows: 16.7 / (16.7+10.7) = 0.60 replaced by a nice round 2-out-of-3, and 10.7 / (16.7+10.7) = 0.39 with 1-out-of-3.

For an applications of these results, see The Next Superbowl.


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