Treatise on an NFL OT decision [B]Treatise on a NFL Playoff game overtime decision[/B]
[I]This treatise is somewhat difficult to follow so careful reading is necessary. I assume the reader knows elementary probability theory and is acquainted with the new NFL rules regarding overtime.[/I]
Let’s examine a possible decision point in an NFL Playoff game in overtime on the first possession, namely Team A going for a touchdown (TD) close to the opponent’s goal line versus kicking a field goal (FG). Further, let’s isolate the case of a 4th and goal situation. A TD wins the game outright while a FG prolongs the game and allows Team B to win or tie after the ensuing kickoff. Two choices are examined here: 1) go for the TD or 2) go for a FG.
Let pTDM be the probability that a TD is scored if it is attempted and pFGM be the probability that if a FG is attempted, it is good. Let pPTD be the ultimate probability of a win in the case of 1) a TD attempted, 2) a penalty or some other event occurs and 3) Team A retains the ball for another play or gets a first down. Let’s ignore the probability that if a FG is attempted, the play may result in a Team A TD (most likely a botched snap, etc). Let’s ignore the probability that Team A allows a turnover that results in a Team B TD and thus loses the game. A faked FG is treated as a TD attempt. Note that this may be a very smart thing to do depending on the personnel involved since it may increase the probability of a TD.
Let pWTD be the overall probability of a win if we choose 1) to go for the TD and let pWFG be the overall win probability if we choose 2) the FG attempt.
Now we can write that pWTD = pTDM +(1-pTDM)*pWNO_TD and pWFG = pFGM *pW1A+(1-pFGM)*pWNO_FG where pWNO_TD is the probability of a subsequent win if no TD is scored by Team A on the TD attempt, pW1A is the probability of a subsequent Team A win after Team B’s first possession and pWNO_FG is the probability of a subsequent win if the Team A FG attempt is not good.
[B]So now we are ready to see if we can make any sense of this stuff.[/B]
We seek to find what values of pTDM are needed to make going for a TD better than the more “conservative” approach of kicking the FG and hoping your defense can hold Team B.
To this end, we simply ask for: pWTD > pWFG. This becomes:
pTDM + (1-pTDM) * pWNO_TD > pFGM *pW1A + (1-pFGM) * pWNO_FG
With some algebra we get:
[B]pTDM > (pWNO_FG - pWNO_TD + pFGM * (pW1A - pWNO_FG)) / (1 - pWNO_TD )[/B]
A pretty obvious result …LOL
Let’s examine pW1A which is the combination of Team A’s probability of winning 3-0 on the 1st possession (denoted p_3-0W1) and winning (denoted pW2A) if Team B ties the game up with a FG (denoted p_3-3T1). Remember in this case, Team B must score or it loses the game outright, thus on a 4th down Team B will always go for it or attempt a FG.
So pW1A = p_3-0W1 + p_3-3T1 * pW2A
Also pWNO_TD = pWxTD + pPTD, the two cases of a win by missing the TD.
[B]Example:[/B]
The following values are estimates from a variety of NFL data and outright guesses.
pFGM = 0.98 : Short yardage FG conversion percentage
p_3-0W1 = 0.53 : Chances of stopping Team B with no score (on 1st poss)
p_3-3T1 = 0.32 : Chances of holding Team B to a FG (on 1st poss)
pW2A = 0.57 : Chances of Team A winning after 3-3 tie, close to same as old rules!
pWNO_FG = 0.51 : Slightly better than 50% chance of still winning the game with good (for Team A) field postion from missed FG
pWxTD = 0.56 : Better than 50% chance of still winning the game with better (for Team A) field postion from failed TD
pPTD = 0.04 : Rare chance of a pass interference or such penalty with subsequent win
gives pTDM > 27.1% which indicates a TD should be tried on a 4th and goal from inside about the five yard line. Note that the NFL average two-pt conversion rate from the 2 yd line is around 44%.
[B]The main decision sensitivity for choosing to go for the TD is that it must outweigh the effectiveness of the defense in stopping Team B on its first possession.[/B] In the example, I use a 53/32/15 mix for a Team A win/tie/loss if the Team A FG is made. If we use a 65/24/11 mix (better defense), then pTDM > 45.3% ! So in this case, going for a TD from the 2 yd line probably would not be advisable.
Who knows, you may be able to take some of this analysis and impress your SuperBowl party guests with it. I invite comments/thoughts on the treatise’s usefulness, modeling, assumptions, refinements and/or possible errors (heaven forbid!).
