The situation:
Green Bay Packers have the ball at 4th down and Goal at the 2 1/2 yard line in overtime. This is the first possession in OT, so a field goal does not win the game for the Pack outright but gives the Vikings a shot at matching. Scoring a touchdown ends the game right there for the home team Packers.The question: What decision gives the Packers the best chance to win the game? Going for the touchdown, or kicking the field goal? How do we decide this?
The first question we have to ask is what factors do we have to consider to make this decision? There are many, many factors that can go into this calculation. For example, how good is your offense? How good is the opponent's offense? How good is your kicker? Do you have momentum on your side? It would be impossible to evaluate the effect of each of these factors perfectly, but ultimately, they boil down into one of 4 factors:
1) What is the probability of making the field goal?
2) What is the probability of scoring a touchdown?
3) What is the probability of winning the game, if you go for a touchdown and miss?
4) What is the probability of winning the game if you attempt a field goal?
The coaches know (or SHOULD know) this information very, very well. Even if they can't perfectly quantify it, their gut instinct is usually pretty good at figuring out these percentages. However, if they're off, then the future calculations might be off as well.
The second part to this decision is how does the math shake out once we figure out what are the relevant factors. Even if one decision is "too risky", is the other decision the better choice, or is it even riskier? While we won't be able to 100% accurately figure out the percentages, we can at least attempt to decide the best course of action, given the information we have.
Ultimately, we need to look at the chances of winning (or Win Probability, abbreviated WP) of each decision.
if WP(Field Goal) > WP(going for it), then the Packers should have kicked.
if WP(Field Goal) < WP(going for it), then the Packers should have went for it.
if WP(Field Goal) = WP(going for it), then it doesn't really matter either way.
Conventional wisdom is to do the "safe" thing and kick the field goal, taking the points, forcing the visiting Vikings to match the score. Even if the Vikings do score, it might only be a field goal, giving the Packers another chance at winning in what is now to be a sudden-death period.
Head Coach Mike McCarthy does the "safe thing" and elects to kick the Field Goal here. It's successful, but the Vikings drive down the field to kick a tying field goal, and after a few other non-scoring possessions, the game ends in a tie. Since NFC North rivals Bears and Lions both lost, this moves the Packers (and the Vikings) a half game up in the standings, but was it the right move?
The Factors:
The folks over at Advanced NFL Stats has already broken this down, but I want to do a further analysis of the 4 broad factors that must be considered.1) What is the probability kicker Mason Crosby will make a 20 yard field goal?
The game was played at Lambeau Field in Green Bay, WI. Game time temperatures was under 20 degrees Fahrenheit in the outdoor stadium. Even with the outdoor elements, a 20 yard field goal is basically as long as a Point After Touchdown try (PAT), which has a very very high success rate. In actuality the rate is about 98%, but let's just simplify it to a 100% success rate.
2) What is the probability of scoring a touchdown if the Packers go for it here?
The Advanced NFL stats guys rated this at 55%, an average number of 4th and 2 results league wide. (The game recap lists the ball at the 2 yard line, however the ball was actually spotted between the 2 and the 3 yard line. While this has little impact on the FG try, the extra 18 inches can make the difference between success and failure on a TD try).
Of course, that number was for average teams. There are other numerous factors that go into this calculation. The quality of your offense vs. the opponent's defense is one factor. With starting QB Aaron Rodgers out with an injury, the Packers had to rely on Seneca Wallace and Scott Tolzien before turning to Matt Flynn due to injury (Wallace) and ineffectiveness (Tolzien). Flynn, though recently discarded by the Seahawks, Bills and Raiders, had just led the Packers from a 7-20 deficit to a 23-23 tie against these Vikings, so "momentum" would indicate the probabilities being a little higher in this case.
The Packers running back Eddie Lacy had 110 yards on 25 carries up to that point, a 4.4 yard average, though he most recently carried for 1 yard on 2nd down. However, it would later be revealed he left the game to an asthma flareup so it's possible the Packers knew he might be ineffective. Backup running back James Starks had 2 carries for 36 yards at that point (18 yard average) though 34 of those were gained on a single carry.
The defense is also gearing up for a stop, expecting a run. Plus, there's no chance they need to defend against a long bomb given the field position, so they have less field to defend.
For now, let's leave this number at 55%, but we'll see in the future how this number changes given the actual Green Bay offense at the time.
3) What is the chance the Packers win, if they go for the TD and miss?
A miss would put the Vikings needing just a field goal to win the game, so we only need to look at the chance the Vikings would score, not necessarily scoring a TD. One benefit of the Packers failing here is the Vikings would be pinned near their own goal line. The Advanced NFL Stats Win Probability calculator indicates an average team starting 1st and 10 from their own 2 yard line has a 24% chance of winning the game, which combines chances of scoring on that drive, plus subsequent win chances if they fail. This would give the Packers a 76% chance of winning even if they missed the TD.
4) What is the chance the Packers will win the game if a FG is attempted and made?
This question is slightly different and more complicated from the previous one, given there are 3 distinct possibilities here. The Vikings could drive for a TD resulting in a Vikings, the Vikings could fail to score resulting in a Packers win, or, the Vikings could drive for a tying field goal, giving the Packers the ball back with the next score winning the game. From the previous calculations, Advanced NFL Stats calculates the Packers win the game 79% of the time if they kick the FG here.
Later on, we can account for the fact that the Vikings kick returner Cordarrelle Patterson has run the ball back for 57, 26, and 28 yards giving the Vikings above average field position each time. However, until that, we'll use their 79% calculation.
The Calculations:
Now that we have some numbers, we really have to determine what the best course of action is. We need to look at the Win Probability of each event separately and then compare them together to figure out which is the better option.
1) Kicking the Field Goal
P(FG)*P(Packers Win with FG) + P(Missed FG)*P(PackersWin missing FG)
Since we've decided the probability of a field goal is 100% in this spot for simplicity, the chance for the Packers to win this game is simply the Probability the Packers Win given a Field Goal which was determined in factor 4 to be 79%.
2) Attempting a Touchdown
P(TD)*P(PackersWin given a TD) + P(MissedTD)*P(PackersWin given Vikings Ball on 2 yard line)
From before the Probability of scoring a TD here is 52% and if made, the Packers win 100% of the time. The probability of the Packers missing the TD is therefore 48%, and the probability of winning, given the Vikings have the ball on the 2 yard line was estimated to be 76%. So the full calculation looks like this:
(.55)(1) + (.45)(.76) = .8848 or 88.48%.
Calculations Summary:
WP kicking a FG: 79%
WP going for it: 89.2%
The Conclusion
As it turns out, there's a 10% more likelihood the Packers would have won the game if they went for it. Even though talking heads like to call going for it on 4th down a gamble, it wasn't the case at all. It was the high percentage play to win the game, though there was also a likely chance to win the game had the Packers just kicked it.Adjusting Variables:
Okay, so maybe you hate my numbers. Maybe you rationalize the chance of scoring a TD on the 2 or 3 yard line isn't that high, or maybe you think Christian Ponder has a much higher chance of leading his team to victory than I do. Let's see how making certain changes affects win probabilities.Case 1: Hey, Christian Ponder and the Vikings offense are way more awesome than you think
If that's the case, there are 2 numbers we will need to adjust here. It's the chances the Packers will win A) after a kickoff (if the Packers kick a FG) and B) if the Packers miss a TD try. Both those numbers will come down, and we'll make it come way down for exaggeration purposes.
Lets move A from 79% all the way up to 60%, let's move B down from 76% to 50%
The expression for going for it now looks like this: (.55)(1) + (.45)(.50) =
Packers WP kicking a FG: 60%
Packers WP going for it: 77.5%
Giving the Vikings offense a better chance actually makes the difference much more pronounced, that the Packers should definitely go for it in this case, though with a good offense it's clear the overall chance of winning goes down.
Case 2: Hey, the Packers suck with Flynn. Why do they have such a high 4th and goal on the 2 scoring rate?
Okay, let's change the chance Flynn and the Packers punch it in from 55% down to 45%. The expression for going for it (using the original numbers) now looks like this:
(.45)(1) + (.55)(.76) = 86.8%. Slightly less than the original win percentage of 89.2%, but still well above the WP of 79% when kicking a field goal.
Case 3: Patterson is awesome, he'll return to midfield!
The number we need to change now is the probability the Vikings can win the game after a Packers FG, since we now thing Patterson will run the ball up to midfield. The Advanced NFL Stats Calculator lists the probability of winning, down 3 in OT as 34%, giving the Packers a 66% chance of winning.
(.55)(1) + (.45)(.66) = .847 or 84.7%. STILL greater than the 79% chance of winning when kicking the field goal.
What is the Break Even Point?
Okay, so instead of continuously changing the numbers around, why don't we just use algebra to find at what point it makes more sense to kick the field goal? What we'll have to do is set up an equation with a variable and see at what point do our variables make it break-even where it doesn't statistically matter whether or not you kick or go for it. The problem is there are three variables to account for.x = The chance the Packers win kicking a FG
y = The chance the Packers score a TD on this play
z = The chance the Packers win given they miss scoring a TD.
The break even equation is this: x = (y) + (1-y)(z)
(Again, we're assuming a 100% chance of making the FG if attempted)
How do we individually figure out what the effect of each variable is? We can assign reasonable numbers to 2 variables and solve for the third.
Let's work with our old assumptions: x = 79%, y = 55%, z = 76%
Example 1: Change the chance of the Packers win, kicking a FG.
x = (.55) + (.45)(.76)
x = .892
The Packers should kick the field goal if the chances of winning after kicking a field goal is greater than 89.2%.
Example 2: Change the chances the Packers win, given they miss scoring a TD.
.79 = (.55) + (.45)(z)
z = (.79 - .55)/.45
z = .533
The Packers should go for it unless the chance of winning given they fail on scoring a TD are lower than 53.3%
Now, one of the issues is in both example 1 and example 2, you're looking at how good the Minnesota offense is vs the Green Bay defense. If x goes up, it's very likely z goes up and vice versa. These two numbers are positively correlated.
Example 3: Change the chances the Packers score a TD on 4th and goal on the 2 yard line.
.79 = y + (1-y)(.76)
.79 = y + .76 - .76y
.03 = .24y
.125 = y
This indicates the Packers should go for it unless the chances of scoring a TD on this play are less than 12.5%! This is probably a really surprising result for most football fans, and even football fans who believe coaches should be more aggressive.
Don't believe me?
Crunch your own numbers. Try it out. See how good the Vikings offense needs to be to make kicking the FG the better decision? What do you think? Leave a comment!1) Give the Vikings a better offense, and see what the break even rate is. For example, let's say the Packers win 90% of the time after a FG and 50% of the time after missing a TD. How often would the Packers have to make a TD to make going for it the right decision?
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