Capstone

Kick It or Go for Two?
Jared Heck
Department of Mathematics and Computer Science
Westminster College, Pennsylvania, USA
heckjr22@wclive.westminster.edu
December 16, 2015
Abstract
In this paper we will analyze whether teams in the NFL should kick the
extra-point or go for two depending on time remaining and score in the game.
After a rule change put in place in 2015 the option of going for two has gotten
alot of attention. It is being attempted at an all time high. We will provide a
chart to see if teams are attempting the two-point conversions at the right time
of the game. Also we will be outlining our model or chart versus other charts
being used still today.
1 Introduction
In 2015 the National Football League (NFL) introduced a rule change. This change
took what used to be an “automatic” one point play after a touchdown and increased
the difficulty of success. The change may interest people enough to watch the extra-
point conversion instead of changing the channel, getting another beverage or using
the facilities after a touchdown. In years prior, the kick was attempted from the two
yard line, making the ball travel 20 yards through the uprights. This year, after a
touchdown, the point after touchdown (PAT) is being moved from the two yard line
to the 15 yard line, ultimately making it a 33 yard field goal attempt. The PAT dates
back to the beginning days of football stemming from the root sport, rugby. However,
the two-point conversion is a little more than two decades old to the NFL. It was first
implemented at the college level and then adopted by the NFL years later. The 1994
NFL season featured the “go for two” option. Teams could attempt to go for two
points by scoring a touchdown in 1 play, from the two yard line.
Dating back 5 years prior to the 2015 season (with the 20 yd. attempt) kickers made
an average 99.4% of their kicks. A missed kick was rare; the shorter distance could be
a psychological benefit or a physical benefit by not giving the ball the chance to alter
its course due to natural movement, wind, or a kick slightly off course. The field goal
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posts are a fixed distance apart but as the kick gets further away, the angel between
the posts gets smaller. However, a little over half way through the 2015 season (33
yrd. attempt), the success rate is down to 94.8%.[5]
Figure 1: NFL Yearly Average Success of PAT
Year Percentage Successful
2010 99.1%
2011 99.4%
2012 99.5%
2013 99.6%
2014 99.3%
2015* 94.8%
In the first two weeks of the rule change, there were more PAT’s missed than all of
2014 regular season, making the decision to go for two a little more enticing for head
coaches in the NFL. In 2015, the two-point conversion still remains about a 50/50
proposition like previous years, but the frequency of attempts have increased due to
now, a more difficult alternative for less points. [5]
Figure 2: NFL Yearly 2pt. Conversion Attempts
Year Attempts Percentage Successful
2010 50 52%
2011 50 46%
2012 58 50%
2013 69 48%
2014 59 47%
2015* 54 50%
These results were exactly what the NFL intended. The NFL is trying to maximize
profit and draw in as many fans or customers as possible. One way to do that is to
create uncertainty in the outcome of a game or even, in this case, a play. They wanted
to see a play become less predictable and a more exciting two-point conversion play
be attempted more. However all parties involved do not feel as comfortable with the
rule change. Head coaches in the NFL are being criticized when they do and do not
go for two. In some situations during a game it is obvious whether to go for 1 or 2.
Example: Consider this scenario in a football game. Team A just scored to tie the
game, say 16-16, with no possessions remaining in the game and have the option to go
for two or kick the extra-point. Team A will obviously choose to kick the extra-point
because they just need one point to win the game, and have a higher success rate for
that play (95.2%).
However, there are situations that are not so obvious. The “status quo” of the
NFL and media coverage has masked the idea of going for two in not so obvious
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situations. In 2013, the NFL game between Denver and San Diego who had just
scored a touchdown to trail Denver by 9, acclaimed sports announcer Jim Nantz
made the comment awaiting the extra-point, “If the book says go for two, take the
book and burn it”.[9] “The Book” refers to the calculated decision making teams
should refer to throughout the game. Nantz made these comments, but did not stop
his blabbering to ponder why it said that. In an online article, Slate writer Josh
Levin says for certain scenarios “The two-point conversion takes a 50/50 proposition
and makes it a winning strategy.”[8]
Example: Consider this scenario, Team A just scored with 6 minutes left in the 4th
quarter to cut their opponents lead to 14-6 and the decision to kick the extra-point
(7) or go for two (8). Team A decides to kick the extra-point to make the score 14-7.
We will analyze Team A’s decision in Section 4 and decide if they made the right
choice.
Based on some of the numbers from Figure 1 and Figure 2 if a team converts a
two-point conversion 47.5% of the time (.475 ∗ 2 = .95 expected pts.) and to date in
2015 extra-points are successfully kicked 94.8% of the time (.948 ∗ 1 = .948 expected
pts.), wouldn’t they be better off going for two all of the time? These expected values
seem to lean towards going for two. Showing that if the probability of success holds
up over time then going for two will slightly out produce the extra-point option. The
expected value is not the only thing to consider, (i.e. Example 1), for example if the
team only needs one point to win. Then we need to find a formulation that considers
the elements of the game that are important to who will win; score, time, and who
has the ball.
The correct decision can be critical to a teams success in that particular game. As a
coach you have to put your team in the best position to win the game, week in and
week out. Countless hours of practice, film study, and game plan are put in weekly to
achieve a win. One game could determine a teams playoff fate. So it is important that
coaches dont fall into the “status quo” of kicking the extra-point, and make the right
call when it comes to putting points on the board.[?] We will provide the answers to
scenarios like Example 1 in Section 4. Also, we will look at the process of calculating
the best answer and different factors that impact these critical plays throughout the
paper. Then we will compare our decision charts of 2015 vs. 2014 to outline and
rationalize the differences.
2 History of the Extra-Point(s)
The NFL has been around for about a century, and the fine tuning of skills has evolved
the game as a whole. Specifically the extra-point has evolved from the beginning days
too. The goal posts used to be placed on the goal line. Due to interfering with play
they were moved back to the end line in 1974, where they remain today. Kickers in the
mid 1900’s were much different than the modern day kicker. For example, Lou “The
Toe” Groza played offensive line for the Cleveland Browns. Oddly enough he was the
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teams place kicker as well. Groza, a 1974 Hall of Fame inductee, kicked straight on
using his toe. Over his 21 year career Groza made 97.2% of his extra-points. During
this time Groza was one of many kickers that were lineman as well. The continued
pursuit of perfection has changed the style of kick used for ﬁeld goals. The European
inﬂuence on the league introduced the soccer style kicker in the NFL in the 1960’s.
Currently, 100% of kickers in the NFL are soccer style kickers. As time has elapsed
the success of extra-points average has gotten closer and closer to perfection. This
rule change has altered the progress and has seemingly resurrected the idea of going
for two. [5].
In 2015, the Pittsburgh Steelers are thriving going for two, converting 4 out of 6
on the year. The Steelers made it a point at the beginning of the season to go for
two more. Currently they have attempted the tw- point conversion the most and
converted it the most also. Most NFL coaches base their decision using their own
intuition or using a chart created by Dick Vermeil, an iconic coach, in the 1970’s when
he coached at UCLA[2]. Vermeil’s chart is still very prominent in the NFL.[6] The
chart is simple to use, it tells you what you should do if you trail or lead by x points.
The problem is, the chart only bases the decision on the score of the game. [2]
Figure 3: Dick Vermeil’s Conversion Chart
By only basing our decision on the score of the football game, we are not accounting
for important factors like number of possessions remaining based on time, the success
of our teams extra-point and two-point conversion rate, and the other teams scoring
success when they possess the football. There is a need for a chart that is still easy
to use but that can calculate a more precise result. After all, every team is trying to
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gain an edge in winning.
3 Sackrowitz
“The outcome of several football games each year depend crucially on the extra-point
strategy.”- Harold Sackrowitz[1] In the year 2000, Sackrowitz, a statistics professor
at Rutgers University, published a paper Refining the Points-After-Touchdown De-
cision. This article got national recognition appearing in The NY Times Magazine
and was mentioned in a Sports Illustrated article featuring football analytics.[7] In
his paper, Sackrowitz used dynamic programming to analyze coaches decisions. His
programm took into account lead score differential and time remaining in the game
and produced a chart for easy use. The chart contradicted some situations that could
be observed in a weeks worth of NFL games.
This type of programming uses formulas to calculate the probability of winning if
you go for 1 or go for 2 after a touchdown or when Team A or Team B turn the
ball over via field goal or no score. Sackrowitz used league averages to represent the
percentages of touchdown, field goal, and no score. Sackrowitz made a chart using
these league averages. The program chooses go for 1, or go for 2 depending on which
option gives a higher probability of winning the game. It does this by working from
the end of the game to the current state or by using backward induction. Here is an
example of the formulation to calculate the winning percentage at a specific state of
a football game.
1 pt. conversion
Pr(Team A wins with a 1 pt. conversion at v(6,d,t))= pa1[Pr(TD by B)∗w(6,d-5,t-1)
+ Pr(FG by B)∗w(3,d-2,t-1) + Pr(NS by B)∗w(0,d+1,t-1)] +(1−pa1)[Pr(TD by B)
∗w (6,d-6,t-1) + Pr(FG by B)∗w(3,d-3,t-1) + Pr(NS by B)∗w(0,d,t-1)]
2 pt. conversion
Pr(Team A wins with a 2 pt. conversion at v(6,d,t))= pa2[Pr(TD by B)∗w(6,d-4,t-1)
+ Pr(FG by B)∗w(3,d-1,t-1) + Pr(NS by B)∗w(0,d+2,t-1)] +(1−pa2)[Pr(TD by B)
∗w (6,d-6,t-1) + Pr(FG by B)∗w(3,d-3,t-1) + Pr(NS by B)∗w(0,d,t-1)]
[1]
The probability that is highest (1pt. or 2pt.) is what Team A should choose to
attempt. Team B however, would choose the lowest probability because the number
is referring to Team A’s success.
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4 Version 1.0
To achieve our goal of developing a more accurate chart, we will be building on Sack-
rowitz’s dynamic programming model. In our Version 1.0, we feature the ability to
make the program team-specific. All that is required is the scoring percentages of
the team that will be using the chart and those of their opponent. We are able to
adjust these so we can represent different match-ups. An advantage of team-specific
percentages is analyzing what situations depend on both teams ability to score. In
addition, the overtime variable, which measures winning percentage in overtime, can
be individualized. The variables we used to create Version 1.0 are shown in Figure 4.
Figure 4: Variables
Variables Definition
pa1 Team A’s probability of successful PAT
pa2 Team A’s probability of successful 2 pt. conversion
TDa Team A’s probability of scoring a touchdown next possession
FGa Team A’s probability of a Field Goal next possession
NSa Team A’s probability of a possession resulting in no points
OTa Team A’s probability of winning in overtime
pb1 Team B’s probability of successful PAT
pb2 Team B’s probability of successful 2 pt. conversion
TDb Team B’s probability of scoring a touchdown next possession
FGb Team B’s probability of a Field Goal next possession
NSb Team B’s probability of a possession resulting in no points
k scoring result of previous possession by Team A (v) or B (w)
d lead(+) or deficit(-) referring to Team A
t possessions remaining in the game; estimated 2:30 per possession
v(k,d,t) probability of Team A winning in state k,d,t
w(k,d,t) probability of Team B winning in state k,d,t
First we start with the trivial cases at the end of the game. This is when there are no
possessions remaining and we can determine a winner based on our value of v(k, d, 0).
These base cases require minimal calculations. Scenarios like this are not hard to
imagine even if we add a couple of possessions (time). We can predict the remaining
scenarios in our head. However once we start to add more possessions the number of
possibilities get very large and we need a computer to calculate them. In our dynamic
program we considered all cases where d = [−50, 50] with up to t = [0, 18] possessions
remaining. We recreated this program starting with our base case of no possessions
remaining in the game, and the possible outcomes the team possessing could achieve
(i.e. Touchdown, Field Goal, No Score).
Here are the calculations for all scenarios where Team A possessing the ball then
turns it over via score or no score, with no possessions remaining.
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v(0,d,0) After No Score
• for d from 1 to 50 do v(0, d, 0) = 1 (the probability of Team A winning is 100%)
• for d from -50 to -1 v(0, d, 0) = 0 (the probability of winning is 0%)
• v(0, 0, 0) = OTa (in this case at the end of the game both teams are tied and
the probability of Team A winning is their overtime success rate)
v(3,d,0) After Field Goal
• for d from -50 to -1 v(3, d, 0) = 0 (the probability of winning is 0%)
• v(3, 0, 0) = OTa (in this case at the end of the game both teams are tied and
the probability of Team A winning is their overtime success rate)
v(6,d,0) After Touchdown
• v(6, 0, 0) = max(pa2+(1-pa2)*OTa, pa1+(1-pa1)*OTa)
• v(6, -1, 0) = max(pa2+(1-pa2)*0, pa1*OTa+(1-pa1)*0)
• v(6, -2, 0) = max(pa2*OTa+(1-pa2)*0, pa1*0+(1-pa1)*0)
• for d from -50 to -3 v(6, d, 0) = 0 (the probability of winning is 0% because you
can only earn up to 2 pts. on extra-point and Team A is down by 3 or more)
We calculated these scenarios for w(k,d,0) or Team B’s probability of winning also.
The only thing that diﬀers is since the score is always referring to Team A, Team
B will want a lower probability so they will choose the minimum values in the cases
above as their optimal strategy.
From here we were able to then expand oﬀ of our base case and look at the scenar-
ios that were not so clear cut. These scenarios occur when we let d and t evolve
throughout the game. As mentioned before we let d = [−50, 50] with up to t = [0, 18]
possessions, essentially simulating the decision making for most of a game. For each
of these scenarios we calculated the winning percentage for Team A after a touchdown
if they kicked the extra-point or they went for two. Our program suggested the higher
of the two in the form of a chart. These charts can be compiled before the game and
at the hand of a coach.
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Figure 5: Good vs. Good
Example: Back to our second example in the introduction, with Team A down 8
pts. after a Touchdown and the decision to go for 1 or 2 with 3 possessions left in the
4th quarter, they kicked the extra-point. According to our calculations the decision
to try for 1 is the wrong decision. Our chart, as seen above, suggests that Team A
has a higher probability of winning the game if they simply go for 2. Regaurdless the
team is going to need another touchdown to have a chance to win. So our program
tells us that they will win more times if they go for two than go for one. If they do
not get it, on their next touchdown they will have to go for two again to send the
game into overtime and try to win there. However if they are able to convert, then
their next touchdown will be followed by going for 1 to win the game. This is all
assuming that the other team will not score again.
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4.1 Chart Analysis
In our charts 1 represents kicking the extra-point and 2 represents going for two. Note
that when a team scores to go ahead by 1 or 2 with no possessions left, they should
elect to take a knee, automatically ending the play and game. They should do this
because if the other team were to get the ball via blocked kick, fumble, interception
and return it for a touchdown durring the duration of the extra-point play, they would
be rewarded 2 points and either tie or win the game.
An advantage of making Version 1.0 team specific, is the ability to simulate match-
ups between teams. The team names are not important, so we classified them as
Good vs. Good, Good vs. Bad, Bad vs. Good, and Bad vs. Bad. By doing this we
can further the analysis of our program and analyze what changes as the ability of
the teams in the match-up changes too. We kept the extra-point and 2pt. conversion
rates the same for each team. In addition to our own analysis, this feature allows
teams to quickly make a chart for their up coming opponent.
Good vs. Good
In the match-up of two of the highest scoring offenses in the NFL was what we con-
sidered our base or normal chart. This chart (Figure 5) is what we will compare our
other match-up charts to
Bad vs. Bad Throughout this chart it is very similar to our base Good vs. Good
chart.
Good vs. Bad This chart is very compatable to the Good vs. Good chart mid
to late game situations. It is earlier when Team A can kick the extra-point even
when they are losing. This is logical because over the course of the game the scoring
percentages are going to hold up more times than not, so teams should not panic if
they know they have a better scoring offense than an inferrior opponent. (See Figure
7 in Section 7)
Bad vs. Good The chart is littered with more two’s than any of the other match-
ups. This could be a consequence of Team A (Bad) not scoring very often, so when
they do, they have to try and keep up with the higher scorring offense of Team B
(Good). (See Figure 6 in Section 7)
2014 vs 2015
Since the rule change has had an obvious effect in the success rate we also analyzed
these same match-ups using the 2014 extra-point percentage of 99.3%. The situations
that are most interesting are those that are not obvious and the chart or program
suggests go for two in both the 2015 and 2014 version. Meaning, no matter if your
kicker is automatic (2014), the 50/50 proposition is still the winning strategy. In total
71 of the decisions, marked by highlight, were opposite from our 2015 chart to the
2014 chart. (See Figure 8 in Section 7)
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5 Version 2.0
Looking to improve Version 1.0, we recognized that if a team scores with one posses-
sion left, the program assumed that the other team would possess the ball for the final
possession of the game. However, the team that just scored could choose to onside
kick if they were still losing. If the onside kick is successful they would repossess the
ball, which could change the extra-point strategy and give them chance to win the
game. Consider this example,
Example: Team A just scored to cut the deficit to 5 points with one possession
left or v(6,-5,1). Version 1.0 assumes that Team B would occupy the remaining
possession suggesting Team A to kick the extra-point, assuming the game is over.
However, Version 2.0 is more realistic and accurate. Team A should go for two and
onside kick to try and get the ball back and extend or win the game. (See Figure 9
in Section 7)
Version 2.0 addresses this problem.
While it sounds good to be able to possess the ball immediately following a score,
NFL teams recover attempted onside kicks about 20% of the time.[10] We created
a program that would choose to kick the ball deep, onside kick, or indifferent. The
program was similar to that of Version 1.0, now producing two charts. One chart of
point after decision, and now an additional kickoff chart. This added a new layer to
our program, which we had to make sure the decisions it was making were dependent
on the both chance of winning when kicking off and after a score. Throughout the
game and our program we represented the option to do any of the two when kicking
off. The 80% chance that the receiving team covers the onside kick, they start that
drive with the ball around the opponent 45 yd. line, which is great field possition.
To enter field goal range from there, a team would need to gain about ten yards.
In addition they have a shortened field for a touchdown. As a result, we increased
the scoring percentages if the team recieving recovers the ball. These conditionally
increased variables makes sure that we account for the risk of the team kicking an
onside kick; meaning that they only do it when necessary. On the contrary, the ball
being kicked deep, the receiving team has to now drive the length of the field, keeping
the original variables in place.
The program with Version 2.0 changes is now making the decisions that we expected
it to be when a team is now able to regain possession by onside kick. It now decides
that when down 5 points with 1 possession remaining, they elect to go for two. (See
Figure 9 in Section 7) Also, we know that the program is working correctly because
when the team is down 10 with one possession remaining, they go for two and then
onside kick. In the scenario when a team is down 8 with one possession left they
elect to go for two. This scenario is closely related to that of being down by 8 with 3
possessions left and choosing to go for two. The only difference is now you are relying
on getting an onside kick and then scoring a touchdown. Whereas with 3 possessions
left, a team is counting on the other team not scoring and then getting the ball back
to go score. (See Figure 9 in Section 7)
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In terms of when to onside kick vs. kick deep. Our program is making the correct
decisions too. (See Figure 10 in Section 7 Note: 1-kick deep and 2-onside kick)
The kick chart or program suggests when a team is tied or winning they should only
choose to kick deep, which makes sense because you dont want to increase a teams
scoring chances if you are winning or tied. However when the program suggests to
onside kick is when the team is trailing and later in the game. For example if a team
is trailing from 1 to 8 points with one possession left they should always onside kick
because that is the margin where they could still win or take the game into overtime.
6 Conclusion
The extra-point rule change has thrown a wrench in the decision making after a touch-
down in the NFL. We are expected to see an all time high of both extra-point kicks
missed and two-point attempts in the 2015-2016 season. However these attempts do
not allways occur when our charts or programs suggest.The blaim for teams missing
the concepts of going for two is not allways on the coaches. I’m sure some would like
to attempt it more often. The head coach wants to win but he also doesn’t want to
make a controversial decision if the people in the front office (his bosses) don t agree
with going for two. The “norm” needs to be broken so that teams are able to preform
optimally. When the “norm” is broken we need to recognize if teams are going for
two, is it because they recognize the winning advantage it gives, or is it because it’s
the next best thing to their unreliable kicker. Statistics and probabilities have proven
usefull in other major sports such as baseball and basketball. There is no reason that
football should be any different.
6.1 Future Analysis
Future analysis could include adjusting the time of possessions to make it more
rounded or smooth. Not all of the possessions are 2:30. Something like an alter-
nating time of possession variable may fix this problem. Also adding the different
types of no score possibilities to the program. For example, an interception or a
blocked punt would increase a teams scoring percentage compared to a punt downed
inside the 10 yd. line. When a team doesn’t score then you would need to look at
how often their opponent intercepts, blocks a punt, or goes 3 and out, just to name
a few. Finally an interesting topic would be to analyze whether teams in over time
should choose to receive or kickoff if they win the toss.
11

References
[1] Sackrowitz, Harold. “Reﬁning the point(s)-after-touchdown decision.” Chance
13.3 (2000): 29-34.
[2] Lombardi, Michael. “Conversion Confusion: When Going for Two, Wait ’til the
Fourth.” NFL.com. 12 Oct. 2011. Web. 6 Oct. 2015.
[3] Wenz, Michael, and Joren Skugrud. “Tackling the Chart: Two-Point Conversions
and Team Diﬀerences in Football.” CHANCE 24.1 (2011): 29-35.
[4] Skugrud, Joren and Wenz, Michael, “Status Quo Bias and Two Point Conversions
in Football” (March 31, 2010).
[5] “Pro Football Reference.” Pro-Football-Reference.com. Web. 6 Oct. 2015.
[6] Mike Miller, Personal Communication. Oct, 2015
[7] Campos, Paul, and Jonathan Chait. ”Sabermetrics for Football.” The New York
Times Magazine. 12 Dec. 2004. Web. 7 Dec. 2015.
[8] Levin, Josh. ”The Strange Case of the NFL Coaches Who Don’t Care about
Winning.” Slate. The Slate Group, 11 Jan. 2013. Web. 10 Dec. 2015.
[9] Wagner, Kyle. ”The Problem With Bitching About Fourth Downs And Two-Point
Conversions.” Regressing. 13 Nov. 2011. Web. 10 Dec. 2015.
[10] Burke, Brian. ”Advanced Football Analytics (formerly Advanced NFL Stats):
Onside Kicks.” Advanced Football Analytics (formerly Advanced NFL Stats):
Onside Kicks. 21 Oct. 2009. Web. 10 Dec. 2015.
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7 Charts for Decision Making
Figure 6: Bad vs. Good
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Figure 9: 2.0 extra-point Chart
16

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