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Game Theory of Football Play Calling
By
Luke Berndt
October 27, 2014
Math 89S Game Theory and Democracy
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In this essay, I plan to look into the game theory of play calling in a game of football.
Passing plays almost always get more yards per play than running plays, but yet, many more
running plays are called in a game. In addition, studies have shown that teams should go for
two point conversions more often. I plan to talk about all of these points and the affect that the
play call has on a game and what a coach could do for his team to score more points and win
more games.
Zero-Sum Game
A zero-sum game is a situation where a gain for one side means that the other side will
have a corresponding and equal loss (4). Therefore, a zero-sum game is almost any contest or
situation where there is one winner and one loser.
An example of a zero-sum game can be shown in the terms of football. The offense can
either call a run or a pass play which both will gain five yards if they are successful. The defense
can also either call a run or a pass play which both will result in the offense losing five yards if
they are successful. The only way the offense can gain their five yards is if they call the opposite
play as the defense (Offense runs, Defense plays for the pass). However, if both sides call the
same play (Offense runs, Defense plays for the run) then the offense will lose yards (3). This can
be seen in the following chart (format for yardage gains is (Offense, Defense)):
Play call
Defense Run
Defense Pass
Offense Run
(-5, 5)
(5, -5)
Offense Pass
(5, -5)
(-5, 5)
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If football were actually this simple, play calling would not be an issue and teams could
call a play with the flip of the coin if they pleased because the probability is ½ (3). However,
many other factors are involved with the calling of a play, whether that is time of game, place
on field, or skill level of certain players. Suppose the offense in the previous example is replaced
with a new offense that has an exceptional passing game and instead of gaining five yards on a
passing play, they gain 10 (3). Here is how the zero-sum game would look now:
Play Call
Defense Run
Defense Pass
Offense Run
(-5, 5)
(5, -5)
Offense Pass
(10, -10)
(-5, 5)
Now the probability that we will run comes out to 2/5 (3). This means that instead of
passing half of the time like the first offense does, the second offense passes only two out of
every five times even though they are better at passing the ball. The reason that this happens is
because of the interactive nature of a zero-sum game (3). If the defense knows the offense has
a dominant passing attack, they will call pass plays more often to try and stop the offense’s
better passing. This leads to the offense running more so they can put up more yards on the
defense (3). Football fans may hear this from the commentators with comments such as “they
need to run the ball more to open up the passing game,” or “the defense should know what’s
coming, they pass almost every play.”
In a perfect world, two players or teams in a zero-sum contest will act rationally and
make choices that minimize their maximum possible loss. This “minimax” solution also means
that players make play selections that do not follow a predictable pattern that might give their
opponent an edge (2). The examples above showed the offense using “minimax” strategies.
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They figured out how often they should pass the ball, using the Nash equilibrium equation. The
calculations to find how often the second team’s offense should throw the ball is shown below
(3):
The offense will run with probability q and pass with probability (1-q). We know that the
equilibrium value of q may be found by making the opponent indifferent between his two
strategies. So:
If Defense defends against the Pass: Expected gain
= 5q - 5(1-q)
= 10q - 5
If Defense defends against the run: Expected gain
= -10q + 5(1-q)
= 5 - 15q
At equilibrium, we must have these expected gains equal to each other, so:
10q-5 = 5 - 15q --> 25q = 10 --> q=2/5
Play Calling for Normal Plays
From the 2001 to the 2005 season, teams only passed the ball around 56 percent of the
time. A pass in this period on average gained .55 yards more than a run, was nine percentage
points more likely to yield a first down, and led to scores with a 3.8 percent probability. Runs
had only a 2.8 percent scoring probability (2). Studies have shown that if a team went from
passing 56 percent of the time to passing 70 percent of the time, they would score an
additional 10 points over the course of a season, which is three percent of total scoring for an
average team (2). If passing is this much more productive than running, why are teams still only
passing 56 percent of the time and not more? This problem is attributed to mediocre,
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inaccurate quarterbacks, bad weather and playing conditions, and the safety that the run has in
terms of less of a chance to turnover possession of the football to the other team.
In addition to not passing enough, teams in the National Football League are too
obvious as to when they are going to pass and when they are going to run the football. Teams
go against the “minimax” theory and do not call the play that is unexpected by the defense.
From the seasons of 2001 to 2005, when a team passes on one play, they are 10 percentage
points less likely to pass on the second play (2). However, this statistic is only for successful
passes. If a team is unsuccessful on one play that they pass on, then they are 14.5 percentage
points more likely to switch to the run on the next play, even though they have regained
control of down and distance and position on the field (2).
If defenses knew about these tendencies and how offenses are not going by “minimax”
strategies, they could better anticipate what play is about to be run against them. If they
somehow learned how to do this, they could give up an average of 10.8 fewer yards per game.
This would translate to one point per game less given up which would be 16 points on the
season. 16 points on the season would lead to an extra half of a victory per year (2). The
statistics for defenses gaining an understanding of the game theory in football are slightly
greater than that for offenses to gain an understanding of the perfect “minimax” theory.
The studies done by Kenneth Kovash and Steven Levit (2) address the National Football
League as a whole. Breaking these statistics up team by team would both require a different
“minimax” formula for each of the 32 different teams, but would also bring in a lot more
outside factors. Different teams have different strengths and weaknesses, play in different
playing conditions, and face different opponents with their own different strengths and
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weaknesses. For example, the New Orleans Saints have a strong quarterback and receiving
corps, play indoors with zero weather to worry about, and play in the National Football
Conference South with some of the weaker defenses in the league. Their “minimax” equation
would be much different than say the Kansas City Chiefs, who have an explosive running attack,
play outdoors, and play in a tough defensive conference.
These statistics have also changed since 2005 because of a change in rules to favor the
passing game. Fans like to see high-powered passing teams that will score a lot of points. In
addition to this, the National Football League is trying to put a stop to all of the concussion
problems they have had in the past. They have implemented rules that have given both
quarterbacks and receivers more protection. These rules force defenses to play off receivers
and not hit the quarterback as much, which allows for easier passing and less need to run the
football.
Two Point Conversions
The average success rate for a two-point conversion in the National Football League is
around 49 percent and kickers make around 98 percent of extra points they take (1). The
expected points for an extra point are .98 (.98 x 1) and the expected points for a two point
conversion are .98 (.49 x 2). While these expected points are almost exactly the same (numbers
are rounded), teams should still almost always go for two points after a touchdown instead of
kicking a field goal. This equality is not enough evidence for coaches call the "safe” play in all
game situations and opt to kick a PAT (Point After Touchdown) instead of going for two.
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If team x is down 12 points after a touchdown (the first six points; team X was initially
down 18 points), then team X should always attempt the two point conversion (1). If you go for
two points and succeed, you are down 10 points and now know that a touchdown and field
goal will tie the game. If you go for two points and fail, you now know that you need two
touchdowns to win. If you kick the extra point you will be down 11 points and might later kick a
field goal that will turn out to be meaningless (1). You want to determine as early as possible,
what kind of scores and how many of them you will need to remain in the game. The only time
that a team should not go for two is if they are down seven points late in a game. In this
situation if they score a touchdown they are down one point and can either have a 98 percent
chance to tie the game, a 49 percent chance to win the game, or a 51 percent chance to lose
the game. This is one of the only situations where it is in a team’s best interest to kick an extra
point instead of going for a two going conversion. Many people believe that two point
conversions should be saved for end of game, dire need situations, but this is not the case. In
fact, it is the exact opposite. Earlier in a game, teams should always go attempt a two-point
conversion, and they should only kick an extra point in late game, game deciding situations (1).
However, these percentages are not completely accurate. Two point conversions’
success rates are only inferior to those of extra points because of poor play calls made by
coaches down by the goal line. Over the past 20 years, running the ball has led to a 60 percent
success rate for two point conversions (1). A jump from a 49 percent success rate to a 60
percent success rate only further proves that teams should go for two points after a touchdown
in almost every situation of a football game. In addition, not all National Football League
kickers’ success rates are 98 percent. For example 2011 Dallas Cowboys kicker, David Buehler’s
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extra point success rate was only 94 percent (1). If both of the success rates were changed, this
would lead to the expected points coming out to be .94 (.94 x 1) for extra points and 1.2 (.60 x
2) for two point conversions. Extra point’s expected points for the 2011 Dallas Cowboys would
have been an entire .8 points per touchdown higher. In 2011 the Cowboys scored an average of
2.4 touchdowns per game. Their scoring statistics if they went for two every time they scored a
touchdown based on expected points for a two-point conversion would come out to:
Points per game off of touchdowns – (2.4 x 6) + (2.4 x 1.2) = 14.4 + 2.88 = 17.28
Points per season off of touchdowns – 17.28 x 16 = 276.48
If they would have kicked an extra point after every touchdown that season, their scoring
statistics based on expected points for extra points would come out to:
Points per game off of touchdowns – (2.4 x 6) + (2.4 x .94) = 14.4 + 2.256 = 16.656
Points per season off of touchdowns – 16.656 x 16 = 266.496
Attempting a two-point conversion after every touchdown in 2011 could have led to a 9.984
(round up to 10) increase in points over the course of the season. 10 points is a substantial
amount of points and can make or break a season considering the Cowboys have a historically
bad record in games decided by three points or less. An increase in .624 points per game by
going for two point conversions would have jumped the Cowboys from 15th in total points per
game to 11th in total points per game in 2011. An increase in .624 points per game may not
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seem like much for the average fan, but it has a great impact in how well a team does over the
course of a season.
Attempting a two point conversion and play calling in general over the course of a game
and a season has a great affect on how well a team does and how many points they put on the
scoreboard or stop the other team from putting up. If coaches paid more attention to the
statistics of a game and the game theory of football, their teams could do much better as a
whole.
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One Page Summary
A zero-sum game is a situation where a gain for one side means that the other side will
have a corresponding and equal loss. Therefore, a zero-sum game is almost any contest or
situation where there is one winner and one loser.
In a perfect world, two players or teams in a zero-sum contest will act rationally and
make choices that minimize their maximum possible loss. This “minimax” solution also means
that players make play selections that do not follow a predictable pattern that might give their
opponent an edge.
A “minimax” solution is determined using the Nash equilibrium strategy, which tells a
player or team how often they should throw a certain move, or run a certain play. This Nash
equilibrium strategy tells a player or team exactly how they should act throughout the course of
a game or situation for them to minimize their maximum possible loss.
Statistics in the NFL have shown that teams should be attempting two point conversions
after a touchdown much more often than they attempt to kick a field goal. The expected points
based on the success rate for the two choices come out to be .98 for an extra point and .98 for
a two-point try. However, these statistics are not completely accurate, as they do not take into
account the poor play calling of coaches going for two, and the success rate for kickers on
different teams varies. If coaches were to call a run play for their two-point attempt, there
would be a 60 percent success rate instead of 49 percent, and if you look at the Dallas
Cowboy’s kicker, the success rate drops to 94 percent from 98 percent. This gives expected
points of 1.2 for two point conversions and .94 for extra points, which is a substantial difference
over the course of a season and plays a huge role in decision-making.
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Bibliography
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2. Belsie, Laurent. "Game Theory and Major League Sports." The National Bureau of Economic
Research. N.p., n.d. Web. 27 Oct. 2014. <http://www.nber.org/digest/oct09/w15347.html>.
3. Shor, Mike. "Game Theory and Business Strategy." Game Theory Readings: Mixed Strategies in
American Football. N.p., n.d. Web. 27 Oct. 2014.
<http://www2.owen.vanderbilt.edu/Mike.Shor/courses/gametheory/docs/lecture05/Football.html>.
4. "Zero-sum." Merriam-Webster. Merriam-Webster, n.d. Web. 27 Oct. 2014.
<http://www.merriam-webster.com/dictionary/zero-sum>.