Fantasy Football: Optimizing Permutations of Teams Based on Statistics
Derick Owens and Amanda Zimecki
Gene Tagliarini, Professor
Abstract
The idea of fantasy football has been around since
the early 1960s and, with the invention of the
internet, has become one of the biggest industries in
sports. Millions of people participate in the game,
making it extremely competitive. Many, if not most, of
those participating are intending on making money,
as it is a form of gambling. And, of course, in any
form of gambling, the odds are against the player.
We’ve set out to design an algorithm to increase the
odds the player has of winning in a week-by-week
fantasy draft using DraftKings. We will discuss the
type of algorithm we will be using and how the
algorithm will increase the odds of the player
winning.
1 Defense/Special Teams (D/ST)
1 Tight End (TE)
1 Flex (RB, WR, or TE)
Given that there are 32 starting players per position
(since there are 32 teams), that yields 32 (QB) * 32
(RB) * 31 (RB) * 32 (WR) * 31 (WR) * 30 (WR) *
32 (TE) * 32 (D/ST) * (30 [RB] + 29 [WR] + 31
[TE]) (Flex) permutations, or 87,063,684,710,400
possible permutations of teams.
extremely large demographic. Its impact on society
varies in results, from what can be considered
We will be using a neural network to give each
player a rating based on 2014’s statistics imported
from fftoday.com. We will then retrieve the player’s
dollar value assigned by DraftKings and add it as an
attribute or statistic to each player. This value will
assist us in a knapsack implementation of the
problem, as DraftKings provides a $50,000 salary cap
for each player. We will also create a ratio referred to
as “true value,” which will be determined by dollar
value*our rating. The produced rating will always be
less than 1, except for those cases in which the player
is elite. The true value, when summed for a complete
and drafted team, will be less than or equal to the
salary cap. But how will we produce these
permutations of teams?
positive to what can be considered negative.
Gambling has an impact on the economy and various
parts of society. Its addictive nature can ruin people’s
lives.
We will be using a greedy implementation of the
knapsack problem to produce possible permutations
of teams. The closer to the salary cap of $50,000,
theoretically, the better.
Key Words: Gambling, gaming, fantasy football,
algorithms, computer science, programming, neural
network, knapsack problem.
1. Introduction
Gambling is a multi-billion dollar industry with an
That’s part of the reason why we’re developing an
algorithm – to make it easier. We want to level the
playing field for players of fantasy football.
We will be combining a neural network with a
knapsack implementation. DraftKings.com uses a
$50,000 salary cap to draft a nine-player team. The
team’s positions consist of:
1 Quarterback (QB)
2 Running Backs (RB)
3 Wide Receivers (WR)
However, it will be possible that a permutation of a
team’s dollar value will be very close to the salary
cap, but the true value will be much lower since each
player’s dollar value is being multiplied by that same
player’s rating produced by the neural network to
produce its true value. In order to draft the
theoretically “best” team, the user would use the
permutation with the highest summed true value.
2. Neural Network
•
+/- .1 each standard deviation (b)
below/above the position average
Each calculated statistic must then be multiplied by a
threshold. The threshold we decided on is the average
player’s statistic, multiplied by two, and is calculated
by the following:
•
Our neural network will create ratings for each
position: QB, RB, WR, TE, and D/ST. We will be
using a different rating system for each position, as
each position has different statistics that determine
the quality of the player. As discussed in the
introduction, the statistics will be the most recent
available statistics in 2014 and will be retrieved from
fftoday.com. Once retrieved, the rating for each
statistic must be rated for the player. Statistics will be
calculated in the following manner:
Tp = Sp/[(ΣSq/q)*2], where:
• T = Threshold
• S = Statistic (being calculated)
• q = All Players
• p = Individual Player
Note that it is possible for a player, if the player
excels in a certain statistic, to earn a score greater
than the “soft” maximum of 1. This is possible if he
exceeds the threshold of twice the average player.
Therefore, if the player excels in such a manner in
enough of the statistics, he can be defined as an elite
player and a rating greater than 1 is computed.
Yards per season, TDs per season, Games Played,
Touches, Targets:
•
R = (Sp/(ΣSq/Σq)) * w, where:
• S = statistic
• p = individual player
• w = weight
• q = all players
• R = individual player rating
Turnovers:
•
R = 1-(T*0.05) * w, where:
• R = individual player rating
• T = Turnovers
• w = weight
Note: It may seem that .05 is a “magic number” in
the calculating the turnover rating, but the concept is
quite simple. Given that the player has 0 turnovers in
a season, that player gets a perfect score of 1 for the
turnovers rating. However, if he has 20 turnovers, he
has a score of 0. It is possible for the player to receive
a negative rating if the player has more than 20
turnovers during the season.
Catch Percentage:
•
R = a (+/-) .1(b), where:
• a = Base score of all players
average
2.1. Neural Network Implementation
Our neural network implementation will be using
weights produced, at this point, by the software
developer’s football knowledge. While these appear
to be “magic numbers,” unless there is some form of
back propagation, we have no choice but to use
“magic numbers.”
There are some issues with the possibility of an
implementation using back propagation. The results
are limited to one: the results of the points produced
by the permutation (which are, of course, known only
after the games occur) vs. the best possible
permutation (which the odds of producing are 1:87
trillion). This means that implementing a weightlearning algorithm would be difficult, or even
impossible, considering there is one unique
permutation to one unique result. There are also too
many unaccounted factors to possibly produce an
algorithm that learns weights. We’ll define such
factors as “football knowledge.”
Football knowledge refers to recognizing external
factors in football: tracking injury reports, coaching
history between two opponents (how well do the
coaches know one another?), a player or team’s
performance in certain weather conditions, a player
or team’s performance in certain types of venues
such as turf or natural grass fields, etc. This list could
go on for quite a long time, and it would be
extremely strenuous to program football knowledge,
let alone set weights that would operate on an
efficient level for those factors.
Therefore, it is better, in our opinion, to focus on the
hard statistics and optimize the permutations for the
user, so the user can use their football knowledge to
make the best choice out of those optimized
permutations produced by our algorithm.
2.2.1. QB Rating Implementation
Quarterbacks will be given their rating based on the
following statistics and respective weight:
STATISTIC
Total Yards (Passing +
Rushing)
Total
Touchdowns
(Passing + Rushing)
Games Played
Turnovers
Completion %
WEIGHT
.20
.25
.35
.15
.05
2.2.1. RB Rating Implementation
Running backs will be given their rating based on the
following statistics and respective weight:
STATISTIC
Total Yards
Total
Touchdowns
(Receving + Rushing)
Games Played
Touches
Targets
Turnovers
WEIGHT
.20
.20
.25
.15
.10
.10
2.2.1. WR Rating Implementation
Wide Receivers will be given their rating based on
the following statistics and respective weight:
STATISTIC
Total Yards
Total
Touchdowns
(Receiving + Rushing)
Games Played
Turnovers
Targets
Catch %
WEIGHT
.20
.20
.25
.10
.15
.10
2.2.1. TE Rating Implementation
Tight Ends will be given their rating based on the
following statistics and respective weight:
STATISTIC
Total Yards
Total
Touchdowns
(Receiving + Rushing)
Games Played
Turnovers
Targets
Catch %
WEIGHT
.15
.25
.25
.10
.15
.10
2.2.1. D/ST Rating Implementation
Defense/Special Teams will be given their rating
based on the following statistics and respective
weight:
STATISTIC
Turnovers Forced
Points Allowed
Total Touchdowns
Sacks
Yards Allowed
WEIGHT
.20
.20
.25
.10
.15
3.1. Knapsack Implementation
Our knapsack implementation has a maximum value
of $50,000, which is the salary cap set by DraftKings.
It is extremely unlikely that we “fill,” the knapsack,
but that isn’t the objective. Our knapsack, although
extremely important, simply gives us our dollar value
permutations. One may believe that the “best”
permutation is one that completely filled the
knapsack. While that is ideal, it doesn’t necessarily
improve the odds of winning. You could be filling
the knapsack with dirt rather than gold. How do we
make it more likely to be picking up gold rather than
dirt?
Once we have the possible permutations produced,
we will multiply each player’s dollar value by the
player’s rating, producing that player’s true value and
sum those to produce the team’s true value:
•
V = d * R, where:
• V = true value
• d = dollar value
• R = our produced rating
•
VT = Σ(Vp), where:
• V = true value
• T = team
• p = player
Note: There is a possibility that, if a team consists of
enough elite players, that the sum of the true values
exceeds the salary cap of $50,000. This, however, is
extremely unlikely, as DraftKings sets their dollar
values for the players according to their performance,
and the elite players are set at a much higher dollar
value than non-elite players.
Once we have generated a true value for each
permutation, we can then display to the user the top
permutations available to choose from. There will
surely be thousands and thousands of permutations
available to the user, some of which have the same
true value. The user will then have to use his football
knowledge to make the best selection out of the
permutations we produced with our algorithm.
4. Algorithm Complexity
The complexity of the entire algorithm is at least
O(n), where n is the number of players. The
complexity of the neural network is O(n) since the
purpose of the neural network is the compute the
rating values of each player. In the “black box” of the
neural network, it is simply computing arithmetic in
order to produce the rating.
Once we have ratings produced, it is then time to
organize those player ratings into permutations. We
will use the greedy implementation of the knapsack
problem to get the permutations that are closest to the
salary cap as possible. The complexity of this
algorithm depends highly upon the implementation
we use.
If a brute force algorithm is used, the complexity will
be 2n where n is the number of players. If we could
produce an implementation of the knapsack problem
using dynamic programming, however, the
complexity could be maintained at O(n).
Brute force is a very straightforward approach and
would be an exhaustive search, where we would
create all permutations of teams and produce the
optimal solution(s).
Dynamic programming, however, works by breaking
this problem down into smaller problems and using
those smaller problems to construct possible
permutations in a bottom-up format. If the
generalized permutation with one or more missing
players doesn’t maintain a given threshold X with the
maximum value of the player from that missing
position, we can throw out the entire permutation and
get rid of an entire position, then try again with
another position. This method could reduce the
complexity dramatically since the method would be
crossing out a “column” of position players and a
“row” of an entire permutation. Furthermore, this
process can be repeated until the threshold of X is
met by all permutations, so the optimal solution set is
what would be left over.
We could further reduce the computational
complexity by adding a GUI with checkboxes beside
the name of each player. If the user knows directly
who they want in a certain position, they could check
the box of that player. Since the knapsack would be
“partially filled” with that player, that position would
no longer be needed to be filled in the knapsack.
Based on the fact that there are 32 starters for each
position, the complexity would be reduced a
maximum of 232, all with a simple Boolean value and
user input.
5. Rating Results
Our rating results matched very well with those of the
experts on ESPN.com. We as well noticed slight
variations which was what we were looking for. This
discrepancy is what makes our player’s value unique
in comparison to professional analysts. Figure 1
provides a visual representation of our highest valued
Running Backs compared to those of the sport’s
experts. This graph is also applied to all of the other
positions to provide us with a visual model showing
our values uniqueness. We are very pleased with our
results and we look forward to continuing with
implementing a GUI and future enhancements to the
algorithm.
Figure 1