Analysis of two NFL teams with coincident scores in the same week

ComptrBob

EOG Master
Prologue: JK commented that it would be interesting to see my analysis for the HRWager sponsored contest involving two NFL team posting identical scores. The following is a write-up of my analysis for attempting to pick two teams with a contest advantage. I've tried to keep the text understandable and not get too verbose so at least a few readers can get through it. :)

Coincidental NFL two team score analysis

History: Recently EOG.com offered a contest with a lone question: Which two NFL teams will record identical scores in Week 14?

Problem statement:
Choose two different NFL teams who will match their scores in a given week's play (and if not match, then will minimize the absolute value of the difference in scores) in a given week. We want to maximize our chances in a contest with an unknown number of contestants each of whom will select one unique pair of teams (denoted as Team A and B). Common sense says we want 2 teams whose scoring patterns are similar. Because football scoring is "lumpy" (majority of scores are 3 and 7), we expect that identical scores by two teams are fairly frequent.


Methodology:
There are two primary variables, denoted TotA and TotB, which are the scores of Teams A and B respectively. We use game theory and postulate the existence of a payoff function of these two independent variables that serves as an approximate metric of our chances of winning. Maximizing that function is one way of determining a solution to this class of problem.


Nomenclature: Let PtAn = probability of TotA = n, and PtBn= probability of TotB = n.

Analysis:
Let?s assume we have modeled a discrete probability distribution (DPD) for the scoring patterns of Team A and B in their specific games. We might consider eliminating the case of Team B being Team A?s opponent since the chances of a tie are so rare, however this isn't necessary.


The scoring model may be very elaborate since it could take into account the bettable team total, line, total, offensive stats, defensive stats, team opponent, matchups, injuries, weather, etc. Or it may be quite simple just using a template of the NFL "average" team score distribution and making some adjustments to various key number frequencies to reflect either higher or lower than average scoring.

Let?s compute the probability Pexact (case of TotA = TotB) given that we have a DPD for the scoring patterns (assume the scores are independent events):

Pexact = summation [PtAi * PtBi] from ?i=0 to max score

Now look at the case where Abs (TotA ? TotB) = 1, denoted as ?Off By 1?

Poff_by_1 = summation [PtAi * (PtB(i-1)+PtB(i+1) + PtBi * (PtA(i-1)+PtA(i+1) ] from ? i=0 to max score

? And the general term given by:

Poff_by_n = summation [PtAi * (PtB(i-n)+PtB(i+n) + PtBi * (PtA(i-n)+PtA(i+n) ] from ? i=0 to max score

Let the payoff function be of the form:

F (TotA, TotB) = Pexact + a1 * Poff_by_1 + a2 * Poff_by_2 + a3 * Poff_by_3 ? Eqn (1)

Where a1, a2, a3, ? constant coefficients are weighting factors to be determined.


Clearly if we choose the constants appropriately and can maximize F, we have an excellent chance at getting at least a near-optimal pair of teams. As the number of contestants goes up, we expect that we have to reduce the size of the constants a1,a2,a3, etc. Also note that when we look at terms like Poff_by_7 and higher, we might consider making a7 and beyond zero or even negative to penalize cases (we know the probabilities are always nonnegative) that result in large discrepancies.

Let?s consider the average scoring distribution for the NFL over the last 10 years, 2004 through 2013 YTD.

The six highest frequencies are: 17 (7.2%); 20 (6.8%); 24 (6.5%); 27 (5.6%); 10 (5.4%); 13 (5.3%) ... the other values can be obtained from any database of NFL scores. The max score, which we used above, turns out to be 62.

With both teams having the ?standard? DPD we can compute an ?average? Pexact which turns out to be 4.01%. Poff_by_1 turns out to be 9.35%, Poff_by_2 = 6.72% and Poff_by_3 = 13.75%.

Incidentally, if we use only 5 years of data, (2008 ? 2013), these percentages are almost the same, Pexact = 4.02%, Poff_by_1 = 9.40%, Poff_by_2 = 6.60% and Poff_by_3 = 13.70%. However, scoring is up in the last 5 years, and the six highest frequencies are: 17 (7.3%); 24 (6.8%); 20 (6.8%); 27 (5.9%); 31 (5.1%); 13 (5.0%).

Now we need to estimate the number of contestants to see how often someone will hit the exact TotA = TotB and how many times we will go to Off By 1, By 2, etc?

Assume 40 contestants is approximately correct for the contest being run. Using the binomial distribution, we find that the exact score match will be hit by at least one contestant 21.45% of time. ?Off by 1? hits 18.79% if there is no exact match. Off by 2 hits 16.77% of the time if the first two cases aren?t hit, etc? We expect the winner?s difference will be two points (or less) more than half of the time.

We want to choose the terms a1, a2, a3, etc to produce smaller and smaller contributions to the payoff function since the lowest the difference, the more likely it is to win the contest, thus let?s choose a1 = 0.3, a2 =0.2, a3 = 0.05, and a4 = 0, etc, which leads to Faverage = .0401 + 0.3*.0935 + 0.2*.0672 + .05*.1375 = .0885

This is the benchmark value for our payoff function F. F for 5-year data (2008-2013) is the same to 3 significant digits.

If we choose, we can incorporate team total derived by using the line and total for the specific game (using Week 14 of 2013).

The following table is sorted from low to high:

Team Mean Score
Browns,Rams,Vikings 17.5
Colts 18.5
Jets 19.0
Dolphins,Seahawks,Titans 19.3
Bills 19.5
Jaguars,Redskins,Panthers,Raiders 21.5
49ers,Falcons 21.8
Giants,Steelers,Texans,Buccs 23.0
Chiefs 23.8
Cards 24.0
Saints 24.3
Bears,Bengals,Cowboys,Ravens 24.5
Chargers,Packers,Lions 25.8
Eagles 28.3
Patriots 28.5
Broncos 31.3

We can use these team totals and in addition, an NFL scoring model if we have one, to produce all 32 distributions (there are over 900 combinations of two teams). Perform the calculation of the payoff function F with Eqn (1) for each pair of teams, which is straightforward programming or can be done with a spreadsheet. As expected, teams with similar team totals have larger values of F. Teams at the extremes tended to have larger variances and smaller values of F.

There were many good combinations, it turns out that the Cards/Saints pair gave a value of F very near the maximum, at .0928. This selection doesn?t guarantee a significant edge over all other contestants since that depends on their choices as well. It does have about a 5% better payoff function than the ?average? 2-team distribution.


Results:
The actual contest ended up with 39 contestants. The result of Week 14 was that the Cards scored 30 and the Saints 31, two ?mid-value? scores (31 is 7[SUP]th [/SUP](4.9%) and 30 is 15[SUP]th[/SUP] (2.6%), both scores went over their projected team total and happened to be almost identical, tying for first place in the contest with 6 others, a higher number of very close guesses than expected (should have been about 5.2 contestants that either had identical or ?Off by 1? results using the ?average? scoring distributions).


Conclusion:
The methodology and results appear to give a fairly significant advantage over blindly picking two teams. It does not give a large advantage, but when this method is combined with a good scoring model, it does give good sets of candidate team pairs. It provides an example using a payoff function which is widely used in more complicated optimal control theory and game theoretic problems.




 
Last edited:

Voodoo

EOG Addicted
Re: Analysis of two NFL teams with coincident scores in the same week

Not going to pretend to have absorbed all of the math stuff but something
did seem intuitively logical - that the lower scoring projected teams would
make the best partners. If your team total projection is 20 or less, Id just
use two of those and assume there would be less variance in their results
then teams that had high, but similar, team totals like 30.
 

trytrytry

All I do is trytrytry
Re: Analysis of two NFL teams with coincident scores in the same week

well yea

F (TotA, TotB) = Pexact + a1 * Poff_by_1 + a2 * Poff_by_2 + a3 * Poff_by_3 ? Eqn (1) :houra
 

PapaParlay

EOG Enthusiast
Re: Analysis of two NFL teams with coincident scores in the same week

Not going to pretend to have absorbed all of the math stuff but something
did seem intuitively logical - that the lower scoring projected teams would
make the best partners. If your team total projection is 20 or less, Id just
use two of those and assume there would be less variance in their results
then teams that had high, but similar, team totals like 30.

Low scoring teams tend to also be more inconsistent. Look at the jets, averaging 19.0 but scoring anywhere from 3 to 37 and everywhere in between depending on which Geno Smith (or other QB) decided to show up that day.
 

WVU

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

PCBob must be a blast at parties
 

trytrytry

All I do is trytrytry
Re: Analysis of two NFL teams with coincident scores in the same week

PCBob must be a blast at parties

this and the "pinny lean" work well with the ladies
"This problem is similar to having 32 shotguns with slightly different bores and sights, and thus spray patterns"
 

Tto827

EOG Dedicated
Re: Analysis of two NFL teams with coincident scores in the same week

this and the "pinny lean" work well with the ladies
"This problem is similar to having 32 shotguns with slightly different bores and sights, and thus spray patterns"
:LMAO

You mean you don't encounter that problem on a daily basis Try :LMAO

Interesting post though Bob, was fun to try and follow along for as far as I could.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

Low scoring teams tend to also be more inconsistent. Look at the jets, averaging 19.0 but scoring anywhere from 3 to 37 and everywhere in between depending on which Geno Smith (or other QB) decided to show up that day.

Yes, this is a case where "human intuition" is wrong about low scoring teams scoring more consistently low than mid-scoring teams consistently scoring mid-level values.

Just a small sample, but it illustrates the point, the lowest 10 team totals (1st-10th) from Week 13 had an average absolute value of the difference of actual points versus team total of 9.8 points, the middle 10 teams (12th-21st) had an average 6.5 points, and top 10 teams (23rd-32nd)had an average of 9.4.
 
J

joeybagadonuts

Guest
Re: Analysis of two NFL teams with coincident scores in the same week

Interesting stuff, Bobby.

:cheers
 

John Kelly

Born Gambler
Staff member
Re: Analysis of two NFL teams with coincident scores in the same week

ComptrBob retains his status as EOG's resident math guru for yet another year.
 

John Kelly

Born Gambler
Staff member
Re: Analysis of two NFL teams with coincident scores in the same week

Yes, this is a case where "human intuition" is wrong about low scoring teams scoring more consistently low than mid-scoring teams consistently scoring mid-level values.

Just a small sample, but it illustrates the point, the lowest 10 team totals (1st-10th) from Week 13 had an average absolute value of the difference of actual points versus team total of 9.8 points, the middle 10 teams (12th-21st) had an average 6.5 points, and top 10 teams (23rd-32nd)had an average of 9.4.

I got fooled here.
 

Tto827

EOG Dedicated
Re: Analysis of two NFL teams with coincident scores in the same week

I got fooled here.
Technically incorrect but in more layman terms that should help if your being serious....

Average absolute value meaning the average difference in the amount of points actually scored by a team vs. their listed team total.

So for the 10 lowest team totals, each scored an average of plus/minus 9.8 points more or less than their listed total.

You only used week 13 as an example here Bob, please don't waste time on it as I wouldn't know how to utilize the info anyway, but do you know off the top of your head if long term this holds true? In college ball, with some of the huge score lines put up, I'd almost guarantee that the higher team totals have more variance.

But thinking now, in the NFL teams don't score more than ~55 so there won't be those huge +/- 40 points swing on a given game like you could see a few times a year with Baylor, Oregon, etc.
 
Re: Analysis of two NFL teams with coincident scores in the same week

:cheers

Hey JK, can you transfer my free $100 wager for winning that contest to ComputerBob. I am done wagering for the time being, and ComputerBob has helped me immensely from reading his posts over the years
(not enough to win lol, but his posts have helped a lot)
 

Tto827

EOG Dedicated
Re: Analysis of two NFL teams with coincident scores in the same week

Quite a gesture Inkwell

Hope you enjoy the holidays, you prolly deserve to
 

ChiTownJoe

EOG Dedicated
Re: Analysis of two NFL teams with coincident scores in the same week

Cursory glance, but Bob just scratched the surface.

"The scoring model may be very elaborate since it could take into account the bettable team total, line, total, offensive stats, defensive stats, team opponent, matchups, injuries, weather, etc. Or it may be quite simple just using a template of the NFL “average” team score distribution and making some adjustments to various key number frequencies to reflect either higher or lower than average scoring."
 

John Kelly

Born Gambler
Staff member
Re: Analysis of two NFL teams with coincident scores in the same week

:cheers

Hey JK, can you transfer my free $100 wager for winning that contest to ComputerBob. I am done wagering for the time being, and ComputerBob has helped me immensely from reading his posts over the years
(not enough to win lol, but his posts have helped a lot)

Consider it done, Inkwell77.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

Just a small sample, but it illustrates the point, the lowest 10 team totals (1st-10th) from Week 13 ...

Technically incorrect but in more layman terms that should help if your being serious....

Average absolute value meaning the average difference in the amount of points actually scored by a team vs. their listed team total.

So for the 10 lowest team totals, each scored an average of plus/minus 9.8 points more or less than their listed total.

You only used week 13 as an example here Bob, please don't waste time on it as I wouldn't know how to utilize the info anyway, but do you know off the top of your head if long term this holds true? In college ball, with some of the huge score lines put up, I'd almost guarantee that the higher team totals have more variance.

But thinking now, in the NFL teams don't score more than ~55 so there won't be those huge +/- 40 points swing on a given game like you could see a few times a year with Baylor, Oregon, etc.

I meant to say the week of the contest, Week 14, not 13, I did not try to "cherry pick" a specific week ..lol.

To be clear, the average absolute value of the difference is not the same as the average difference. Say, two teams actual vs. listed team total are -7 and +7 respectively. The average difference is zero (-7 +7)/2, while the average of the absolute values is (7+7)/2 = 7.

I don't know if the lowest 10 team total has a larger long-term discrepancy (average abs) than the middle 10 or top 10. I suspect it might, and even using the two team payoff function F, close low pairs aren't a whole lot worse than pairs with middle team totals. Obviously, if 2 teams ever existed in the NFL that averaged say 10 points and had scores that only landed on 0,3,6,7,10,13,14,17,20,and 21, they would most likely produce the best value of the payoff function F.

College football, obviously, has a quite different scoring pattern. To solve the optimal "exact match" problem for college, you would use the same technique, but with much different data.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

Cursory glance, but Bob just scratched the surface.

"The scoring model may be very elaborate since it could take into account the bettable team total, line, total, offensive stats, defensive stats, team opponent, matchups, injuries, weather, etc. Or it may be quite simple just using a template of the NFL “average” team score distribution and making some adjustments to various key number frequencies to reflect either higher or lower than average scoring."

Very astute reading, Joe. I update a scoring model for each team for each week of the NFL. I could write a book on the subject. The probability distribution for scoring allows me to construct bets on the game so its pretty much my "holy grail".
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

Consider it done, Inkwell77.

I very much appreciate the favorable comments, but this isn't necessary. Please keep the account with Inkwell77.

I usually don't even enter contests because they can be a distraction, but the HRWager radio contest is usually pretty quick and interesting. The work I did here was really out of intellectual curiosity. Like a lot of things, if you don't practice analysis through mathematics, you get rusty at it. I'm glad it was fun for at least a few readers to wade through.
 

ChiTownJoe

EOG Dedicated
Re: Analysis of two NFL teams with coincident scores in the same week

Bob, awesome stuff. I would be interested in any book you put together, if you ever get around to putting one together.

how did you determine a1-a4 values?
 

John Kelly

Born Gambler
Staff member
Re: Analysis of two NFL teams with coincident scores in the same week

Me too. I'm the dummy in the term dummy-down.
 

pantherman

EOG Enthusiast
Re: Analysis of two NFL teams with coincident scores in the same week

The first post in this thread is the single best post I've ever read on this site. Really great analysis Bob.

FWIW, while I was in Vegas, we looked into creating offerings on NFL score distributions, but it was thought that the audience was too niche.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

ComptrBob and I chat often.

He still confuses me.

Me too. I'm the dummy in the term dummy-down.

I try to be as clear as I can when discussing these rather detailed subjects. I'm always up for answering questions. :)

One thing that bothers me about our education system is that many math teachers do not convey how logical, simple, and "cool" math can be when you take it step by step. I was very fortunate to have some great teachers along my educational path.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

Bob, awesome stuff. I would be interested in any book you put together, if you ever get around to putting one together.

how did you determine a1-a4 values?

The a1, a2, a3, ...an factors are used in the payoff function to "clamp down" the system to optimize the desired result. Sometimes they are even called clamping parameters. First off, I chose a payoff linear function (wrt to 'Poff_by_n's) for simplicity, usually this works best, but not always.

Then in a contest of 40, I wanted to optimize for a result where the winner was either exact or off by 1, so the a1 = 0.3 sets the contribution of the Poff_by_1 term to slightly less than the 0.0401 Pexact term. The a2 and a3 are chosen to makes these terms contribute something, but way less than the first 2 terms. The a4 ... set to zero to target winning by "off by 0,1,2, or 3". If we had 100 contestants, the a1-a4 values would be even smaller to try to optimize for Pexact.
 

ComptrBob

EOG Master
Re: Analysis of two NFL teams with coincident scores in the same week

The first post in this thread is the single best post I've ever read on this site. Really great analysis Bob.

FWIW, while I was in Vegas, we looked into creating offerings on NFL score distributions, but it was thought that the audience was too niche.

Thanks. Unfortunately, you are right that the audience for NFL score distributions is too niche. Sometimes I look at the Superbowl props involving scoring distributions, but mostly for fun. :)
 

John Kelly

Born Gambler
Staff member
I remember a talented EOGer (PANTHERMAN) proclaiming to me that ComptrBob's analysis of this topic was pure genius.

Where is this talented EOGer now?

He's an NFL executive crunching salary cap numbers for a team west of the Mississippi.

He one day will hold the position of NFL General Manager.

#EOG'sMostAccomplishedAlum
 
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