## Super Bowl squares revisited

Posted by Doug on January 30, 2008

**See also: **PFR Super Bowl Squares mobile app

Three Super Bowls ago, I wrote this post over at Sabernomics. In it, I looked at your probability of winning a squares pool with any given square. For example, I found that in a one-unit-per-square pool, either of the '0/7' squares would have an expected value of about 3.8 units. Compare that with, say, a '5/6' square, which has an expected value of 0.22, or the lowly `2/2' square and its expected value of .04. Because it was all the data I had at the time, I only considered the last digits of the final scores of games, but someone correctly pointed out in the comments that most pools also give prizes for (the last digits of) the cumulative scores at the end of each quarter.

Well, now I have score-by-quarter data for the entirety of the NFL's 2-point-conversion era (1994--present), so it's time for an update.

I'm sure there are lots of ways to do this, but a bit of googling indicates that a standard payout structure is something like 10% of the pot after each of the first three quarters, and 70% for the final. This doesn't alter things too drastically, but it does have a couple of effects.

- The '0/7' squares enjoy an even larger advantage over an average square. The '0/7' squares have an expected value of about 4.9 under this scheme.
- The '0/0' square starts climbing the charts.

More than 20% of all games are in a '0/0' situation (remember, that includes 10-0 and 10-10 as well as 0-0) after one quarter. At halftime, about 7.5% of all games are a '0/0.' So the more weight you put on the intermediate stages, the better the '0/0' square looks. Here is a chart that shows the expected value of a given square after each quarter, along with a final column that shows the expected value under a 10/10/10/70 system:

+-----+-----+------+------+------+------+------+ | | | q1ev | q2ev | q3ev | q4ev | ev | +-----+-----+------+------+------+------+------+ | 7 | 0 | 11.8 | 5.6 | 4.7 | 3.9 | 4.9 | | 0 | 0 | 20.5 | 7.5 | 4.4 | 1.9 | 4.5 | | 3 | 0 | 9.2 | 5.1 | 3.4 | 3.3 | 4.1 | | 7 | 7 | 6.9 | 6.3 | 4.2 | 2.2 | 3.3 | | 7 | 4 | 1.3 | 3.0 | 3.3 | 3.4 | 3.1 | | 7 | 3 | 4.7 | 4.5 | 3.3 | 2.0 | 2.7 | | 4 | 0 | 3.5 | 3.6 | 2.6 | 2.1 | 2.4 | | 4 | 1 | 0.0 | 0.5 | 1.6 | 2.3 | 1.8 | | 3 | 3 | 3.1 | 3.2 | 3.3 | 1.2 | 1.8 | | 4 | 3 | 0.9 | 2.3 | 2.3 | 1.5 | 1.6 | | 7 | 1 | 0.1 | 1.5 | 2.0 | 1.8 | 1.6 | | 6 | 0 | 1.1 | 2.2 | 1.7 | 1.5 | 1.6 | | 4 | 4 | 0.2 | 1.8 | 2.3 | 1.5 | 1.5 | | 6 | 3 | 0.3 | 1.5 | 1.5 | 1.7 | 1.5 | | 1 | 0 | 0.3 | 1.2 | 1.3 | 1.5 | 1.3 | | 7 | 6 | 0.5 | 1.7 | 1.6 | 1.0 | 1.1 | | 3 | 1 | 0.1 | 0.9 | 1.0 | 1.0 | 0.9 | | 8 | 1 | 0.0 | 0.0 | 0.0 | 1.3 | 0.9 | | 8 | 0 | 0.0 | 0.4 | 0.8 | 1.0 | 0.8 | | 6 | 4 | 0.0 | 1.1 | 1.2 | 0.8 | 0.8 | | 9 | 7 | 0.1 | 0.5 | 0.7 | 0.8 | 0.7 | | 6 | 1 | 0.0 | 0.4 | 0.5 | 0.9 | 0.7 | | 9 | 3 | 0.1 | 0.4 | 0.5 | 0.7 | 0.6 | | 9 | 0 | 0.2 | 0.7 | 0.5 | 0.7 | 0.6 | | 7 | 5 | 0.0 | 0.2 | 0.4 | 0.8 | 0.6 | | 8 | 7 | 0.0 | 0.0 | 0.0 | 0.8 | 0.6 | | 1 | 1 | 0.0 | 0.0 | 0.0 | 0.8 | 0.6 | | 5 | 0 | 0.1 | 0.2 | 0.4 | 0.7 | 0.6 | | 8 | 3 | 0.0 | 0.0 | 0.0 | 0.7 | 0.5 | | 9 | 4 | 0.0 | 0.3 | 0.6 | 0.6 | 0.5 | | 7 | 2 | 0.1 | 0.3 | 0.5 | 0.5 | 0.5 | | 6 | 6 | 0.0 | 0.6 | 0.5 | 0.5 | 0.5 | | 8 | 4 | 0.0 | 0.0 | 0.0 | 0.8 | 0.5 | | 4 | 2 | 0.0 | 0.2 | 0.4 | 0.6 | 0.5 | | 2 | 0 | 0.1 | 0.4 | 0.6 | 0.6 | 0.5 | | 9 | 6 | 0.0 | 0.0 | 0.0 | 0.7 | 0.5 | | 9 | 1 | 0.0 | 0.0 | 0.0 | 0.6 | 0.4 | | 3 | 2 | 0.0 | 0.1 | 0.3 | 0.5 | 0.4 | | 8 | 5 | 0.0 | 0.0 | 0.0 | 0.6 | 0.4 | | 5 | 4 | 0.0 | 0.0 | 0.0 | 0.5 | 0.4 | | 8 | 6 | 0.0 | 0.0 | 0.0 | 0.5 | 0.3 | | 6 | 2 | 0.0 | 0.1 | 0.1 | 0.4 | 0.3 | | 5 | 3 | 0.0 | 0.0 | 0.0 | 0.5 | 0.3 | | 9 | 2 | 0.0 | 0.0 | 0.0 | 0.4 | 0.3 | | 8 | 8 | 0.0 | 0.0 | 0.0 | 0.4 | 0.3 | | 5 | 1 | 0.0 | 0.0 | 0.0 | 0.3 | 0.2 | | 6 | 5 | 0.0 | 0.0 | 0.0 | 0.2 | 0.2 | | 2 | 1 | 0.0 | 0.0 | 0.0 | 0.3 | 0.2 | | 5 | 2 | 0.0 | 0.0 | 0.0 | 0.3 | 0.2 | | 9 | 8 | 0.0 | 0.0 | 0.0 | 0.3 | 0.2 | | 5 | 5 | 0.0 | 0.0 | 0.0 | 0.2 | 0.1 | | 9 | 5 | 0.0 | 0.0 | 0.0 | 0.2 | 0.1 | | 8 | 2 | 0.0 | 0.0 | 0.0 | 0.2 | 0.1 | | 9 | 9 | 0.0 | 0.0 | 0.0 | 0.2 | 0.1 | | 2 | 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | +-----+-----+------+------+------+------+------+

Here's how to read that. Take the top line for example. If you have one of the two '0/7' squares, then your expected value is 11.8% of the first-quarter pot, 5.6% of the second-quarter pot, and so on. With the 10/10/10/70 system, your overall expected value would be about 4.9.

As I was poking around the web looking for info on standard payout schemes for these kinds of pools, I came across this page. One of the commenters there suggests using not the final digit of each team's score, but the final digit of the *sum* of each team's score. So a 17 would be an 8, a 22 would be a 4, and a 38 would be a 1.

I was too lazy to check this one out quarter-by-quarter, but just looking at final scores, this scheme produces a much flatter expected value curve.

+------+------+------+ | | | ev | +------+------+------+ | 7 | 4 | 2.0 | | 8 | 2 | 1.6 | | 4 | 1 | 1.5 | | 6 | 3 | 1.5 | | 8 | 6 | 1.4 | | 9 | 6 | 1.4 | | 6 | 4 | 1.4 | | 8 | 5 | 1.4 | | 7 | 0 | 1.3 | | 4 | 0 | 1.3 | | 6 | 5 | 1.3 | | 8 | 7 | 1.3 | | 4 | 3 | 1.3 | | 3 | 0 | 1.2 | | 5 | 3 | 1.2 | | 6 | 1 | 1.2 | | 7 | 6 | 1.1 | | 8 | 3 | 1.1 | | 8 | 1 | 1.1 | | 8 | 4 | 1.1 | | 9 | 3 | 1.1 | | 5 | 4 | 1.1 | | 5 | 2 | 1.1 | | 7 | 3 | 1.1 | | 5 | 1 | 0.9 | | 9 | 8 | 0.9 | | 3 | 1 | 0.9 | | 7 | 7 | 0.9 | | 7 | 5 | 0.9 | | 9 | 7 | 0.9 | | 9 | 4 | 0.9 | | 3 | 2 | 0.9 | | 4 | 2 | 0.8 | | 7 | 1 | 0.8 | | 9 | 1 | 0.8 | | 8 | 0 | 0.8 | | 5 | 0 | 0.8 | | 6 | 0 | 0.8 | | 6 | 6 | 0.7 | | 1 | 0 | 0.7 | | 9 | 5 | 0.7 | | 3 | 3 | 0.7 | | 7 | 2 | 0.7 | | 9 | 2 | 0.7 | | 9 | 0 | 0.6 | | 5 | 5 | 0.6 | | 2 | 1 | 0.6 | | 4 | 4 | 0.6 | | 1 | 1 | 0.6 | | 6 | 2 | 0.6 | | 2 | 0 | 0.5 | | 8 | 8 | 0.5 | | 0 | 0 | 0.2 | | 9 | 9 | 0.2 | | 2 | 2 | 0.0 | +------+------+------+

Look where '0/0' is now!

I don't think *fairer* is the right word, but this seems to me to be a clearly more interesting pool. It is less determined by the random assignment of squares and more determined by the random actions that happen as the game unfolds. And that's how it ought to be.

For the extra geeky, one way to improve this (in my opinion) would be to, at the end of the game, flip a coin to determine whether the criterion to be used is "last digit" or "last digit of sum of digits." In other words, say the game ends up at 23-14. If the coin comes up heads, it's a '3/4'. If it comes up tails, it's a '5/5'. The point of the pool is to keep people interested. With the coin flip rule in place, I'd guess that, even into the fourth quarter, just about anyone (except the poor suckers with '2/2' and '9/9'; they're beyond help) could invent a reasonable scenario whereby he or she collects the prize.

So if 2/2 wins the pool, will it be the largest Super Bowl upset of all time?

It'll be the largest Super Bowl Squares upset of all time. The name, I think, does it justice as only we would even care about it.

I participated in an excellent version of this last year (too bad the game itself sucked) in which we bid for the squares after the numbers had been assigned. The guy who bid $20 for one of the 7/0 squares was really ticked when 4/6 (which went for $1) gobbled up the 1st and 2nd quarters. We did it like a silent auction during the pregame...it was a lot of fun!

What would the numbers look like if the payout per quarter was equal for all at 25%, with the 4th quarter being the final score (if OT)?

we did "squares" at work. it was a late start and we were unable to fill all the squares after we collected all the money. what do we do now if it lands on an empty suare?? any ideas appreciated. thanks.

Doug,

Adding up the numbers, I get total evs for q1ev of 65.2%, q2ev of 58.3%, q3ev of 52.5%, and q4ev of 54.7%. If these are percentages, shouldn't they add up to 100%?

genamb,

when I've run into that before, I've done one of two things:

1.) Carried that quarter's money over to the next quarter. Ala a state lotto.

2.) Made empty squares "common squares" and that quarter's money was divided evenly among all participants. Requires an even number of participants.

thanks mike! I will suggest that. I like the lotto idea! does anyone else have any suggestions???

What I'd do is just increase the other prizes proportionately. Instead of each quarter being worth $121.25, the 3 quarters with an actual winner will instead be worth $161.67. If every quarter hits an empty square, just declare the pool a nullity and return everybody their money.

thanks newstotom! that is also a good idea- it will make it interesting. We have one day to figure it out and I appreciate all input!!!

NewsToTom:

Expected Value is the probability multiplied by ther potential payout; so these #s will not add up to 100% as they do not represent pieces of a pie so to speak.

Right Doug?

NewsToTom:

I noticed that too. What you have to do is double the percentage for all non-double squares. That is, double the value for 70, 74, 36, etc., but not for 00, 77, 33, etc.

-Eddo

Thanks for these odds! I managed to get in a pool where the squares weren't random and was able to make some easy money ðŸ™‚

Gibybo, glad to help.

NewsToTom and Paul, Eddo has it right. If you imagine a 0-7 line just under the 7-0 line (and with identical ev), a 0-3 line under the 3-0 line, and so on, then it would add up to 100 (+/- rounding error).

These odds are great and can be used to teach (or steal) from the statistically under-informed masses.

I took the high moral ground and used these numbers in an Excel tool, linked via my name, which will provide a framework within which you can run and analyze a particular pool.

Thank you for the data!

Would there be an easy way to modify these numbers into expected value for a 25/25/25/25 split?

In other words, every quarter has an equal weight.

Another great way to keep it interesting is to do forward and reverse jersey number pay outs. For instance I am in a 60 dollar per square pool that runs the jersey numbers. If a player scores a touchdown (for example player #81) Then NFC 8 AFC 1 would get 60.00 and NFC 1 and AFC 8 would also get 60.00. We do the same for interceptions. It gives some of the crappier squares a chance to win something. Then the remaining money is divided 10% first quarter score, 20% half time, 10% third quarter and 60% final score. It's alot of fun and keeps everyone involved in the game.