## Super Bowl Squares

Posted by Doug on January 26, 2009

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

This is a re-run of a post I ran a year ago:

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.

ADDENDUM: this post at the footballguys message board has the data only for games where one team was a 6.5--7.5-point favorite. It's based on a smaller sample, of course, but it might help you estimate the difference between Steelers7/Cards0 and Cards7/Steelers0

Bear with me; what is a squares pool?

In a squares pool, you buy a square. That square randomly gives you two numbers, in this case, one for the Steelers and one for the Cardinals. At the end of each quarter, if the last digit in the score matches yours, you win. For instance, if you had Steelers 3, Cardinals 4, and at the end of the half, the score was 14-3 Cardinals (or 14-13 or 24-33 or 4-3), you would win the prize for that quarter.

Obviously, having common numbers like 0, 7, 3, and 4 help, and having unlikely numbers like 9, 2, 5, and 8 don't - but there are always exceptions (Patriots 32, Panthers 29 springs to mind). Also, like the article said, every contest is different - the one I'm doing costs $5 a square, and you win $50 if the numbers match at the end of the 1st or 3rd, $150 at half, and $250 at the end of the game.

A few points:

1. 10-10-10-70 is really the "typical" payout structure? Most squares pools I've been in are 25-25-25-25 (with the final 25% going to final score, not score at the end of the fourth quarter).

2. For the "sum-the-digits" expected values, I have a feeling the original inquiry meant to sum the final digit for each team to get a single digit for each quarter. That is, a 7-7 tie at the end of one quarter would be 4 (7+7=1

4). A 28-21 final score would be a 9 (8+1=9).3. I may sound critical, but this is still an awesome post. I remember forwarding the original post to a few friends.

But what squares should I pick???

Haha, nice analysis though!

I love the analysis. I usually don't like the squares game, because I feel its almost like playing the lottery. But this gives a nice angle to it.

Our pools is 12.5%, 25%, 12.5%, 50%, because halftime has more significance than the 1st and 3rd quarters.

Hmm, is the math on this adding up? For the 3rd quarter I get a 14.7% chance of the matching digits and a 37.8% chance of unmatching digits. 37.8% * 2 + 14.7% = 90.3%. (Q2 = 97.2%, Q1 and Q4 are both within 1% of 100%.)

Now, I thought squares had a home team and an away team. This shows no correlation to home or away.

And the squares are based each on the scale of "ending in 0-9"

10 across and 10 up, giving 100 possible outcomes, this list has only 55. Are we to assume all of the missing data including and after 2/2 has 0 value?

Am I thinking of the wrong squares game?

Brian - The list only has 55 because it doesn't distinguish, for example, between the score of 0(home)-7(away) or 7(home)-0(away).

Not counting similar digit scores (i.e. 0-0,1-1,2-2, etc) there are 45 unique possibilities. Factoring in the 10 similar digit scores makes 55.

This is a great post. I have always thought that the normal way to do squares pools is wrong - 2/3 of the people in the pool lose interest after the numbers are posted. A more sporting way would be to use your study and have people buy squares AFTER the numbers are assigned. They would pay more for the better squares. The prices would be assiged in proportion to the findings of your study. And, similarly to the guy at the horse track who picks up the long shot, a guy could buy a bunch of the "lousy-number" squares for bargain prices and get more for his investment if the "long shot" numbers came in.

I also can't believe I just spent this much time on the subject of squares pools!

keith (#9):

A good way to keep interest is to re-draw numbers each quarter. Now, this only works if you have ten or fewer people at your party.

Rather than having a two-score square, we just have people draw one number, zero through nine (without seeing the number they're drawing, obviously). Then, at the end of each quarter, you add up the two scores. Say the quarter ends 28-14. That adds up to 42, so whoever drew "2" wins the pot for that quarter.

It's easy and quick to re-draw ten numbers each quarter - just take a deck of playing cards and pull out the ace through ten of one suit and have each person pick a card.

Thanks to Doug's research, I put together an excel tool to analyze the value of squares contingent on user specified rules

http://www.xlssports.com/2009/01/superbowl-squares.html

I have a question?

Say your running a board and the pay offs are per quarter.. o.k no problem BUT! as time runs out for that quarter and as we seen in the steelers/cardinals at the half the score was steelers 10, Cardinals 7 but time ran out and the steelers scored a td with no time on the clock, ok so that means its 16/7 and now they get the chance to a fieldgoal now its 7/7, so i guess my question is which score actualy gets paid????? (0/7 6/7 7/7)....

The score at the end of each quarter - so 7/7 would get paid off, since the quarter ended 17-7.

And this SB was pretty boring with the numbers that ended up arriving: 3/0, 7/7, 0/7, 7/3.

And I thought I was a nerd. Whew, that's some research. Here's some I did that only uses the last 4 years worth of data -- and it has a little more color! Enjoy.

http://caseyshead.com/2010-super-bowl-squares-breakdown/