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.