How much is your Super Bowl square worth?
Most Super Bowl parties have a mini-betting pool called Super Bowl squares. In this game, a 10 x 10 grid is set up, with each axis ranging from 0 to 9. One axis represents the trailing digit of the home team’s score and one represents the trailing digit of the away team’s score. In most games, each square is sold for $10. At the end of each quarter, if the current score corresponds to your square, you win a chunk of the $1000 pot. For example, if the 1st quarter ends with the home team winning 17–13, the holder of the square (7, 3) would get paid.
In most games, you buy in and are randomly assigned a square, so don’t take this post as gambling advice. But it might be worthwhile to know how much value you can expect out of the square you get assigned.
I scraped Pro Football Reference for the scores at the end of each quarter for the regular season games from the past three seasons. Using these frequencies, I made a heatmap of the frequencies for each score combination.
What immediately stands out is that the highest frequencies scores are combinations of 0, 3, and 7. No surprises there, as field goals are 3 points and touchdowns are 7. Zero can represent both the lack of scoring and the combination of a TD and FG. The most notable low frequencies are all scores that end in 2, 5, and 8. These scores are rare (2 and 8) or impossible (5) to achieve on their own and difficult to get by combining FGs and TDs.
However, the likelihood of hitting your square changes by quarter. When the game starts 0–0 in the first quarter, naturally you are much more likely to hit on a 0–0, 3–0, 7–3, etc. with the initial scoreless state. But as the game progresses and the starting score of the quarter becomes more and more varied, we should expect the end scores of the quarters to get more diverse — and that’s exactly what we see.
We see the first quarter is dominated by the 0s, 3s, and 7s, and while these scores stay the most probable throughout, the disparity between these scores and the rest fade as the game goes on. In the 4th quarter, for example, (4, 7) is one of the most prominent squares. While the same squares are still the most probable, their probability is far lower relative to the probabilities in earlier quarters.
In a Super Bowl squares game where the holder of the correct square at the end of each quarter gets 25% of the $1000 pot (assuming $10 buy-in), we can find the expected value (EV) for each square — effectively how much that square is worth at the beginning of the game. Of the 100 possible squares, only 26 have positive EV (you are expected to win money), while 74 have negative EV.
The disparity in Super Bowl squares is astounding. Randomly getting assigned the square (0, 0) gives you an expectation of making nearly $80 while getting anything in the bottom 10 will leave you with an expectation that would have to be paid out in coins.
Breaking this down by quarter, we see that the first quarter is the big money maker for these squares. As the game goes on, less and less money can be expected each quarter for the more popular squares. The standard deviations of EV for each quarter reflect this, dropping from 6.93 to 3.73 to 2.89 to 1.94. In short, the spread of these expected values drops from quarter to quarter. By the 4th quarter, we have scores that end in 1 and 6 crack the top 10 EVs.
These expected values change if the game is played with a different payout structure. Many games payout 20–20–20–40% to put more emphasis on the winner. Another common structure is 20–30–20–30. These structures (especially the former) diversify the EV by putting weighting the less-predictable 4th quarter scores.
With a 20–20–20–40 payout, the EVs get pushed towards $10. The ranking stays much the same (with (0, 0) leading the way at $69.53) with 28 squares now having a profitable expectation and only 5 squares under $1. Not much of a change, but a bit less extreme as the more diverse 4th quarter scores are more heavily weighted.
How did last night shape up?
While the Patriots won last night, playing the probabilities won too. In most games you can’t choose your squares, but if you could, playing by the book would’ve been the best strategy. Going with a more common 20–20–20–40 payout, the (0, 0) square (expectation $69.53) won the first quarter, good for $200. The square (0, 3) (expectation $44.27) won the second and took home $200. The big winner was the (3, 3) square (expectation $27.86) which won both the 3rd and 4th quarters and won $600. These winners were the 1st, 5th, and 10th most likely squares respectively.
In fact, it’s a good thing to get (0, 0) when the Pats are in the Super Bowl. Five of their nine appearances in the Tom Brady Era have had the first quarter end with the score 0–0.
For a boring Super Bowl, we were rewarded with an equally boring night of Super Bowl squares. And on a night that the most familiar team won the Lombardi Trophy, the most familiar squares paid out.
