After March madness (which I have written about
), the second most popular office sports betting tradition in the United States is Super Bowl Squares (third is probably the Kentucky Derby, which I also wrote about
, although this varies more by region). Super Bowl Squares differs from the other two in being completely random; there is no skill involved.
I’ll describe the most common version, but there are plenty of variants. Players pay a fixed amount (I’m using $50 in my example, but I have seen games from $0.10 per square to $500) and write their name somewhere in a 10 x 10 grid. After all 100 spaces are filled, the digits 0 to 9 are randomly assigned to rows and columns. By tradition, the NFC team is represented by the rows, while the AFC team is represented by the columns, but this makes no difference.
Payouts are based on the last digit of the score of each team at the end of the first quarter, half, third quarter, and game. A common payout structure is one eighth of the pool for first and third quarter, one quarter of the pool for the halftime score, and half of the pool for the final score. With $50 squares $5,000 is collected and -- according to that schedule -- would be distributed with $625 going to each of the winners of the first and third quarter, $1,250 to the winner of the half, and $2,500 to the winner of the game. The same square can win more than one of these payouts.
For example, suppose you picked the upper left square and the number drawn for the first row was 1, and the number drawn for the first column was 3. You would win any quarter in which the 49ers’ score ended in 1 and the Ravens’ score ended in 3. This is an average pick; in fact, it is the most average pick this year. 3s are very common, especially in the early going. 1s are neither common nor uncommon, but occur mostly later in the game.
Note that this pick is not the same as 49ers 3, Ravens 1. That pick wins if the 49er score ends in 3 and the Ravens score ends in 1. This year, it is a slightly worse pick.
Although there is no skill involved since the numbers are assigned randomly, it’s nice to know what your square is worth (or if people buy, sell, or trade squares, it’s valuable to know what your square is worth). The most common valuation method is to compute the average historical payout in the 46 Super Bowls played to date. This is not bad for the most valuable squares, but the sample is too small to get good values for the less common ones. One particular anomaly is there have been seven safeties in Super Bowls, when you would expect fewer than three based on overall NFL statistics. Safeties make a dramatic change to the square values; there seems no reason to assume they should be more common in Super Bowls. Another problem is that the game has evolved over the years, making the early data not representative.
Another common approach is to use recent regular season and playoff games. This gives a reasonable sample size for all but the least likely squares (and perhaps you don’t care whether your 2,2 is worth $0.50 or $2.00; 2,2 is particularly bad because there are no common scores that end in 2 below 42, and if the fates smiled and the game ended 42-42, it would go into overtime and you would still lose). Here the problem is that elite games are different from average games.
It’s also common to take shortcuts in the computation. For example, instead of computing the payoffs, people will sometimes just compute the frequency for each digit being the last digit at the end of a quarter. In that method, 3,1 and 1,3 always have the same value. There are two problems with this (in addition to the data problems mentioned above). The first problem is that the scores affect each other. The biggest reason is games cannot end in ties, so two of the same number like 2,2 is less likely than you would expect if you only knew the frequencies of 2s. There are less important interactions as well. Teams may go for two points, or choose a touchdown attempt over a field goal, or otherwise change behavior based on the score differential.
The second problem is that the distribution of scores differs by quarter. 0,0 is the most common result after one quarter; it has a 14% probability this year (that does not just mean a score of 0,0 -- it could be 10,0 or 10,10 or something else as long as both teams’ scores end in 0). But there is only a 2.5% probability that the final scores will end with 0,0 (there is a 3.5% chance 0,0 will be the score at the end of regulation, but a lot of those are 0,0 and 10,10 ties, which get broken in overtime).
The right valuation method is to simulate the game. You don’t need to do a sophisticated simulation as you would for normal sports betting. I will describe a simple method that is good enough for this purpose.
The two most important parameters are the level of the score and the proportion of touchdowns to field goals. The first matters because at high scores different final digits are more common than at low scores. The second matters because the mix of touchdowns and field goals makes certain numbers more or less common (9, for example, is only likely as a result of three field goals and no touchdowns; you rarely see a missed PAT and a field goal, a touchdown and a safety, or scores of 19, 29, 39 or higher numbers).
As long as you get those two parameters right, you’ll have reasonably accurate valuations. The point spread and over/under give pretty good estimates for score level calibration. For example, this year, with an over/under of 47 and the 49ers favored by 4, Las Vegas is telling us that the 49ers should score 25.5 points on average, with 21.5 for the Ravens (the numbers add up to 47 and differ by 4). While Las Vegas odds are not perfect, they are far more accurate than guessing scores based on average NFL statistics, average Super Bowl statistics, or what the two Super Bowl teams did in previous games.
We can look at regular season statistics for each team to get an estimate for the ratio of touchdowns to field goals. San Francisco scored 44 regular season touchdowns and kicked 29 field goals; for Baltimore, the numbers are 42 and 30, respectively. Those ratios -- applied to the expected point totals above -- suggest that San Francisco expects to get 20 points from touchdowns and 5.5 from field goals, while the Ravens expect 16.5 points from touchdowns and 5 points from field goals.
Now that we have our two important parameters right, we can afford to be sloppier. A simple assumption is that the distributions of touchdowns and field goals are independent Poissons. I won’t go into the math here (it’s easy to look up online), but this assumption allows us to convert an expected number of events into a probability distribution over possible totals. For these numbers, it looks like:
So, for example, we think that there is a 6% chance that the 49ers will score no touchdowns, and a 6% x 16% = 1% chance they will score no touchdowns or field goals and be shut out.
Now let’s consider each possible outcome. Let’s start with the most likely outcome: 2 touchdowns and 1 field goal for each team. We know the probability by multiplying the four individual probabilities: 23% x 29% x 26% x 31% = 0.5%. We’ll assume that each of these scores is equally likely to occur in any quarter. That gives 256 possible combinations; for each combination, we can determine the final digits of each team’s score at the end of each quarter, and the probability will be 0.5% / 256.
There’s one last step because this simulation ends in a tie. To keep things simple, pick one team at random and add 3 to its point total, as this is the most common result of overtime. Now all we have to do is repeat the process for all the common combinations of touchdowns and field goals, and sum the results. It would be tedious to do by hand, but it’s easy on a computer.
I do something a bit more sophisticated: I simulate the game drive by drive. I don’t go to the play-by-play level; I collect statistics on what happens each time either team has the ball (touchdown, field goal, punt, turnover returned for a touchdown, turnover not returned for a touchdown, and less common outcomes). I also have a model for the clock, which changes behavior and can end a drive, and simple models for missed PATs, two point conversion attempts, and safeties. I account for strategy changes with score differential and time. These things matter for some of the less common squares.
My results below are for a $50/box game with $625 for the first and third quarter, $1,250 for the half, and $2,500 for the final score. The most valuable square is 0,0 at $257.95, and any pick with only 0s, 3s, or 7s is good. The worst pick is 2,2 at $1.35. Picks with 2s, 5s, 8s, and 9s are bad. The neutral numbers are 1, 4, and 6. The cells are colored green for valuable, red for not valuable, and yellow for middle, with gradations.
If you’re buying or trading squares, most people will overvalue 0s and 8s this year, especially for the 49ers, while 2, 6, and 9 will generally be undervalued. For example, most people would rather have 0,0 than all the squares that start with 49ers having 6. This year, I think having 6,0, 6,1, and so on up to 6,9 is worth $545, more than twice the $258 of 0,0.
If you want the individual breakdowns, here are the values for Q1, out of $625:
Halftime, out of $1,250:
Q3, out of $625:
And final score, out of $2,500:
If you play a version that rewards the fourth quarter score -- that is, the score at the end of regulation -- here are the values based on a $625 total; however, these values are not added in to the total value table at the top:
So good luck to everyone.