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# Aaron Brown's Super Bowl XLIX Squares

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## A cheat sheet for a classic American gambling extravaganza.

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In 2013 and 2014, I wrote columns describing how to value draws in the ever-popular Super Bowl Squares office pool.

With Super Bowl XLIX scheduled for Sunday February 1, I'm going to update my strategy for the 2015 game; if you need more details on how it works, see the previous columns here and here.

The game is played on a 10 X 10 grid. Players buy one or more squares by paying a fixed amount (I'm using \$50 per square, but it can be any amount) and writing their name on the appopriate square(s).

When all squares are filled, the organizer draws the numbers 0 through 9 at random to assign to each of the rows, and then again to assign to each of the columns. In this way, each square gets assigned a row digit from 0 to 9 and a column digit from 0 to 9.

Suppose your square gets 4 for the row and 8 for the column. By convention, the rows represent the NFC team (Seattle Seahawks this year) and columns represent the AFC team (New England Patriots). You win if the final digit of the Seahawks' score is 4 and the final digit of the Patriots' score is 8. So if the Seawaks win 24-18, you win.

Note that this is different from the square with 8 for the row and 4 for the column, which wins if the Seahawk's last digit is 8 and the Patriots' last digit is 4.

Rules differ on how to award the prize money, but a common pattern is to award one-eighth of the pool based on the first-quarter score, one-quarter based on the halftime score, one-eighth based on the score at the end of the third quarter, and one-half based on the final score of the game.

There is no skill involved in this game; it's pure gambling because the row and column assignments are random. Still, it's nice to know what your square is worth, and sometimes people buy, sell, or trade squares.

The most common method for valuing squares is using the results of past NFL games.

This is not accurate. Games differ quite a bit. If you only use similar games, your sample size is too small for accurate value estimates, especially of the less valuable squares. If you use a large sample, you will be including games played under different conditions, by different kinds of teams, possibly in different eras.

A better approach is to simulate games. You don't need the kind of sophisticated simulation a quantitative sports bettor would use. All you have to do is get a reasonable guess at the most probable numbers of touchdowns and field goals each team might score (we'll talk about missed extra points, two point conversions, safeties and overtime later).

For 2015, the Patriots are a 2-point favorite over the Seahawks, and the over/under (a bet on the total number of points scored by both teams) is 48.5. So the oddsmakers are telling us they expect the Patriots to score 25.25 points, on average, and the Seahawks 23.25. There are reasons these odds differ from the actual probabilities, but they're more accurate forecasts than most people can come up with on their own. They're certainly good enough for this purpose.

The other thing that matters a lot is the propensity of teams to get touchdowns versus field goals. In the regular season and the play-offs, the Patriots scored 52 touchdowns versus 35 field goals. The Seahawks scored 43 touchdowns and 31 field goals. The Patriot ratio is typical of NFL teams while the Seahawks have relatively more field goals. The big difference from last year's Super Bowl, which also featured the Seahawks as a 2 point underdog and had a similar over/under of 48 points, is that the AFC team last year, Denver, scored lots of touchdowns and few field goals.

On the defensive side, the Patriots gave up 32 touchdowns and 29 field goals, while the Seahawks allowed 27 touchdowns and 21 field goals. Both teams have smothering defenses that are not only stingy with points, but are efficient at holding opponents to field goals even when they do score.

Looking at these figures, a reasonable expectation is that both teams expect to get two field goals on average, with the Patriots adding an expected 2.75 touchdowns and the Seahawks adding an expected 2.5.

Of course, you could do a fancy play-by-play simulation factoring in weather, ball inflation, injuries and play calls; but it won't matter much for the distribution of score digits. You'll get almost as accurate a result by pretending the scores are random dice throws, with expected numbers similar to the ones I estimated above. This would not be a smart way to bet on football games, but it's a pretty good way to value Super Bowl Squares.

One thing that does matter is some less common events. Although individually uncommon, they are important for the values of certain squares.

Safeties are one. I believe there's little reason to think any team or game is more or less susceptible to safeties than average; there's just not enough data. So sticking in some average value is fine. I put missed extra points in the same category, unless you're doing weather or injury simulations. Two-point conversions can be covered by simple rules that NFL coaches follow most of the time.

The most important deviations come near the end of games. A team trailing by eight points that scores a touchdown with less than a minute to go will go for two points, not one. And if the game is tied at the end of regulation, one team or the other will add either 3 or 6 (or very rarely 2) points to its total.

Here's how my simulations value this year's Super Bowl Squares. Squares are colored from green (most valuable) through yellow (average) through red (least valuable). Any number above \$50 is good, since the squares cost \$50.

Last year 3,0 was the most valuable square, and in 2013 it was 0,0. These squares are good in 2015 (and always) but the best square this year is 0,7. Zero is usually the best number, but this year 7's are about equally good.

Threes are generally the third best number, but this year fours are almost as good as threes.

Six and one are average numbers. Squares with 2s, 5s, 8s, or 9s are generally bad.

This year, as usual, 2,2 is the worst square (last year that dubious honor went to 6,6). The board is pretty symmetrical this year, in most cases, the order of your numbers doesn't matter much.

In case your game uses a different payout structure, I give the four component tables for the table above: how much you expect to get for the first quarter, half-time, third quarter, and final score payouts, based upon a total payout of \$5,000 (\$50 X 100 squares):

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