__After__the squares are selected, each row and each column is assigned a unique digit, chosen at random, from 0-9. Squares become winning squares after the first three quarters and after the final score by using the ending digit of the score. So, if after the first quarter, the score is Home Team 14 Visiting Team 3, then the square identified by the row with the number "4" and the column with the number "3" is the winning square for that quarter. It is possible that a single square wins more than one time throughout the game. What strategies can you employ for this game in terms of how many squares you can pick and how those squares are configured?

### Best Numbers

If you knew which rows and columns would get assigned which digits, then you'd want the best numbers for each of the quarters (Q1-Q4) and the final score (FS). Here are the probabilities that have shown up using 5220 professional football game box scores (quarter by quarter scores) from 1994 (when the two point conversion rule was adopted) through the entire 2013-2014 season.

This shows that overall for Q1-Q3,FS, the numbers 0 (especially in the first quarter) and 7 are the best, 3 and 4 are good, 6 and 1 are OK, and 8, 9, 2, and 5 are not so good (unless the score that day happens to turn out to be 42 to 35!).

But the overall numbers only tell part of the story. You have a combination of numbers - one number for each team. Depending on which of your numbers corresponds to the favorite and underdog team does matter. Here is an expanded table showing the relative value of all the possible combinations of digits for Q1-Q3,FS. This table shows what your numbers are worth relative to the others. The 0-0 and 7-0 (favorite-underdog) squares are the best and worth about 8x the average square. The 2-2 rarely shows up and is worth only about 1/40th (0.02) of an average square's value.

### The Strategy

The bottom line is that Football Squares a fair game, such that over the long term, it makes no difference as to what squares you select in terms of how many winning squares you are likely to collect. Someone who has no knowledge of football is expected to win just as many squares as anyone else choosing the same number of squares. This bottom line, however, does not mean that you can't use a strategy. A strategy can be used to affect the likelihood of certain outcomes in any one game. In short, you can trade a small increased chance that you won't win any squares for a much larger chance that you'll win two, three, or even (although rare) four squares in a single game.

Using the 5220 professional football game box scores (1994-2013 regular season), over one billion computer simulations have been run where squares are picked, a game is chosen, and rows and columns assigned random 0-9 digits. It turns out there are 60 unique configurations you can select up to five squares given the digits for rows and columns are random. Over a billion simulations are needed because there are 4,124,276,932,608,000,000 total possibilities (10! x 10! x 5220 x 60 - that's over 4 quintillion or 4 billion billion).

At the extreme ends, there are two strategies that emerge. If you pick squares such that each square selected has a unique row and column, you will maximize your chance at winning something (most often only a single square) and minimize your chance, relative to other strategies, that you'll win two or more squares in a single game.

The other extreme is to select a strategy that maximizes your chance of winning two or more squares. You can try to go home the BIG winner (I like this strategy!). This strategy says that you should select squares along the row/column corresponding to the visiting team (or with even stronger results, use the row/column corresponding to the underdog especially if the game is expected to be low scoring). In fact, you should pick as many squares as you can in that row/column and even start on a next row/column as long as you aren't violating any etiquette for picking too many squares.

Consider selecting four squares in the two extreme strategies ... configuration H, picking them all in a column - maximizes winning two or more squares and configuration A, picking them with unique rows and columns - maximizes winning something. Expanded, the probability table looks like this:

The expected squares are calculated by multiplying the number of wins with the associated probability and summing it all up. The numbers for 2 or more wins and winning something (>0) match the values next to the patterns below. All configurations of selecting four squares sum to 16% as each square has a 1% expectation of winning each of the Q1-3,FS (4 squares x 4 opportunities x 1% each is 16%). Notice, that essentially your strategy - how you select your squares - allows you to trade off a small probability for winning something for a much larger probability for winning two or more squares.

### All possible ways to select up to 5 squares

Below is a final tabulation of all possible ways to select 1-5 squares along with the probability of winning 2 or more squares and the probability of winning something. Reading top to bottom and then left to right, the configurations are ordered to maximize your probability of winning two or more squares. The column direction reflects the underdog team and the row direction reflects the favorite. Squares next to each other mean any squares on the same row or column. A square offset from the others mean any square with a unique row and column. Pick one of the extreme configurations or pick one in between - the choice of strategy is yours

### The Bottom Line

In summary, your strategy can't change your total expected value -- only the probability for various outcomes to happen. However, your strategy can definitely change the likelihood for the different outcomes. For me, I want to go home the big winner, so I'll live and die each time I play by selecting them all in a column corresponding to the underdog team. More times, I'll go away without winning, but when I do win, I'll be the big winner of the day! The choice of strategy is yours!

This is a fantastic analysis, but it falls short of answering one very basic question. "Which configuration is the best configuration to play which maximizes your TOTAL EXPECTED EARNINGS. To answer this we need to know the partial percentages for Win = 0, Win = 1, Win = 2, Win = 3, Win = 4, and Win = 5. If we know those individual percentages than we can simply use the following formula for Expected value for each configuration. EV = WP(1)*payout + WP(2)*payout*2 + WP(3)*payout*3 + WP(4)*payout*4. This equation assumes all payouts are equal which may or may not be the case for a persons given pool. For the sake of this investigation let's assume that all payouts are the same. Then the equation simplifies to the following:

ReplyDeleteEV = OnePayoutAmount * [WP(1win) + 2*WP(2wins) + 3*WP(3wins) + 4*WP(4wins)]

Since it seems that you've done most of the leg work already, can you tweak your simulation to produce the WP's (win probabilities) for each of the exact number of wins and then plug into this formula so we can make a comparison of configurations based on TOTAL EXPECTED VALUE. It might turn out that competing strategies will be a wash in the end, but you never know until you examine the data.

Thanks for the comment. The total expected earnings per square selected is the same! There is no way to maximize your total expected earnings. When I started this study, indeed, I was looking for the way to maximize expected earnings but soon realized every square is identical in terms of it's expectation value since the digits for rows and columns are random. No square is preferred and no combination of picking multiple squares is preferred in terms of giving higher expected earnings.

DeleteWhile the expectation value is the same, different configurations have different probabilities for the possible outcomes. In your notation, WP(1), winning a single square, can be relatively high and WP(2),WP(3),WP(4) relatively low OR you can pick a configuration where WP(1) is lower, but WP(2),WP(3), and WP(4) are higher. Total Expected Earning for either configuration, no matter how the payouts are awarded, is exactly the same.

I've modified the post to make this explicit using your suggestion. showing all the probabilities for configurations H and A when selecting four squares.