"For I repeat: there are no 'facts' about the future. Only 'opinions'."

Information about the future is just about always incomplete -- worse: inaccurate or wrong, biased or misleading.  Unless more and better information can be obtained -- make that until more and better information can be obtained -- you must work with what you have.  Superbowl Sunday is coming, which is a metaphor for many kinds of  management decisions that won't wait for all the desired information to materialize.

A recommedation repeated often on these pages is invoked again here with an excerpt from Discovering Assumptions,...

"Discovery is nourished by assumptions. If you don't have enough data, you what? -- assume something. When there is too much of the stuff or it's riddled with contradictions, what else can you do? Assume something."

"But!" bellowed the instructor while commencing an exaggerated scrawl across the top of the blackboard: "Make your assumptions explicit!"

Assumption #1 That the objective of each team will be to maximize its own score independently at all times.  Some assumptions are easy to make. This one simplifies our analysis by setting aside tactical matters arising on the field resulting from relative scores.  A team in the lead will act the same as a team coming from behind.

Assumption #2 That the next Superbowl will not end with a shut-out. Teams in the Superbowl are the best in the league; you would not expect either to be held scoreless.   It is a good idea to make explicit why each assumption is being made. This one avoids a bias toward the digit zero for a losing team that scores less than 10.  The next two assumptions are intended to simplify our analysis...

Assumption #3 That there will be no safeties and no two-point conversions in the next Superbowl.   Knowledge does play a part in assumptions such as this one.  Though not impossible, both of these scoring increments are extremely rare, as anyone familiar with the game will know.

Assumption #4 That there will be no missed conversions in the next Superbowl.  Missed conversions are indeed rare, but, more relevantly, they are irrelevant, since the same scoring effect will result from any pair of field goals, which are not rare.

At this point we have developed the beginnings of a model for the next Superbowl.  A model?

Time out.  Whereas touchdowns are often scored in a sequence of offensive plays following a kickoff return, touchdowns can also result from a turn-over (pass-interception or fumble) anywhere on the field.  Field goals, on the other hand, can be scored only by the team in possession of the ball -- the offensive team -- and only from some limited distance from the goal line.

he first Superbowl was played in Los Angeles between the Green Bay Packers of the NFL and the Kansas City Chiefs of the AFL on Sunday, January 15, 1967.  As of this writing there have been a total of 36 Superbowl Sundays.  In solving The Next Superbowl puzzle, surely you will want to consult the records from all previous games.  Not so fast...

For what purpose might all that information be used?

Well, you say, it is relevant knowledge, isn't it?  Plenty of facts, in fact.

On the other hand, what we really want to do here is formulate a model that will predict events for which we have limited knowledge and no facts at all. Unlike the weather and other trendy phenomena suitable for extrapolation, least significant digits in football scores are, from game to game, utterly independent.

The best use of those Superbowl records, it seems plain, will be to test the validity of our assumptions, the quality of our reasoning -- hey, the predictive power of our model.

We are not predicting the past, so we dare not be influenced by events from the past that have no causal connection to the next Superbowl.  Accordingly we elect to defer consultation of the record-books until after we make all of our assumptions and develop the model.  We can always use that information later to adjust our assumptions.  The technical term is "tweaking."

For now, let us pretend it is Saturday, January 14, 1967.

Relative score is the sine qua non for making wagering decisions.  For a football pool, the absolute scores decide the winning squares in the payoff matrix, for we saw earlier that, under our assumptions, certain digits are unlikely to appear in football, especially in low scores, whatever the score of the other team.

Here are two numerical assumptions derived by -- well, derived by SWAG.  The acronym stands for "Stupid Wild-Ass Guess."  You cannot get more explicit than that.

Assumption #7 That the winning team will outscore the losing team by a margin of 2-to-1.

ou will recall that the puzzle gave us another clue ("the team in blue is favored to win").  Upsets are not infrequent, though, so what we have here is not so much a clue as an additional puzzle condition.  Instead of being given the names of the teams before the next Superbowl, we are told that the winner's name will be assigned to the matrix for the columns and the loser's name will be assigned to the rows (after the game is over, obviously).

The SWAGs in Assumptions #6 and #7 have the winner scoring no more than 30 points maximum and the loser scoring no more than 15 points.  The losing team, for example, might score anywhere between 3 and 15 (a shut-out having been precluded by Assumption #2).  The winning team, under our assumption, can accumulate up to 30 points, so long as its final score is higher than that of the losing team when the clock runs out.

Throughout the game both teams are afforded what we have been calling "scoring opportunities" at which, if successful, points are accumulated. Each successful scoring opportunity could be the last one in the game; accordingly, the next Superbowl can be represented as a set of successful scoring opportunities.  Indeed, any number of "next Superbowls" can be thus simulated.  Forget about other statistics (running and passing, interceptions and fumbles).

But in the next Superbowl, how many points, 7 or 3, will be accumulated at each successful scoring opportunity?  Nobody knows, of course.  When the range of numerical values for an unknown is known but nothing else, people often resort to a random process and rely on what is popularly known as The Law of Averages.

Assumption #8 That the scoring increments accumulated by a team at each successful scoring opportunity can be selected by a random element.  In effect, we can flip a three-sided coin... Excuse the interruption -- "three-sided coin"?  Sure.  Think of a cylinder whose diameter equals its height.  Around the periphery is printed the phrase "field goal," and the two faces each read "touchdown."  Neat, huh.

One of the most practical tools for modeling nowadays is the spreadsheet (see, for example, Easy as Pi).  The author has created a Spreadsheet Model for Solving the Next Superbowl Puzzle.

ere are the results from a thousand Superbowls.
 
 

Each of these percentages represents the likelihood -- the chance -- of the respective cypher appearing as the least significant digit in the final score.  A blank entry means that under our assumptions, the corresponding cypher will appear so infrequently in the next Superbowl that they have been dropped from consideration (columns 4 and 7, along with rows 1, 5, and 8).

Taken as products of percentages in all possible combinations, these numbers represent the joint likelihood of winning $100 in each of the corresponding squares.  Any product less than 1.0% means we can discard that combination as being less likely than one in a hundred (pure chance).

Your best choice, apparently, is 6-to-7, which will appear in final scores of 16-7, 26-7, or 26-17.  The chance of that outcome is given as the product of 36.2% * 33.0% = 12.0%, which is 12 times more likely than "luck of the draw."  The next best is 6-to-4, which will appear in final scores of 16-14 or 26-14 and enjoys a likelihood more than 8 times what uniformly distributed cyphers would imply.

 

Winner Digit	Loser Digit	Joint Chance %	Total Wager	Cumulative Chance %
6	7	11.9	1	11.9
6	4	8.5	2	20.3
6	0	7.4	3	27.7
6	3	6.9	4	34.6
9	7	6.5	5	41.1
3	7	5.5	6	46.6
9	4	4.6	7	51.2
9	0	4.0	8	55.2
3	4	3.9	9	59.1
2	7	3.8	10	62.9
9	3	3.8	11	66.7
3	0	3.4	12	70.1
3	3	3.2	13	73.3
0	7	3.0	14	76.3
2	4	2.7	15	78.9
2	0	2.3	16	81.2
2	3	2.2	17	83.4
0	4	2.1	18	85.5
0	0	1.8	19	87.3
0	3	1.7	20	89.0
6	6	1.4	21	90.4
5	7	1.1	22	91.5

 

	0	1	2	3	4	5	6	7	8	9
0	19		16	12			3			8
1
2
3	20		17	13			4			11
4	18		15	9			2			7
5
6							21
7	14		10	6		22	1			5
8
9

As a matter of more than idle curiosity, you will want to check the validity of our assumptions against Superbowl records.  Have fun (see History of the Superbowl).

ere is the payoff matrix with all 36 past Superbowls superimposed.

 

	0	1	2	3	4	5	6	7	8	9
0	19		16	12			3			8
1
2	18
3	20		17	13			4			11
4			15	9			2			7
5
6							21
7	14		10	6		22	1			5
8
9

• Your selections would have paid off up to 12 times, depending on how many of those particular squares you purchased year after year.  Exclamatory punctuation is advisable.
• As indicated by the darker squares, five winning combinations were duplicated and one combination (4-to-7) appeared in the final scores of three Superbowls.  You would have missed them all.  Bummer.
• The most recent Superbowl (XXXVI) ended with a score of 20-to-17 (0-to-7 in the least significant digit department), and you would have won -- but only if you had purchased at least 14 of your 22 preferred squares.  Still, that was 7 times better than chance odds.  Hey, get a load of this...
• That 0-to-7 combination in the final score of Superbowl XXXVI had never appeared throughout the history of the Superbowl.  Even with access to the records for all 35 games, you would be clueless.  Not only that, but...
• If, for the Superbowl XXXVI pool, you had purchased the squares for all the combinations that had ever appeared in the past, it would have cost you $28 to find out that the recordbooks were no help to you.  So much for extrapolation, huh?
• Finally, if, year after year, you purchased the squares for every combination that had appeared in all previous years, you would have wagered a total of $554 and won six times, for a total of $600.  That's barely better than an even bet (1-to-1 odds).  Our model would have done better.  How much better?

This tabulation postulates 13 suppositions -- that you purchased the indicated squares for $1 each, summing up to the indicated number of squares as determined by entries in the column marked "Preferences."  Kind of a mouthful.  See if these comments help...

• If you had bet a buck a year on just your favorite square (6-to-7), you would have won only once (Superbowl III in 1969), losing ever since.  Overall, you got $100 in return for $36 in wagers (almost 3-to-1 in Las Vegas lingo).
• If, starting back in 1967, you had purchased your first three preferences every year (6-to-7, 6-to-4, 6-to-0), you would have won twice (Superbowls III and XX), pocketing $200 for $108 in wagers.  That's not great, but 2-to-1 is a whole lot better than chance (3 in a hundred).
• Careful inspection of the table shows that, provided you put at risk $16 per year over 21 years (1967-1988), you would have won 8 times, receiving $800, for a cost of $336 (almost 5-to-2 odds).  If you don't quit in 1988 (uh, why would you?) but continue purchasing the same 16 squares, by 2002 you would have invested $576 and won $1,000 (better than 3-to-2).
• Finally, you will see in the table that by repeatedly purchasing all 22 "Preferred" squares in the solution to The Next Superbowl puzzle, you would have won a total of $1,200 on $792 in wagers (still 3-to-2 odds) -- this, despite the fact that Preference number 22 (5-to-7) never came up.  Maybe next year.

s expressed above, "The best use of those Superbowl records...will be to test the validity of our assumptions, the quality of our reasoning -- hey, the predictive power of our model."  Inspection of box scores in the official Superbowl records shows the following:

Assumption #1 Teams maximize scores independently -- confirmed as postulated, with the come-from-behind team taking slightly more risk (see Assumption #3).

Assumption #2 No shut-outs -- confirmed.

Assumption #3 No safeties and no 2-point conversions -- a total 1,615 points have been scored throughout the history of the Superbowl; fewer than 14 have come from safeties and 2-point conversion (respectively 5 and 2 of each, less than 1%.

Assumption #4 No missed conversions -- the record shows a 95.8% success rate.

Assumptions #5 Two-to-one touchdowns versus field goals -- a total of 192 touchdowns have been scored in Superbowls and 97 field goals.  Would you believe 1.979-to-1?  Hoo-hah!

Assumptions #6 and #7 Those danged SWAGS -- they are the most doubtful.  We want to examine them closely in the light of the historical records.  Meanwhile...

Assumption #8 Use of random element -- although our model applied retrospectively did seem to give quite satisfactory results, we must not fail to notice that half a dozen combinations of least significant digits appeared repeatedly over the years, and our model did not select any of them.

This scattergram depicts the historical record of all 36 Superbowls.  The total score for each game is plotted on the horizontal axis, and the ratio of the winning team's score to the losing team's score is plotted on the vertical axis.  It looks like it might have been produced by an ink-sneeze, doesn't it.

• The SWAG in Assumption #6 is emphasized as a vertical line, setting an upper limit on the total scores for modeling purposes at 45 points.  Clearly that does not comport well with the historical record, since half the games scored higher than 45 points.  How embarrassing. Exclamation point declined. The vertical dashed line tweaks Assumption #8, allowing the model to encompass all but 5 of the games in history.
• The SWAG in Assumption #7 is emphasized as a horizontal line, imposing a 2.0-to-1 ratio on the final scores.  Not too bad.  The average ratio is found to be 2.1 for all 36 games.  By merely eliminating those two humiliating routs that occurred in 1972 (ratio 8.0-to-1) and 1990 (ratio 5.5-to-1), the historical average would indeed come out 2.0 on the nose.
• In our model, the effects of tweaking Assumption #6 are as follows: Increase the upper boundary for the winning team from 30 to 60 points and 15 to 30 for the losing team.

Having tweaked Assumption #6, we see that higher scores mean more opportunies for each cypher.  This time, there are 28 squares worth betting on, as shown in the following tabulation:
 

Winner Digit	Loser Digit	Joint Chance %	Total Wager	Cumulative Chance %
7	7	4.7	1	4.7
8	7	4.3	2	9.0
9	7	4.1	3	13.1
1	7	3.9	4	17.0
3	7	3.9	5	20.9
4	7	3.9	6	24.8
0	7	3.6	7	28.3
7	4	3.1	8	31.4
7	0	3.0	9	34.4
8	4	2.8	10	37.2
8	0	2.7	11	39.9
9	4	2.7	12	42.6
1	4	2.6	13	45.2
3	4	2.6	14	47.8
4	4	2.6	15	50.4
9	0	2.6	16	53.0
1	0	2.5	17	55.5
4	0	2.5	18	58.0
3	0	2.4	19	60.4
0	4	2.4	20	62.7
0	0	2.3	21	64.9
5	7	1.3	22	66.2
6	7	1.3	23	67.5
7	3	1.3	24	68.8
2	7	1.2	25	70.0
7	1	1.1	26	71.1
8	3	1.1	27	72.2
9	3	1.1	28	73.3

The payoff matrix for the football pool may now be marked to show the Order of Preference for each of your wagers.  Inasmuch as this exercise applies the solution to The Next Superbowl puzzle retrospectively, all 36 past Superbowls are already superimposed...
 

	0	1	2	3	4	5	6	7	8	9
0	21	17		19	18			9	11	16
1								26
2
3								24	27	28
4	20	13		14	15			8	10	12
5
6
7	7	4	25	5	6	22	23	1	2	3
8
9

• By picking a higher value for Assumption #6 (maximum total scores), your selections would have paid off up to 14 times.  That's merely one more than the 13 times we got using the lower SWAG -- and it would have required investing in at least 25 squares per game, instead of 22.  Exclamatory punctuation is almost unavoidable.
• Using our initial Assumption #6, we missed all five winning combinations that were duplicated plus the one combination that appeared in the final scores of three Superbowls.  The higher total in Assumption #6 snagged three of them, including that triple (4-to-7).
• The most recent Superbowl (XXXVI) ended with a score of 20-to-17 (0-to-7 in the least significant digit department), and you would have won -- this time if you had purchased at least 7 out of your 28 preferred squares (compared to 14 out of 22 with the lower SWAG).

On the Monday after Superbowl Sunday, having spent the previous day on your couch getting advertised to, you will take $72 to the bank.  Either that or you must pony up $28 out of your lunch money to pay off the winner.

What, do you suppose, is the likelihood of that?

Donella H. and Dennis L. Meadows, in their landmark 1972 book, The Limits to Growth: Report for the Club of Rome's Project on the Predicament of Mankind, defined a model as an ordered set of assumptions.  Their modeling of the future continues to be both acclaimed and vilified.

An average is merely one estimate of "central tendency."  So is a median.  Deviations often matter more.  An average of 22.5 is not a particularly attractive representation of all the values between 15 and 30.  Distributions can be weird, too.  Oh right, and fractional results are often unusable:  The average of two 7s and one 3 is 5.67, for which our payoff matrix offers no square.

You can assign values in some arbitrary order (7,7,3,7,7,3...), but it's not cool to project events as occuring in any particular pattern when no particular pattern can be justified.  A randomizing element is often used with good effect.  In the solution to Randomly Wrong you will find a Monte Carlo style of model, in which random numbers operate on a key variable. That's what we shall do here. {Return}
 
 

Sophisticated solvers might want to play with the model that produced the results reported here.  If so, send an e-mail request to puzzles@niquette.com. {Return}
 
 

Among the messages commenting on The Next Superbowl puzzle was one I received just in time for the 2003 contest, which included a query that has given me an idea for applying the solution to conventional sports pools.
 

he expected value of each square can be estimated from the "tweaked" model by converting those "Joint Chance" percentages into dollars.  Accordingly, if you do not own them already (as a result of purchasing random assignments), those first three choices (7-7, 8-7, 9-7) appear to be worth more than $4.00 each.

Your coworkers, having not studied the solution to The Next Superbowl puzzle, ought to be happy to accept an offer of, say, $2.00 for each of them, which means they have doubled their money before the opening kick-off.  Meanwhile, for a total investment of only six bucks, you're looking at better than a 13% chance to win a C-note.

Remember, a corporate executive often has no other choice but to believe his or her business model when making decisions -- often huge, critical decisions -- about the future.

There is a fallacy in this reasoning, of course.  You still have to win based on the gridiron results, and there's a 2/3rds chance that you won't.  Meanwhile, nine of your coworkers are no longer at risk, having already pocketed their winnings. They have renounced their interest in "The Gambler's Playoff."  Thus, those who sell you their squares won't experience the same excitement while watching the game as you will.

Finally, if you do lose, one of your coworkers will have won that $100, and he or she only put up a buck.  Which of you will have had more fun on Superbowl Sunday?