Copyright ©2002 by Paul Niquette. All rights reserved. 
xcerpting
from
the title page of
Puzzles
with a Purpose, "Our purpose is to
celebrate processes and principles..."
This puzzle gives us an
opportunity to illustrate
practical methods used in making predictions 
predictions with incomplete
information, in fact. Well, let's be careful with
the word "fact,"
taking into account a principle expressed in The
Future is Not What it Used to Be...
"For I repeat: there are no 'facts' about the future. Only 'opinions'."Which is not to say, there are no facts at all. Football has its rules, and each team has its records. However, in The Next Superbowl puzzle, the identities of the teams are not yet known. 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."ere goes... 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 shutout. 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 twopoint 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?ur assumptions lead us to expect that at every scoring opportunity, each team, whoever they are, will most likely score either seven points (a touchdown followed by a successful conversion) or three points (a field goal). Many unknowns will influence those decisions as the teams interact with each other. Time out. Whereas touchdowns are often scored in a sequence of offensive plays following a kickoff return, touchdowns can also result from a turnover (passinterception 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.Assumption #5 That, out of every three successful scoring opportunities, a team will make two touchdowns and one field goal. Before continuing here, you may want to explore "The Price of a Point." This assumption merely appropriates the results from the solution to that puzzle. 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?A football pool is much like a casino game in which the winner is determined by a wheel with numbered frets, each corresponding to a square in the payoff matrix. As we have already noted, however, those frets are not all equally spaced. 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 recordbooks 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 iss a game played against the clock, football necessarily imposes an upper limit on the scores for both teams. Only so many touchdowns and field goals can take place in the time allowed. The total depends on the offensive and defensive strengths enjoyed by both teams taken together. Taken separately, those same qualities determine the closeness of the contest. 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 WildAss Guess." You cannot get more explicit than that.Assumption #6 That the total score accumulated by both teams in the next Superbowl will be no more than 45 points. Assumption #7 That the winning team will outscore the losing team by a margin of 2to1. ou will recall that the puzzle gave us another clue ("the team in 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 shutout 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 threesided coin... Excuse the interruption  "threesided 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 6to7, which will appear in final scores of 167, 267, or 2617. 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 6to4, which will appear in final scores of 1614 or 2614 and enjoys a likelihood more than 8 times what uniformly distributed cyphers would imply.Indeed, there are 22 squares worth betting on, as shown in the following tabulation: The payoff matrix for the football pool may now be marked to show the Order of Preference for each of your wagers... By the way, this is the solution to The Next Superbowl puzzle. Well, one solution, anyway. 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).
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...
Assumption #1 Teams maximize scores independently  confirmed as postulated, with the comefrombehind team taking slightly more risk (see Assumption #3).he SWAGs in Assumptions #6 and #7 imposed an arbitrary lid on the teams of 15 and 30 points. Have you looked at those game records? Some tweaking music, please. Superbowl History: Ratio vs Total Score 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 inksneeze, doesn't it.
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:A few observations about the performance of our tweaked model...
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.Over the years, few critics have understood that the idea is to challenge the assumptions underlying an analysis not the results. Now, it is a long way from The Next Superbowl puzzle to "the Predicament of Mankind." Or is it? {Return} 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 email request
to puzzles@niquette.com.
{Return}
{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 (77, 87, 97) 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 kickoff. Meanwhile, for a total investment of only six bucks, you're looking at better than a 13% chance to win a Cnote. 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.If you really believe the "tweaked" model, then you would go around offering $2.00 to your coworkers for the first nine choices. That might get you as much as a 34% chance at $100, for only $18 put at risk. 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? 



