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Topic: Black Box Football (Read 5269 times) |
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william wu
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Black Box Football
« on: Oct 9th, 2003, 5:11am » |
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Somewhere in outer space, aliens are picking up a communications signal coming from Earth. When demodulated properly, the signal turns out to consist of nothing but audio of final scores in American football games, starting as early as 1994. The signal sounds like this: "14 to 7. 24 to 16. 10 to 7. ...". And every once in a while, the numbers are interrupted with the creepy explanatory message: "These are final scores in American football games."
The aliens don't know what football is, except that it is some kind of competitive game played between teams of Earthlings. And they understand the idea of scores in games, and that the team with the higher score after a game is the winning team. Using only these rudimentary understandings and the information in the signal, they would like to determine what the possible scoring increments are in the game of football.
Given some finite segment of the signal, how should the aliens go about estimating the possible scoring increments?
Note: The scores start as early as 1994 because I think that's when the most recent change in scoring rules occurred, with regards to 2 point conversions. In any case, assume the scoring rules are fixed for all games whose final scores are described in the signal.
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| « Last Edit: Oct 9th, 2003, 5:21am by william wu » |
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towr
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Re: Black Box Football
« Reply #1 on: Oct 9th, 2003, 6:02am » |
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I don't know anything about football, so I might as well be an alien..
So supposing any sort of increment could occur, I don't think there is a definitive way to find out.. Suppose there are two-point increments and 4-point increments, you can't distinguish a 4 point-increment from two 2-point increments..
You could try to find the minimum increment by looking at the minimum (>0) ever scored in a game, supposing that's from someone scoring only once.. But chances are that doesn't occur.
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Mike_V
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Re: Black Box Football
« Reply #2 on: Oct 9th, 2003, 9:45am » |
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I think it might be possible to infer 4 point increments even in the midst of 2 point ones. This depends on the circumstances of course. For example if you got scores something like 4-0, 16-14, 16-12, 18-16, 20-20, 20-12, you could "estimate" (as the question states) that there is a 4-point increment and a 2-point increment.
So, in American football, you will of course see an unusual number of multiples of 7. My first try would be to find the largest factor of each score that is most common. (So in the above example, it would be 4.) Then look at the scores modulo that number and repeat.
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James Fingas
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Re: Black Box Football
« Reply #3 on: Oct 9th, 2003, 10:43am » |
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The distribution of scores will give you a very good idea of what scoring increments are possible. If we assume that the different scoring increments are independent and that each is generated by a Poisson distribution, then we model the score function as the sum of n scaled Poisson distributions.
Points = [sum]i=0..nSi*Ni
where Si is the number of points you score for thing i, and Ni is a Poisson random variable with parameter [lambda]i. There are n different things you can do to score.
The pdf of the scores will look like the convolution of these n scaled Poisson distributions. To figure out what n, Si and [lambda]i are, you would take the (discrete) fourier transform of the pdf of the scores (or the histogram, which estimates that pdf), which would look like the sum of scaled fourier transforms of Poisson distributions, and then try out likely decompositions from there. This would give you the full picture of n, Si, and [lambda]i.
To just find Si, in the frequency domain you will see a number of peaks in the spectrum. The locations of these peaks are equal to the inverses of the possible scores Si.
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For those of you who think football is a game played primarily with your feet (silly people), here's the scoring:
2 pts for a safety (very rare)
3 pts for a field goal (very common)
6-8 pts for a touch down, usually 7 (common). Occasionally 6 when they miss the kick afterwards. It's only 8 when the score is such that it's worth taking a big risk in the hopes of one more point (and that's more common when both teams have already scored, so a final score of 8 is very rare).
Analyzing the frequency of all the final scores, you could easily deduce the existence of a rare 2 pointer and common 3 and 7 pointers. The rarity of a final score of 2 eliminates the likelihood that all those 7 point games were formed by a 3 and 2 2's.
The only difficulty would be figuring out whether there is a 6 or 8 pt score, and eliminating the possibility of rare higher scoring opportunities.
Because of the way strategy affects which scores a team will strive for at any point in a game, you can't simply correlate the commonness of 2,3, and 7 as final scores with the frequency of each in the total in a high scoring game. But I suppose really smart aliens who have a very large number of final scores to analyze could work some of those high scoring games backwards, explaining any frequency anomalies by giving each score both a difficulty and a risk factor. But could they figure out risk strategy well enough to infer the rare 6 and 8 pointers?
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James Fingas
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Re: Black Box Football
« Reply #5 on: Oct 9th, 2003, 11:16am » |
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I don't think 6 pointers are quite as rare as that. It's not completely uncommon to miss the conversion. As for 8 pointers, I don't know how you score them, but if they're very rare, then they would almost definitely get lost in the noise, and be undistinguishable from 6 and 2 pointers.
The nice thing about using the fourier domain is that it gets rid of all those correlations for you. If there truly is no correlation between scoring 6 points and scoring 2 points, then the 8 point scores will show up. However there will always be some random correlation between the 6 and 2 point scores, and until you get enough scores that the amount of 8 scores are larger than this sample-size-induced correlation, you will not be able to be sure that a team can score 8.
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SWF
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Re: Black Box Football
« Reply #6 on: Oct 9th, 2003, 9:34pm » |
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This riddle is reminiscent of another football riddle in the Medium section which is still unsolved after 9 months.
For the problem at hand, I happen to have a frequency distribution of scores from the last 2.5 years of NFL scores, and would be interested in results of some the proposed solutions using this data. There were no scores of 2 points in that time, but might want to put in a small non-zero number under 2, since it happens occasionally. It would also be interesting to see if it can be determined that 2 is a possible increment with a 0 for its frequency (the 0.1% of games with 11 points should give it away). For this list, in each row first number is the score, and 2nd number is the percentage of the time this final score is obtained.
0 1.8
1 0.0
2 0.0
3 2.7
4 0.0
5 0.0
6 2.4
7 3.8
8 0.0
9 1.6
10 6.1
11 0.1
12 1.0
13 6.4
14 4.4
15 1.6
16 2.5
17 6.1
18 0.7
19 1.0
20 7.3
21 5.2
22 0.7
23 4.5
24 6.6
25 1.4
26 2.2
27 5.2
28 3.0
29 1.0
30 2.7
31 4.2
32 0.6
33 0.8
34 2.9
35 1.3
36 0.4
37 1.4
38 1.9
39 0.4
40 0.4
41 0.9
42 0.7
43 0.1
44 0.4
45 0.3
46 0.0
47 0.1
48 0.4
49 0.4
50 0.0
51 0.0
52 0.2
53 0.0
54 0.0
55 0.1
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william wu
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Re: Black Box Football
« Reply #7 on: Oct 26th, 2003, 7:11pm » |
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on Oct 9th, 2003, 10:43am, James Fingas wrote:| If we assume that the different scoring increments are independent and that each is generated by a Poisson distribution, then we model the score function as the sum of n scaled Poisson distributions ... The pdf of the scores will look like the convolution of these n scaled Poisson distributions. To figure out what n, Si and [lambda]i are, you would take the (discrete) fourier transform of the pdf of the scores (or the histogram, which estimates that pdf), which would look like the sum of scaled fourier transforms of Poisson distributions, and then try out likely decompositions from there. This would give you the full picture of n, Si, and [lambda]i. To just find Si, in the frequency domain you will see a number of peaks in the spectrum. The locations of these peaks are equal to the inverses of the possible scores Si. |
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This is a cool idea, but wouldn't the fourier trasnform of the pdf of the scores will look like the product (not sum) of scaled fourier transforms of the distributions, since convolution in the probability density domain is multiplication in the transform domain? Thus the peaks of the constituent distributions may not be easy to see if you're looking at the product of curves.
Incidentally, the person I got this problem from said this problem is a neat metaphor for particle physics. Scientists started with the atomic masses of various elements (final scores), and they wanted to know what the were the possible masses of the particles (possible scoring increments) that could be added up to make the atom masses. After enough observation of "final scores", they deduced that there would probably be three scoring increments possible: the masses of protons, neutrons, and electrons. But after observing even more behavior (e.g. 2-point safety), they discovered that perhaps these particles could be dissected into even smaller particles (like quarks).
What do you think of this metaphor? I don't think the analogy fits very well. The masses of the subatomic particles were never determined by staring at lists of elemental masses? Maybe you could consider the experiments performed in the early 20th and late 19th centuries as elaborate football games involving various different elements which resulted in certain scores, but it seems like a stretch to me.
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| « Last Edit: Oct 26th, 2003, 7:22pm by william wu » |
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towr
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Re: Black Box Football
« Reply #8 on: Oct 26th, 2003, 11:58pm » |
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The discovery of the electron weight was the first thing I thought about when I read this, but it was much simpler since there is only one weight there, and you can thus simply use the lowest common denomenator..
And the weight of 'the atom' was easy since they just noticed it was allways (nearly) a multiple of the hydrogen atom. Distinction between neutrons and protons is also easy enough because they're seperable, and you can then again find the weight independantly.
So overall I don't think it's a very good metaphore.
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| « Last Edit: Oct 27th, 2003, 12:00am by towr » |
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Re: Black Box Football
« Reply #9 on: Oct 27th, 2003, 5:58pm » |
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First of all, the error in our current known values for the masses of protons and neutrons is, I believe, still orders of magnitude larger than the mass of an electron.
Second, since one of the "scores" (hydrogen) was clearly that of an individual proton (once the matter of isotopes was figured out), there was no need for a lot of analysis to determine what this mass was. An examination of the spectroscopic traces of other isotopes of hydrogen led quickly to the masses of neutrons too.
The mass of electrons was calculated from their charge, and from the charge-to-mass ratio, which could be measured by how much their trajectory curves in response to a magnetic field.
Which brings us to the measurement of the charge of an electron. If you have not heard of Milikan's oil drop experiment (it pains me to consider the possibility that you have not been taught it), I would suggest reading up on it. It is a fairly easily understood example of a fundamental scientific experiment. Milikan's results were a list of numbers representing charges on various oil drops. Since his task was to find a single number that all the values were multiples of, it was considerably easier than the task of our armchair-quarterback aliens, who must figure out several possible increment values.
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william wu
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Re: Black Box Football
« Reply #10 on: Nov 4th, 2003, 10:52pm » |
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Yes I am familiar with these experiments, and I originally agreed that it was probably a bad metaphor. However, to play devil's advocate, couldn't we argue that Mendeleev performed work similar to this football game? He listed the elements according to increasing atomic weight, and noted various patterns and periodic properties. Including: "chemically analogous elements have either similar atomic weights (Pt, Ir, Os), or weights which increase by equal increments (K, Rb, Cs)". Furthermore, he predicted possible "scores" which were not present in the data set he had at the time, such as the atomic weight of Aluminum. All this happened around 1869, preceding the aforementioned experiments of the early 20th century.
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| « Last Edit: Nov 4th, 2003, 11:03pm by william wu » |
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Re: Black Box Football
« Reply #11 on: Nov 5th, 2003, 1:16am » |
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You could certainly argue it, but you can argue anything..
He allready had the weights of all known elements, and after grouping them in a table according to properties he could simply look at the empty spaces and conclude the element that goes there would be one hydrogen-weight less than the one to the right, or one more than the one to the left..
Also scores in a football game don't have periodic chemical-like properties afaik 
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