The Hidden Complexity of the Olympics

Chad laments that we don’t hear that much about the decathlon any more, because Americans aren’t really competitive. I also think it’s a shame, because any sport in which your score can be a complex number deserves more attention.

Yes, it’s true. The decathlon combines ten different track and field events, so to come up with a final score we need some way to tally up all of the individual scores so that each event is of approximately equal importance. You know what that means: an equation. Let’s imagine that you finish the 100 meter dash in 9.9 seconds. Then your score in that event, call it x, is x = 9.9. This corresponds to a number of points, calculated according to the following formulas:

points = α(x0x)β   for track events,

points = α(xx0)β   for field events.

That’s right — power laws! With rather finely-tuned coefficients, although it’s unclear whether they occur naturally in any compactification of string theory. The values of the parameters α, x0 and β are different for each of the ten events, as this helpful table lifted from Wikipedia shows:

Event α x0 β Units
100 m 25.437 18 1.81 seconds
Long Jump 0.14354 220 1.4 centimeters
Shot Put 51.39 1.5 1.05 meters
High Jump 0.8465 75 1.42 centimeters
400 m 1.53775 82 1.81 seconds
110 m Hurdles     5.74352    28.5    1.92    seconds
Discus Throw 12.91 4 1.1 meters
Pole Vault 0.2797 100 1.35 centimeters
Javelin Throw 10.14 7 1.08 meters
1500 m 0.03768 480 1.85 seconds

The goal, of course, is to get the most points. Note that for track events, your goal is to get a low score x (running fast), so the formula involves (x0x); in field events you want a high score (throwing far), so the formula is reversed, (xx0). Don’t ask me how they came up with those exponents β.

You might think the mathematics consultants at the International Olympic Committee could tidy things up by just using an absolute value, |xx0|β. But those athletes are no dummies. If you did that, you could start getting great scores by doing really badly! Running the 100 meter dash in 100 seconds would give you 74,000 points, which is kind of unfair. (The world record is 8847.)

However, there remains a lurking danger. What if I did run a 100-second 100 meter dash? Under the current system, my score would be an imaginary number! 61237.4 – 41616.9i, to be precise. I could then argue with perfect justification that the magnitude of my score, |61237.4 – 41616.9i |, is 74,000, and I should win. Even if we just took the real part, I come out ahead. And if those arguments didn’t fly, I could fall back on the perfectly true claim that the complex plane is not uniquely ordered, and I at least deserve a tie.

Don’t be surprised if you see this strategy deployed, if not now, then certainly in 2012.

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45 Responses to The Hidden Complexity of the Olympics

  1. Aaron F. says:

    I think this is my favorite CV post so far. Thanks for brightening my day. 🙂

  2. jmchez says:

    What, the heck?!

    Wouldn’t it just be easier to compare the athlete’s performance as a fraction of the world record for that event. So a 100 meters run in 9.9 seconds would score 9.69/9.9×1000 or 979 points. A 15 foot pole vault (4.57 meters) would be 457/614*1000 or 745 points and so on. That way a high school kid could keep track of his points against Olympians or against his country’s high school record.

  3. James Chisholm says:

    More interesting is that, to a physicist’s level of accuracy, the power laws fall into three categories: running events (~2), jumping events (~3/2) and throwing events (~1).

    This is screaming out for a Theory of Decathlon Unification.

  4. Ben says:

    Thanks for the laugh at the end.

  5. Murk says:

    @jmchez – it might be easier, but it wouldn’t be fairer. They’d probably based the power on the expected variability between high class athletes for that event – to make the total range of scores available for the 100m about the same as that for the javelin.

    http://en.wikipedia.org/wiki/Decathlon_scoring_tables

    “Beginning in 1920, the IAAF considered, at least, the following criteria for a legitimate decathlon scoring table:[2] (1) The table should reflect the fact that, at higher levels of performance, a unit gain (such as a decrement of 0.01 second in sprint times) is more significant than at lower levels of performance, because of the physiological limitations of the human body. (2) The scores for different events should be comparable, in a manner such that equal skill levels in different events (however difficult it is to define such a concept) are rewarded with equal point levels.”

  6. Murk says:

    Regarding the complex number bit of the original post – I’ll bet there is some bit of small print that says (if x>x0 for track, or x<x0 for field, then the score is Y)

  7. Dan says:

    Some great commentary on Olympic scoring.

    What I would like to know is how Gymnastics can justify five significant digits (a score like 16.025) that are based on human judges giving out scores to two significant digits (a score like 9.1)?

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  9. Brian says:

    jmchez: well, they can still do that with this system, they just have to know a little algebra! (or use, say, a score calculator).

    The scoring almost *has* to be nonlinear, because the marginal cost of improvement is different at different times – dropping a 18 second 100m time to 17 is much easier than going from, say, 10.9 to 9.9. Ideally, I’d think the reward should roughly correspond to the fraction of athletes with times in that region. Of course, everything is tweaked for historical reasons, and to make sure specialists can’t dominate.

    According to this wiki page the throwing events used to have sublinear powers, essentially based on the idea that running speed and throwing speed should scale, but range scales like the square of velocity. The factor of two seems to have remained. Not sure how jumping factors in, though, James!

  10. Freiddie says:

    Haha. Hacking into the “flaws” of the decathlon formula!

    Does it ever bother you that you can’t say A < B in complex numbers?

  11. Eugene says:

    What if they take the *imaginary* part?

  12. Eugene says:

    Further thought : I am surprised that the gaussian didn’t show up anywhere in their calculation of scores.

  13. Simon DeDeo says:

    In the end, man races against himself, so it should be possible to “unify” the scores by simply referencing to how probable the exceeding of expected values was. Perhaps better to use one of the extreme value distributions instead of the Gaussian?

    You could even reference them to different populations (e.g., the athelete’s home country) so that a fourth place finish by someone from Luxembourg would easily beat a first place finish by someone from China. Thus formalizing the usual Cool Runnings chatter.

    One also would like to have some non-linear relationships here (e.g., two second-place finishes count more than a first place and a third place.) That way you could have a running of the couplings along with the athletes. Ha! I wrote this comment just for that joke.

  14. SamuelRiv says:

    I’m a decent athlete, but a lightweight who hadn’t thrown a shotput since the 8-pounders in jr. high. I competed in an intramural track meet in college and thought I’d do a full decathlon (which I actually did quite well in) sans pole vault. But when it came down to the full-weight shotput, I wasn’t prepared for how heavy it would be and actually threw a negative distance (inside the circle)! If they were doing proper decathlon scoring, I think I would have broken the system.

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  16. TimG says:

    It seems like the simplest solution would just be to change the formula to:
    ?(x - x0)|x - x0|^(beta-1)

    The idea of a complex number score is pretty amusing, though. 🙂

  17. TimG says:

    Oops! That should be:
    a(x - x0)|x - x0|^{(beta-1)}

  18. ollie says:

    Ah, they were sloppy in their function definition.

    They mean, of course, to use a conditional function which looks like:

    f(x) = …… if x gte 0
    f(x) = 0 if x lt 0

    This definition preserves continuity but not differentiability at x = 0.

    Hence, were we to come up with some sort total energy use function, it would be a tough problem to maximize one’s score while holding the energy function to a fixed value.

  19. Elliot says:

    Given the scoring for women’s gymnastics this year, I would suggest that the use of imaginary numbers is far from a theoretical construct.

    e.

  20. ollie says:

    Elliot, boxing too. I’d just say that I wouldn’t want to get hit by some of those “non-scoring” punches.

  21. Phil says:

    LOL, and an idea is born.
    Here i am, an irreverent, maybe jaded 200m runner who will, realistically probably finish last ( since someone has to ) and it’s my last Olympic appearance or maybe the best scenario is i have a slight injury which guarantees this loss, whatever. So, no medals, no sweet lucrative sponsorship deals with Nike after the olympics etc. So what can i do to salvage something from the situation ?

    Run the race in s u p e r s l o w m o t i o n !

    The uproar over such a publicity stunt would be considerable and certainly no small amount of ireverence and courage would be needed since these athletes are representing their countries and all so, this is why an injury basis is the ideal (perhaps ONLY ) “justification” for such a stunt. Nevertheless, if sponsorships and historical notoriety have a real world and legitimate role or value in the Olympic universe then i think such a stunt could succeed admirably.

  22. Isabel Lugo says:

    Dan (#7),

    Gymnastics scores aren’t really correct to the thousandth of a point. All the scores (in the Olympics, where there are six judges and the middle four count) are multiples of 0.025, since they’re obtained by averaging four scores which are each multiples of 0.1 (for the “execution”) and then adding another score which is a multiple of 0.1 (the “difficulty” of the routine). It’s just that they’re reported with three digits after the decimal point so that the numbers can be given exactly; it would perhaps be more confusing if they were rounded to two places after the decimal point, because then it would not be obvious that the gap between 16.00 and 16.03 is the same as the gap between 16.03 and 16.05.

    I’m not sure why they don’t just add up the scores instead of averaging them, though. I suspect it has something to do with there being different numbers of judges in different competitions.

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  24. BlackGriffen says:

    Clever, but the simplest solution is probably to throw an (implicit?) step/Heaviside function in there.

    Although the conditional with an absolute value mentioned above makes perfect sense, too.

  25. BlackGriffen says:

    Also, the individual event scores aren’t distributed throughout the complex plane. As long as you have a way to pick a unique branch of the logarithm/fractional power law the scores will be arranged along a single curve in the complex plane and it is possible to uniquely define an ordering along that curve. You could then define the score as a real parameter from some parameterization that lands you at the point on the curve designated by the above equations. The simplest parameterization is given above by TimG and ollie.