Mathematical Modeling of Attempt Selection | Jeff Russell - Page 5

# Thread: Mathematical Modeling of Attempt Selection | Jeff Russell

1. Member
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I am the only person who took the actual meat of the article -- the mathematical model -- and nontrivially engaged with it. Except for an early commenter who misinterpreted the math.

The model may include a "proficiency" parameter, but it still isn't realistic for truly advanced lifters, due to the assumption of independent lifts. Not to mention that the numerical results become increasingly imprecise as the reliability curve becomes steeper.

I am rather disappointed. I posted my derivation not for any serious purpose, but rather to bring the mathematically-inclined members of the SS community (of which there are at least a few) out of the woodwork to casually sling some equations. I imagine Jeff intended the same.

I've been met with lots of grumpiness and little erudition, even though I know the SS community is capable of both. If you're not interested in the mathematical content of the article, don't worry yourself with my posts in this thread. (Why would you even click the title?)

2. Who has disappointed you by not appreciating your math? I have already told you why you are being received poorly.

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I'm looking for pleasant mathematical dialogue (if there's any to be had) not appreciation or validation. If this can actually help people acquire or demonstrate strength, that is great too.

I'm sorry if it sounded like I wanted to change the flow of meets. Apparently the constraint of increasing attempts is very important. Also, everyone seems fundamentally opposed to the slightest chance of missing an opener. So I respected those in my analysis of the 3-attempt meet. The optimal algorithm is as follows.

Given: baseline weight B and proficiency parameter λ
1. Attempt B. According to the model, you hit this with 100% probability. Congratulations.
2. Attempt B + w, where w is a function of λ in the red curve below. (It unfortunately no longer has a nice closed form, being the root of an exp-log equation.)

3. If missed, reattempt B + w. If hit, attempt B + w + 1/λ .

So, the article's lifter (B=485, λ=0.06) would attempt 485, 503 and, if successful, 519. These are lower numbers than the article suggested, because the exponential curve drops faster than the article's.

I am not sure I would describe the 2nd attempt B+w as "conservative". The blue line plots the increment 1/λ the lifter would make if 2nd were their last attempt. It essentially coincides with w; in that sense, it is conservative. However, the exponential model gives a very high probability of missing that lift.

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Proof of above. Also corrects an error in my post a few pages ago.

5. Fascinating.

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Originally Posted by Shiva Kaul
I am rather disappointed.
Take heart; we're just a very pragmatic crowd. I'll see if I can add a bit more to the theory here.

A non-monotonic meet is a hypothetical that makes me uncomfortable as well. I haven't run any calculations though my own model on this possibility, but I wouldn't be surprised to find little to no statistical advantage to it. There are other practical concerns at work there: it changes the fatigue considerations, something our models don't account for anyway, and makes a mess of the meet itself in various ways, as Rip points out. There are other hypothetical rule changes that could increase average performance (4th attempts, "do-overs", solid fuel boosters, etc.) but these aren't of much relevance unless the mathematics are their own reward. For Shiva and myself, maybe this is the case.

Moving on to the shape of the probability curve, I don't think I would advocate an exponential curve as you've used here, Shiva. It doesn't intersect or converge to zero for a "1RM" very well. Admittedly, I have pulled some other shapes out of my butt in similar fashion for the simulation (link again here). The only one that is any better than a guess is the cumulative normal distribution, which is what you get when you integrate the normal distribution. My reasoning was that the lifter's quality of execution would follow the normal distribution (being the sum of independent factors), and that a cumulative distribution of this result would shape the probability curve. This was not laid out in detail in the article, because it's a bit in the weeds and I'm not fully confident in it. But it has some properties that are nice, including quick convergence to near-guaranteed failure and a slight chance of failure even at light weights (no weight is 100% guaranteed). You can see a version of it as the fourth option in the simulation. It's worth pointing out again, as the article does, that using this function doesn't produce estimates greatly different from simpler curves. See also the error function.

You're right to note that my model does make rather aggressive attempt suggestions. The article does a little bit of work to address this near the end, outlining some other factors the model doesn't account for that could lead an athlete to temper their approach. One of the main reasons I built this model was just to see how it compared to the conventional wisdom for attempt selection, and happily it wasn't too far off. But anyway I wouldn't take it as any kind of strict recommendation from me, much less the Starting Strength organization.

People really, really don't like missing openers. I still suspect that most casual competitors go too light on their openers out of fear of this, rather than as a result of careful statistical balance. This is probably logical for people who just compete for fun (like me); if you screw up your meet, it's no fun anymore. A more serious competitor will take a bit more risk since bombing out and finishing last are valued similarly to them. Seeing more aggressive openers pop out of the model was one of the first interesting things I noticed. I'm reasonably well convinced there is some insight on offer there.

A question I played with during my research was "how do we value bombing out"? The meet rules straightforwardly score that as a zero, but that doesn't mean the the lifter will assign the same utility to that outcome. You could imagine cases where an athlete didn't care about a wide span of crappy outcomes greater than zero - should we assign "zero" to any lift, for the purposes of our model, that is less than some threshold the athlete is interested in (say, enough to place on the medal stand)? By computing the mean outcome, we're implicitly valuing bombing out as "0", but really that could have more or less value to the lifter. By varying this, you get a kind of "risk tolerance" parameter which might be fun to play with. I didn't take it further.

Lastly there is the matter of changing attempts in response to misses during the meet. I actually did program this into the model (only for the monotonic rules), but the results were underwhelming enough that I omitted them from the final version. The model showed some benefit to going up a bit after missing the opener, but not by very much and not to much benefit. I suspect that the reasons for increasing your attempt following a miss have more to do with situational day-of-meet details than aggregate statistics.

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