Mathematical Modeling of Attempt Selection | Jeff Russell

[Shiva Kaul]

I like the idea of modeling strength as a curve of probabilities. It is rather provocative in that hitting or missing an attempt doesn't yield much information about your underlying strength, for the same reason that flipping a coin once doesn't tell you much about its bias. This might actually be a reasonable modeling assumption, because random, imperceptible changes to bar position, feet width, etc. can affect maximal attempts.

You don't prescribe a functional form for the reliability curve. This makes optimal attempt selection intractable - not numerically, since there are just 3 optimization variables, but analytically: there's no insight into how the attempts are chosen. I propose that we use an exponential curve to model the challenging/interesting range of weights, where one actually has a nonneglible chance of missing an attempt.

1-exp(-λ x) represents the probability that you will miss an attempt at weight B + x, where the base weight B is a sure deal. This curve has a parameter λ. High values denote proficiency: the probability of missing drops very quickly, because you are very consistent. A low value means there's a lot of wiggle room to hit a bigger lift.

You require the attempts to be set nonadaptively, i.e. in advance of competition, disallowing the athlete from, say, changing their 2nd and 3rd attempts after the 1st fails. This is how coaches initially advise their athletes, and seems to accord with the observation that missing doesn't yield much information. However, nonadaptive selections are probably suboptimal. As time ticks out, a losing soccer/basketball/etc. team will play desperately aggressively. This has nothing to do with their ability; it is just an accurate assessment of their risks and rewards at that time. So, let's allow adaptive strategies. As it turns out, these are actually be easier to determine.

Let's start with a 2-attempt competition. Attached is the mathematical derivation of the optimal attempt selection, under the above exponential model with the base weight B and the proficiency parameter λ. It is: attempt B + 1.13/λ. If hit, attempt B + 2.13/λ. Otherwise, attempt B + 1/λ. 1.13 is just an approximation of (1 + e + e W(-2 e^{-2 - 1/e}))/e, where W is Lambert's W function.

In practice, attempts monotonically increase in weight. However, in the optimal strategy, the first attempt might be greater than the second. This actually makes sense: if you hit a challenging 1st attempt, it gives you the opportunity to shoot for the moon on the 2nd. (Note that our model does not account for "warming up" with a first attempt, or the potential injury to body and mind after missing a lift.)

Another way this analysis could be furthered is by considering the variance of the attempt selection, not just the mean. Conservatively choosing attempts so that all of them are successful is probably not a good way to maximize the mean, but it is a good way to minimize variance. This is a bit more respectful of the competition, which is ideally a repeatable demonstration of strength, not just luck of the draw.

If someone is interested, I could derive the optimal strategy for 3 attempts. But first - is this model reasonable? And would anyone actually change their attempts because a mathematical derivation told them to?

[K9ZAZ]

I have not read through all this yet, so I don't have much comment, but this analogy

It is rather provocative in that hitting or missing an attempt doesn't yield much information about your underlying strength, for the same reason that flipping a coin once doesn't tell you much about its bias.

strikes me as wrong. Hitting an attempt DOES tell you something about your underlying strength; what would it mean for there to be probability mass to the right of 100% of your 1rm?

[凤凰来仪]

[QUOTE=Shiva Kaul;1725677]I l

And would anyone actually change their attempts because a mathematical derivation told them to?

Interesting.

Yes, they would if you can show that it works at a meet.

I may look into this further... but may not have time.

[凤凰来仪]

I read the article and have decided I do not have enough experience with meets, 1RM attempts or coaching lifters in general to add any value to this discussion.

General impression is it would be very difficult to create a model that outperforms an experienced coach relying on lifter feedback before/after attempts.

[Shiva Kaul]

Hitting an attempt DOES tell you something about your underlying strength; what would it mean for there to be probability mass to the right of 100% of your 1rm?

Yes, but hitting an attempt does not mean you can hit it 100% of the time, especially if it is a very aggressive attempt that might only be considered at a meet. Due to this uncertainty, the idea of a definitive “1RM” isn’t accurate. The model espoused by the article says hitting a lift at weight w is like flipping a coin with bias f(w), where f is a decreasing function of w, and takes values in [0,1]. An accurate estimate of the bias requires flipping the coin a few times.

[Shiva Kaul]

Apparently USSF requires attempts to be nondecreasing, which might preclude an optimal strategy like the one derived above. Is this just for practical reasons, or is it meant to enforce some kind of discipline on attempt selection? USAPL does not seem to have this requirement.

Amusingly, the calculations in the originally posted article were carried out in 1976. See section 2 for the optimization formulation and the outcome probabilities, and section 7 for the numerical experiments. The author suggests an intuitive but intractable quadratic form for f in section 3.

[stef]

"Apparently USSF requires attempts to be nondecreasing, which might preclude an optimal strategy like the one derived above. Is this just for practical reasons, or is it meant to enforce some kind of discipline on attempt selection? USAPL does not seem to have this requirement."

It does. You see that across all the barbell sports, and way back in history. It is not a USSF-specific thing at all.

[Shiva Kaul]

Indeed, but what is the primary rationale? It makes loading the bar and determining flights easier. It might build the drama of the competition. Is it intended to discourage aggressive attempts which have a high probability of being missed?

[Mark Rippetoe]

It is designed to keep the meet running on time, and to help the lifters know when their attempt must come out of the rack.

[Shiva Kaul]

Suppose that non-monotonic (i.e. currently illegal) attempt selection led to to substantially higher totals. Say, 3-5% higher. Would the practical benefits of the current rules be worth the hindrance of performance?