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I don't know if anyone else is interested in this, but for those of you who are and have taken a linear algebra class, I used a Markov chain to analyze what the distribution of players in the different leagues of the Weekly Challenge will be in the long run.
I'm pretty sure the leagues work like this (let me know if I'm wrong): top third promotes to next league (if there is one), bottom third demotes to previous league (if there is one), and middle third remain in the same league.
Based on these distributions, we can make 4 categories: Entry, Bronze, Silver, Gold. The stochastic matrix looks approximately like:
[ 0.67 0.33 0 0 ] [ 0.33 0.33 0.33 0 ] [ 0 0.33 0.33 0.33 ] [ 0 0 0.33 0.67 ]
Finding the steady state solution two this gives that in the long run: 25% of the total players will be in each league, so which league you are in (in the long run) determines exactly which quarter percentile of players you are in.
Of course, it's very important to note that this assumes that the influx of new players each week is insignificant. Maybe to incorporate the influx of new players you could use a system of diff eqs, but I'm not totally sure.
Anyways, just wanted to share this, and I hope it isn't too irrelevant to this sub.
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