## Sunday 29 July 2018

### FE Heroes: Arena Tiers 2.0

A year ago I analysed the tier distribution in FEH and now with the v2.8 update we are going to have a twist, with the new tier 21 [a.k.a. the WHALE TIER] introduced. It is no more than a simple application of queue theory and matrix diagonalization....well no, just solving linear equations to get the new distribution. Just a quick reminder, we can generate the ratio by the following equation

$p_{k+1} = r_{-1, k}^{-1}(\sum _{d =1,2,3} r_{d,k} p_{k-d} - (1-r_{0,k})p_k)$

from the bottom, where $r_{d,k}$ stands for the ratio of going up by $d$ tiers from tier $k$, and $p_k$ stands for the proportionality constant of players at tier $k$ in an equilibrium state.

Quite surprised that they looked extremely similar, right? If you look deep into the chart you will found that to be unsurprising at all. You may even wonder why are they different...

...that's because tier 20/21 are the only tiers with $r_k$ changed, after all.

(Apology for the stupid y-axis.) Numerically with the introduction of tier 21, players moved upward by 2%. There will be 2% more tier 18+ players [tier 18 is important due to the 4 orbs reward], and 2% more tier 20 players, which we, infantry lovers, do not care at all.

So here is the counter-intuitive part: why is there a difference on tiers below tier 20, given that the parameters were not altered? If we hide the flow between tier 20-21 and call the two tier 20+, this new tier actually has the same parameter as the old tier 20:

Old tier 20: 30% stay - 70% down 1

New tier 20: 20% up 1 - 10% stay - 70% down 1
New tier 21: 30% stay - 70% down 1
New tier 20+: 30% stay - 70% down 1

...or is it?

For sure that's not correct - by combining two tiers we brake the memoryless feature of the promotion system: if one stays in tier 21, he will 100% stay in tier 20+ in the next season, and that is not consistent with the global parameter for tier 20+. We can't really solve a set of linear equations correctly without the memoryless feature so the idea of tier 20+ does not really work. One simply go back to tier 20/21, do the math and 2% is really an existing difference.

...but why?

Even though the idea of tier 20+ does not help mathematically, it is true conceptually. Players are allowed to stay in tier 20+ longer [for those in tier 21] comparing with the old tier 20, hence expending the capacity at the top.

And, well, for me I do not bother getting top ranks anymore. I'm happy with my tier 19.5, or even 18.5 teams...