Thread:Obromios/@comment-22439-20160104215407/@comment-25205196-20160319180741

Obromios, as you have not responded and I do want you to lose the page content before it gets deleted I'm reposting it here. I do hope you consider putting it back up as a blog post.

== Introduction == Hay Day is a fascinating game, which almost certainly has mathematicians working on the design of different features. This means that some aspects of the game can be worked out mathematically. This page is for people interested deducing mathematical aspects of Hay Day. As an example, here is a simple derivation of the relative proportion of teams in the different Derby leagues.

Proportion of Teams in Derby Leagues
There are five leagues, denote the number of players in the different leagues as N_1, N_2, N_3, N_4, and N_5, where N_1 is the number in the easiest league and N_5 is the hardest league (the Champions League). At present, 3/15 of the teams are moved up each round and 3/15 are moved down. There are two exceptions to this, In the easiest league, no players are moved down and in the hardest league no players are moved up.

Denote the proportion of players who are moved up or down as A (at present A is 3/15).

In the lowest league, A x N_1 players are moved up each round and A x N_2 are moved down into the lowest league. At steady state we have

A x N_1 = A x N_2, so we can deduce in the long run, we should have approximately

N_1 = N_2,

i.e. there will be equal number of players in the easiest league and the second easiest league.

Now consider the second easiest league. At steady state we have

A x N_2 + A x N2 = AxN_1 + AxN_3,

but N_1 = N_2 so we have

N_2 = N_3.

By induction, this equality of numbers will hold up to the hardest league. Here, at steady state we have

AxN_5 = AxN_4, so at steady state we have approximately that

N_4 = N_5, so

N_1 = N_2 = N_3 = N_4 = N_5, i.e. the number of leagues in each team will be approximately equal.

This holds no matter the number of leagues or the distribution of team ability. It is also independent of the value of A. It will vary if a different proportion is moved up to that proportion moved down.