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## Siyalatas + Morgulis

In advanced game, Siyalatas gives 15 % bonus to DPS per level. With initial higher boost in lower level, the function for Siyalatas is:

Morgulis gives 11 % bonus to DPS per level. The function is quite simple:

Their combined bonus is therefore

Their respective prices are n and 1:

To find the best ratio, we need to use marginal utility (or marginal bonus):

In economics, we maximize output when the ratio of marginal utilities (bonuses) to prices is equal for all:

or

Simplifying gives

For example:

level **100** of Siyalatas corresponds to

level **500** of Siyalatas corresponds to

## Bhaal + Fragsworth ratio

While Bhaal gives 15 % more to Critical Click damage per level, Fragsworth gives 20 % more to overall Click damage per level.

A player with active build has to find a ratio for the two Ancients, having in mind that Critical Click damage in itself is a multiple of base click damage so Fragsworth gives Critical Click bonus as well.

Regardless of Critical Click chance, your total Critical Click damage multiplier is equal to:

where M is your base efficiency, 1 + 0,2F is Fragsworth's multiplier and 1 + 0,15B is the Bhaal's multiplier.

Both Ancients have their cost in Hero Souls the same as their level:

Their marginal utility (how much you get from the next level) is a derivative of their total benefit:

Economics tell us that we maximize such system when the ratio of marginal utilities is equal to the ratio of prices for all given inputs:

so

by simplifying the equation with WolframAplha, we get:

The solution after rounding is pretty much

for levels above 2.

So you should keep the the Ancients on the same level.

## Libertas + Mammon

The maths for these two is the same, as before and gives result:

This means, that at the start, for low levels you should keep them around

And in the later game, they start to level out as the limit of this function is