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Siyalatas Siyalatas + Morgulis Morgulis

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

B_S = 6,4 + 0,15S

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

B_M = 1 + 0,11M

Their combined bonus is therefore

B_{S*M} = (6,4 + 0,15S) * (1 + 0,11M) = 6,4 + 0,15S + 0,704M + 0,0165SM

Their respective prices are n and 1:

P_S = S

P_M = 1

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

MB_S = {\partial B_{S*M}\over\partial S} = 0,15 + 0,0165M

MB_M = {\partial B_{S*M}\over\partial M} = 0,704 + 0,0165S

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

{{\partial B_{S*M}\over\partial S} \over P_S} = {{\partial B_{S*M}\over\partial M} \over P_M}

or

{\partial B_{S*M}\over\partial S} * P_M= {\partial B_{S*M}\over\partial M} * P_S

(0,15 + 0,0165M) * 1 = (0,704 + 0,0165S) * S

Simplifying gives

M = S^2 + 128S / 3 - 100 / 11

For example:

level 100 of Siyalatas Siyalatas corresponds to M = 100^2 + 1280 * {100 \over 3} - {10 \over 11} = approx. 14 258

level 500 of Siyalatas Siyalatas corresponds to M = 500^2 + 1280 * {500 \over 3} - {10 \over 11} = approx. 271 324

Bhaal Bhaal + Fragsworth Fragsworth ratio

While Bhaal Bhaal gives 15 % more to Critical Click damage per level, Fragsworth 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:

CH = (1 + 0,2F) * (1 + 0,15B) = M * (0,2F + 0,15B + 0,03BF)

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

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

P_F = F

P_B = B

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

{\partial CH\over\partial F} = 0,2 + 0,03B

{\partial CH\over\partial B} = 0,15 + 0,03F

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:

{{\partial CH\over\partial F} \over {\partial CH\over\partial B}} = {P_F \over P_F}

so

{(0,2 + 0,03B) \over (0,15 + 0,03F)} = {F \over B}

by simplifying the equation with WolframAplha, we get:

B = {\sqrt{100 + 45F + 9F^2} - 10 \over 3}

The solution after rounding is pretty much

B = F - 1

for levels above 2.

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

Libertas Libertas + Mammon Mammon

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

L = {\sqrt{4096 + 180M + 9M^2} - 64) \over 3}

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

L = 0,9M

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

L = M

Eg. Libertas Libertas level 1000 corresponds to Mammon Mammon level 1011

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