Skull Allocation & Monster Item Selection

How to optimally distribute skull investments over the different monsters?

Which monster item to choose from a chest if multiple ones are offered?

Skull Allocation - Monster vs Monster

D_total = D_clicks + D_passive = CS * CM * (BD_bob + GSD * 0.05 * D_passive) + D_passive = CS * CM * BD_bob + CS * CM * GSD /20 * D_passive + D_passive = CS * CM * BD_bob + D_passive * (CS * CM * GSD/20 + 1)

D_passive = sum_{i € ntm}(D_i * B_i * B_g) = B_g * sum_{i € ntm}(D_i * I_i * S_i)

D_m = bd_m * (1.035^l_m -1) * (50 + l_m) S_m = bs_m * (1-cr) * 20 * (1.05^l_m -1) l_m = log_{1.05}(1 + S_m / (20 * bs_m * (1-cr))) D_m = bd_m * ((1 + S_m / (20 * bs_m * (1-cr)))^log_{1.05}(1.035) - 1) * (50 + log_{1.05}(1 + S_m / (20 * bs_m * (1-cr))))

dD/dl = bd * ((1.035^l -1)' * (50+l) + (1.035^l -1) * (50+l)') = bd * (ln(1.035) * 1.035^l * (50+l) + 1.035^l - 1) = bd * 1.035^l * (ln(1.035) * (50+l) + 1 - 1.035^(-l)) dl/dS = 1/ln(1.05) * (ln(1 + S / (20 * bs * (1-cr))))' = 1/ln(1.05) * 1/(1 + S / (20 * bs * (1-cr))) * 1/(20 * bs * (1-cr)) = 1/ln(1.05) * 1/1.05^l * 1/(20 * bs * (1-cr)) = 1/(ln(1.05) * 1.05^l * 20 * bs * (1-cr)) dD/dS = dD/dl * dl/dS = bd/bs * (1.035/1.05)^l * (ln(1.035) * (50+l) + 1 - 1.035^(-l)) /(1-cr)

res(lx) = ln(1.035) * (50+lx) + 1 - 1.035^(-lx) dD1/dS1 = dD2/dS2 bd1/bs1 * S1 * I1 * (1.035/1.05)^l1 * res(l1) = bd2/bs2 * S2 * I2 * (1.035/1.05)^l2 * res(l2) (1.05/1.035)^(l2-l1) = bd2/bd1 * bs1/bs2 * S2/S1 * I2/I1 * res(l2)/res(l1) l2 - l1 = log_{210/207}(bd2/bd1 * bs1/bs2 * S2/S1 * I2/I1 * res(l2)/res(l1))

eff = bd * S * I /bs l2 - l1 = log_{210/207}(eff2 / eff1)

Blue Specter : log_{210/207}(11/3) ~= +90.299 Flying Squid: log_{210/207}(17/3) ~= +120.553

Monster Item Selection

X: B * 1.035^log_{210/207}(B) = B * B^log_{210/207}(1.035) = B^(1+log_{210/207}(1.035)) Blue Specter: (11/3)^3.390858 = 81.915 Flying Squid: (17/3)^3.390858 = 358.449

Rank Monster 1 The Tomb King 2 Flying Squid 3 The Black Lich 4 The Big Plague 5 Blue Specter 6 Swarm of Bats 7 Red Knight 8 Giant Zombie 9 Zombie Horde

In this chapter we will investigate two closely related questions:I choose to tackle this with calculus as it's a very elegant way to obtain answers here and the results are more universal and smooth than if approached discretely.The player's total damage output is comprised of passive DPS from non-tap monsters and tap damage done by clicking/tapping and Sloth's Form. Tap damage itself is comprised of two components, Bob's base damage and damage from Golden Shower of Deathness, which is a portion of total passive DPS. Let CM = (1 + cc * (cd-1)) denote the average crit multiplier with crit chance cc and crit damage multiplier cd, CS the click speed in clicks per second, D_i the DPS of i, BD_bob the base click damage of Bob at this current level, and GSD the level of the Golden Shower of Deathness. Then we have:With Bob's base damage scaling much worse than the DPS of non-tap monsters, this portion becomes negligible rather quickly and in fact vanishes completely eventually (due to the limited accuracy of floating point arithmetic), so total DPS is essentially a multiple of passive DPS.Let's have a closer look at the passive DPS. It's the sum of the DPS of all non-tap monsters, including all buffs, both specific for individual monsters and generic. Generic buffs, however, may be factored out. Specific buffs are comprised of Skill damage multipliers (Spectral Reposession, Pastafury, King's Presence as affected by Son of the Lich) and monster item damage multipliers.At this point, to specify D_i, we need to remember how exactly levels (and by extension, skulls) factor into this. This time, we'd like to include the linear factor from the passive skill, but we can factor out and hence drop any factor that is not monster-specific.Now, we'd like to analyze the rate at which damage changes with invested skulls, so we need to take the derivative of the damage D_m in respect to skulls S_m. It's just proportional, but we will see later on that any general factors which we've omitted will not matter for the sake of comparisons. For clarity, the index m will be omitted in the following transformations. And in its final form, even though it's the derivative in respect to skulls, we want it expressed in terms of levels rather than skulls since that's the information immediately visible in the game.This is multiplied with I_m and S_m to include active skill and item bonuses. As of now, the cost reduction factor can be omitted, since all non-tap DPS monsters have equal cost reduction, but this may change in the future.Inspecting this derivative function reveals that it is declining, so for each monster it is true that the more skulls we've already invested in it, the less damage per further skull invested is gained (diminished returns). As such, we can conceive the optimal skull allocation strategy as a flow equilibrium where skulls are invested into the different monsters such that their damage derivative to skulls is equal at all times. We can even attempt to express this equilibrium state by equating two such derivatives representing arbitrary monsters and simplify it approximately to a general, simple rule:Since the residual ratio within the logarithm changes very little for larger levels with what is otherwise a constant difference, one obtains this handy rule of thumb:Furthermore, the developers have chosen to give all non-tap monsters the same base damage to base cost ratio. As long as this is true, the rule simplifies further as for the sake of these efficiencies only the item factor I and active skill factor S of each monster need to be considered, other than that all monsters ought to be kept at the same level.Before monster-specific items come into play, the only monsters that differ from the rest in this regard are Blue Specter through Spectral Reposession and Flying Squid through Pastafury. Assuming for simplicity the permanent uptime of all relevant skills, the multiplier is 11 for Blue Specter, 17 for Flying Squid, and 3 for everyone else. Therefore, before different stocks of monster-specific items come into play, they should be kept this many levels above the other non-tap monsters:When it comes to selecting one of multiple monster items from a chest, we want to choose the one that provides the most damage increase. That is, the one that holds the largest damage proportion after its item damage multiplier is factored out.The base damage values grow by a factor of 10 for each successive slot up to including the Black Lich, then by a factor of 1000. For the extra damage of Squid and Specter due to their special skills , we need to consider both the direct damage bonus and the indirect one due to keeping them on a higher level. Their overall bonus factors therefore are:This means Blue Specter overtakes one but not quite two spots at his place in the hierarchy, between Swarm of Bats and The Big Plague, whereas Flying Squid stays where he is since he cannot quite reach the Tomb King, who still has ~2.79x as much damage.To conclude, this establishes the following hierarchy for selecting monster items from chests:While the direct effect of items does not come into consideration for the monster item selection, their indirect effect (through warranting a higher level) does. This leads to further cementing this hierarchy, the gaps in damage will get larger over time.As a result, in the long run, Tomb King will deal almost and practically the entirety of the total damage.