r/OnePieceTC • u/FateOfMuffins Doktah Carrot Muffins • May 03 '18
JPN Analysis TM Kizaru - The Misconception of Multi-Component Specials (PART 1)
Introduction
SEE TLDR IF YOU DON'T WANT TO READ THE MATH OR EXPLANATIONS
Ever since TM Kizaru's CA and special were introduced, players have been shitting on him, complaining that he is the worst TM unit and not worth farming (although Bandai decided to spice things up this time with the free Legend pulls).
Originally, I wanted to discuss his special and compare TM Kizaru with other TM units overall in this post. However, it's getting quite lengthy so I'm going to split it into 2 posts.
In this post, I want to begin with his special and how multi-component specials (especially health cuts) are seemingly misunderstood by most players. In particular, players seem to read specials such as 1.75x boost as being automatically weaker than specials with 2x boosts - this is NOT the case.
My next post will compare TM Kizaru with other TM units.
Part 1 - Multi-Component Specials
Most units these days have multiple components in their specials, including utility components such as reducing attack down, defense up, damage reduction, healing, etc., in addition to damage components such as attack boosts, orb boosts, HP cuts, nukes, etc.
I will not be spending any time analyzing utility components, as they are incredibly difficult to compare across units, as it is completely situational. It's practically impossible to theoretically say whether a special that reduces despair by 3 turns is better or worse than a special that reduces paralysis. Events may need an orb locker, or multi-turn attack booster, or perhaps a paralysis reducer, but these specials could be completely useless in other events. However, it is still important to note that TM Kizaru has other utility components, namely a 3 turn boost that extends across multiple boss stages, as well as an orb lock which synergizes well over multiple turn burst.
Therefore I will be only focusing on the damage components of specials, specifically excluding nukes (which although may be necessary in certain situations and definitely contributes to how much damage a unit deals, is situational and much more difficult to compare between different units).
Damage-Components of Specials & Health Cuts
Different boosts can stack with each other multiplicatively, but not with boosts of the same nature. However, health cuts are essentially damage boosts that can endlessly stack, both with other health cuts and with other attack boosts, provided the stage does not have a barrier and can be HP Cut.
How do HP cuts work as damage multipliers? A very simplified answer would be to divide the other damage multiplier by the percentage health remaining (or 100% - HP Cut). So a 20% health cut would be equivalent to a 1 / 0.8 = 1.25x damage multiplier.
While that's true for a single turn burst without other assumptions, let's get into the nitty gritty details and assumptions for some specific scenarios. For all the calculations below, we'll assume that each unit in the team has the same attack stat for simplifying purposes.
Definition: Effective Damage (ED) - If an enemy unit has X HP and the team deals Y > X total damage (which includes all damage components of specials, including HP cuts and nukes), the team's effective damage is equal to X.
Note: This excludes the effect of IntHawk and V2 Doffy specials from the remainder of the calculations as their specials are extremely unique and situational.
Example: Enemy unit has 5M HP. Your team after using all specials deal a total of 5.5M, including 1M from a 20% HP cut and other damage of 4.5M. Your effective damage is 5M. The principle is - it does not matter how much overkill damage you deal if you are still able to defeat the enemy unit.
Definition: Equivalent Multiplier (EM) - Suppose with unit A, the team is able to deal an effective damage of Y. The Equivalent Multiplier is the minimum flat damage multiplier X that a replacement unit B provides such that the team with unit B instead of unit A deals exactly Y effective damage.
Example: Enemy has 1M HP. Unit A does a 20% HP cut, after which the team one shots the enemy. ED is 1M. Without any other information, the team with A will deal at minimum 800k in damage after the HP cut. The minimum flat damage multiplier that a replacement unit B needs is then a 1.25x damage boost. Higher damage boosts, like 2x, will result in the same ED of 1M, but it is not the minimum damage boost. The EM is 1.25x.
Equivalent Multiplier for TM Kizaru
Suppose we have TM Kizaru's special, which has a 20% health cut in conjunction with a 3 turn 1.75x adaptable attack boost. Both components contribute to damage, but how much damage does TM Kizaru deal in total?
1) Single turn burst, mono-DEX team
To keep comparisons simple, assume that TM Kizaru is boosted in the team. Further assume that after the health cut, we are now able to 1-turn the enemy. Let X be the EM of a replacement unit. Let C be the damage multipliers from the remainder of the team, including the captain ability and chain multipliers and excluding TM Kizaru and the replacement unit.
Then for TM Kizaru:
- 0.8 * ED = 1.75 * C
- ED = 1.75 / 0.8 * C
For the replacement unit:
- ED = X * C
Equating the two gives:
- X * C = 2.1875 * C
- X = 2.1875
Conclusion
For a single turn mono-DEX team, TM Kizaru provides the highest attack boost @ 2.1875x in the game except for V2 Shanks.
2) 2 turn burst, constant attack boost, mono-DEX team.
Let's suppose that instead of a 1-turn situation, we now need 2 turns to completely burst the enemy, with the ED over 2 turns being the entire enemy's health bar. In this calculation, since TM Kizaru is a 1.75x boost for 3 turns, let's assume that the replacement unit provides a constant EM of X for at least 2 turns.
Let C1 being the remaining damage multiplier for the first turn and C2 being the remaining damage multiplier for the second turn.
Then for TM Kizaru:
- 0.8 * ED = 1.75 * C1 + 1.75 * C2
- ED = 1.75 / 0.8 * (C1 + C2)
For the replacement unit:
- ED = X * C1 + X * C2
Equating the two gives:
- ED = X * (C1 + C2)
- X = 2.1875
Conclusion
In order to replace TM Kizaru for up to 3 turns of damage, you need a unit that provides an EM of 2.1875x for up to 3 turns.
3) Single turn burst, non mono-DEX team.
One unique feature of TM Kizaru's special is that the damage boost is adaptable, similar to V2 Shanks. What if you replace any other type booster (who is boosted in the team) with TM Kizaru? For example, if you decide to replace Colo Sicilian with TM Kizaru in a V2 Akainu team.
In which case TM Kizaru himself is not boosted by the CA and will be the first one attacking. To be simple, let's exclude chain lock or chain boost.
Let C1 be the damage multiplier for the units that are boosted by the team, excluding chain. Let C2 be the damage multiplier for the units that are not boosted by the team (TM Kizaru).
Then for TM Kizaru:
- 0.8 * ED = C2 + 1.75 * C1 * (1.3 + 1.6 + 1.9 + 2.2 + 2.5)
- ED = 1.25 * C2 + 2.1875 * 9.5 * C1
For the replacement unit:
- ED = X * C1 * (1 + 1.3 + 1.6 + 1.9 + 2.2 + 2.5)
- ED = 10.5 * X * C1
Equating the two gives:
- 10.5 * X * C1 = 1.25 * C2 + 2.1875 * 9.5 * C1
- X = 1.25 / 10.5 * C2 / C1 + 2.1875 * 9.5 / 10.5 * C1 / C1
- X = 0.12 * C2 / C1 + 1.98
Since C1 and C2 > 0,
So even if he is not boosted, TM Kizaru provides almost as much damage as a 2x type booster. You can replace literally every other single turn 2x type booster in the game with TM Kizaru.
In fact, the result from 2) follows through as well. TM Kizaru is equivalent to a 3-turn 1.98x type booster FOR ANY TYPE.
In fact, if we compute 3) using chain lock of between 2.5x-3x OR a chain boost of between 0.5x-0.9x, we'd get that TM Kizaru is equivalent to a 3-turn 2.02x-2.05x type booster FOR ANY TYPE.
4) 2 turn burst - Other Units
Similar to example 2. This calculation works for other units with single turn boosts such as Colo Lucy or 6+ Croc. This differs from the TM Kizaru situation above since you are free to use a second unit for the 2nd turn of burst (for example, using Colo Lucy for turn 1 and V1 Doffy for turn 2).
For example, suppose we're looking at Lucy, who contributes a 1.75x orb boost for 1 turn and 20% HP cut. He is the only unit being changed and does not impact the 2nd turn's damage.
Assume C is the remaining damage multiplier for the first turn. Assume D is the total damage multiplier for the second turn. Assume that in the replacement team, the total damage multiplier for the 2nd turn is also equal to D (in the above example, suppose we switch out Lucy for turn 1 but keep Doffy for turn 2).
Then:
- 0.8 * ED = 1.75 * C + D
- ED = 2.1875 * C + 1.25 * D
For the replacement unit:
- ED = X * C + D
Equating the two gives:
- X * C + D = 2.1875 * C + 1.25 * D
- (X - 2.1875) * C = 0.25 * D
- X = 0.25 D / C + 2.1875
Since both C and D must be positive,
Conclusion
So a unit like Colo Lucy actually provides an orb boost with a minimum EM of 2.1875x for a single turn, which actually deals more damage than 2x orb boosters like Doffy, especially since his 20% HP cut is unblockable.
Furthermore, Lucy's EM is actually higher than 2.1875x if you are in situations where you need a multi-turn burst.
Common HP cut & damage boost units
The above math can be extended towards a number of different units, summarized below.
| 1.75x + 7% HP Cut | 1.75x + 10% HP Cut | 1.75x + 15% HP Cut | 1.75x + 20% HP Cut | 2x + 10% HP Cut | 2x + 20% HP Cut | |
|---|---|---|---|---|---|---|
| What Players See | 1.75x | 1.75x | 1.75x | 2x | 2x | |
| Effective Multiplier >= | 1.88x | 1.94x | 2.06x | 2.1875x | 2.22x | 2.5x |
| Examples | Colo Jesus | Valentine Nami | 20 Anni Sanji | TM Kizaru | Franky | 6+ Croc |
| Marguerite | Lucci | Colo Lucy | ||||
| Oars | ||||||
| Daruma | ||||||
| Bellamy |
Comments
- 6+ Croc provides the strongest single turn burst in the game, despite Nami having a 2.25x attack boost.
- Oars is actually the strongest conditional booster in the game, at an Effective Multiplier of 1.88x!
Thanks for reading! If anything doesn't make sense with my calculations or assumptions, or if you have any other questions, feel free to ask!
I'll be doing a Part 2 that compares TM Kizaru with other TM Units sometime soon.
TLDR
Units with 1.75x multipliers are NOT necessarily weaker than units with 2x multipliers
Colo Lucy is one of the strongest universal orb boosters in the game, equivalent to (at minimum) a 2.1875x orb booster
Versus all content that can be HP Cut, TM Kizaru is equivalent to a 3-turn 2.1875x DEX attack booster (+ orb lock)
Versus all content that can be HP Cut, TM Kizaru is equivalent to a 3-turn 1.98x (occasionally 2.02x-2.05x) ADAPTABLE attack booster (+ orb lock)
1
u/FateOfMuffins Doktah Carrot Muffins May 03 '18
Eh university habits die hard. I can assure you that I could have made this a lot more "elongated" than what was presented.
Right before I make my definitions and dive into the nitty gritty details, I specifically stated the simplified calculations, which are but a few lines long in total. The remaining calculations are "elongated" because they were supposed to be in order to show the details.
I'll keep that in mind to highlight this part more clearly and possibly put it in the TLDR.
Yes I agree realistic applications are more important than hypotheticals, but there's a reason why hypothetical theories and mathematical models exist in every aspect of our world in the first place. (You may not see them, but I assure you, they're there in the background). Compared to "realistic" scenarios, theoretical models are much simpler, yet provide a reasonable approximation to "reality".
If you want realistic calculations, I could easily extend the calculations beyond a few lines of equations each. Why did I not do so? Because it will become incredibly complicated and require significantly more effort and more "elongated" equations.
Search up the 80/20 rule, otherwise known as the Pareto principle. 20% of effort will produce an 80% result (which is theoretical model and good enough). If you want the 100% result (realistic), you'll need to put in maybe 5x the effort.
Lastly, doesn't it seem ironic that you want realistic and practical results from the unit instead of fake hypotheticals that everyone has been stating ever since TM Kizaru was announced?
In which case, why not wait till you actually use him in practice before shitting on him based on theory craft?