Zenless Zone Zero: The Final Answer of "PEN"

Hello everyone! Soon after the launch of Zenless Zone Zero, at least two hot topics arose in terms of attribute gains:

1. How is the Pen Ratio gain?
2. How does the In-combat ATK bonus alone affect the building of agents?

In this article, we will follow the "Dilution Principle", which has been proven by a lot of practice in Honkai: Star Rail/Genshin Impact, to quantitatively analyze the attribute gain among the ZZZ, and at the same time, answer the two questions mentioned above that we are concerned about.

Quick Overview of Conclusions of Zenless Zone Zero

1) PEN Ratio is better when it's above 50% - get the Increased ATK and CRIT Rate/DMG done before then.

DEF reduction and Impact dilute each other, higher DEF reduction makes PEN gains lower, and PEN Ratio in general is currently prioritized less than ATK/Increased DMG/CRIT Rate/DMG.

2) The in-combat ATK can be interpreted as a separate multiplier zone - but it doesn't affect the current equipments of agents, it just improves the strength of some of the equipment/buffs.

The Core Skill of Soukaku and effects of 4-Pieces Woodpecker Electro set- which are also additive between buffs that are the same as the in-combat ATK - will be diluted, but not by the Drive Disc Main Stats and Sub-Stats dilution.

3) Higher gains are seen in the agent detailed stats when ATK green/ATK white = 90%.

4) When you play attribute anomaly agents (e.g. Grace Howard, Piper Wheel, Jane Doe), just stack Anomaly Proficiency as high as you can.

5) If you want to combi defense equipments, stack 50% HP first, then stack at a ratio of HP:DEF = 3:2.

PART 01. Known Condition Collation in ZZZ

1.1 Zenless Zone Zero DMG Formula

Damage = Base DMG Zone * Increased DMG Zone * CRIT Zone * DEF Zone * Resistance Zone * Reduced Vulnerability Zone * Daze Vulnerability Zone * Special Multiplier Zone

Base DMG Zone = Multiplier * ATK

ATK = [Base ATK * (1+Off-combat ATK%) + Off-combat Numerical ATK] * (1+In-combat ATK%) + In-combat Numerical ATK

Among these values, in-combat ATK is used to differentiate between ATK buffs that do not settle until after entering combat, such as The Core Skill of Soukaku, EX Special Attack of Lucy, the ATK bonus effects of Drive Disc 4-Pieces sets such as Woodpecker Electro/Puffer Electro/Hormone Punk/Thunder Metal, and W-Engine effects such as Cannon Rotor and Starlight Engine.

In-combat/Out-combat ATK include both percentage form bonuses and fixed value forms.

DMG Increasing Zone = 1 + DMG Increasing

CRIT Zone = 1 + CRIT Rate * CRIT DMG

DEF Zone =1/[(1-PEN Ratio) * （1- Reduced DEF) + 1]

This DEF Zone formula corresponds to a situation where agents and enemy both are level 60 and the enemy DEF is 794.

Resistance Zone = 1- Resistance + Reduced Resistance + Resistance PEN

Reduced Vulnerability Zone = 1+ Vulnerability- Reduced DMG

Daze Vulnerability Zone = Normally 150%

1.2 Main/Sub-stats of S-Rank Drive Discs

Sub-Stats values are usually 10% of level 15 Main Stats values.

For example, the value of a CRIT Rate Sub-Stat is 2.4%.

Definition: Unit value = level 15 Main Stat/24

1.3 Introduction to the "Dilution Principle"

For example, the first equipment which increase 50% ATK for your agent changes your agent's ATK from 1000 to 1500, with a relative boost of (1500/1000-1) = 50%

The second equipment which increase 50% ATK for your agent changes your agent's ATK from 1500 to 2000, with a relative boost of (2000/1500-1) = 33%

The 50% and 33% here are the relative boosts of the two equipment match, and the fact that 33% is less than 50% is the essence of dilution - diminishing relative boosts.

If the second equipment's stat is replaced with 50% increased DMG, your actual DMG goes from 1500 to 2250, with a relative boost of (2250/1500-1) = 50%

It is clearly superior to the "Second equipment with 50% increased ATK " option - this is the basic principle of the "multiplication zone dilution" guide to fitting.

If we change the object of study to a hypothetical piece of equipment that increase 0.001% or even an infinitesimal amount of ATK, and study the relative rate of increase for each new piece of equipment, and plot the "rate of increase y for the xth piece", we can get an intuitive attribute benefit curve.

This will help us understand more intuitively the benefits of each multiplier.

The GR growth rate curves for each attribute are shown below and the corresponding formulas are shown in the graphs.

In general, you only need as high a growth rate as possible.

PART 02. Base Multiplier Area Yield Curve & Interpretation in ZZZ

The "base multiplier zones" are ATK Zone (AT),DMG Booster Zone (DB),CRIT Range (CR), and DEF Zone - PEN Ratio (D-PR), which are characterized by the fact that these four multiplier zones can be affected by stats of Drive Discs and are regular damage multiplier zones.

The stat corresponding to the 1%CRIT Rate is used as the unit of the x-axis.

For example, a point with x=1 in the ATK Zone image would represent a 1.25% ATK percentage bonus at this point;

In the PEN Ratio image, a point with x=1 would represent a 1% ATK percentage bonus at this point ......

2.1 ATK/DMG Booster Zone in ZZZ

ATK and DMG booster are both 30% of the main stats in ZZZ, and the curves are made to overlap perfectly. In practice, it can be memorized like this.Whichever ATK/DMG Booster buff is less is boosted.

For example, if you currently have 50% ATK bonus and 60% DMG Increase, then at that point it is more profitable to boost ATK again. Conversely, if you have 50% DMG Increase and 60% ATK bonus, that time is when it would be better to get the DMG Increase again.

For the in-combat ATK bonus, you can think of it as a new "vulnerability multiplier". This multiplier will only increase the strength of the equipment that has the in-combat ATK bonus, and will not affect the choices of the main or sub stats of our drive discs.

2.2 CRIT Range in ZZZ

The CRIT Range curve is a segmented function: we default agents' CRIT Rate and CRIT DMG to 5 and 50.

First segment: 1% CRIT Rate per 1 unit stack until mob rate = 25%

Second segment: stack (0.5% CRIT Rate + 1% CRIT Damage) for every 1 unit until CRIT Rate = 100%.

Third segment: 2% CRIT Damage per 1 unit, no limit

This is because the panel does the most damage for the same number of stats when (CRIT Rate: CRIT DMG = 1:2).

2.3 PEN(D-PR) in ZZZ

Pen in the denominator in DEF zone, so it is shown to be anti-dilution, i.e. the more you pile it on, the higher the gain.

Anti-dilution doesn't mean that this stat is advantageous. On the contrary, the higher gain of stacking more of these stats means that you stack less and can only earn the low gain area, a total sunk cost.

Why is it called anti-dilution, it is very easy to understand, because increasing defense is beneficial to the game unit, it is positive dilution, reducing defense is anti-dilution.PEN and DEF reduction are multiplied in the denominator to dilute each other.

As you can see in the graph, the benefit of PEN is lower with 20% DEF reduction than without it.

How to interpret this - both PEN and DEF reduction are in the denominator, both are anti-dilution.ATK and DMG Booster as positive dilution is a high gain in DMG Booster after ATK is stacked more;PEN and DEF reduction as anti-dilution is the low gain in DEF reduction after PEN is stacked more.● How to prove it - Limit idea, obviously, at 100% DEF reduction, no matter how much you stack PEN, you can't boost DMG anymore, meaning at this point the PEN attribute gain is 0.

Overall, the gain of PEN is rather average, unless the ATK/DMG Booster is already more than 150%, with Lina, only consider bringing the PEN main stats.About [PEN]:

A level 15 main stat has a PEN of 24%, one-tenth of which is 2.4%. This means that if there is a class sub-stats called PEN, one of PEN stat should be 2.4%. and another PEN of 9 is equivalent to a 1.1% PEN for an enemy with 794 DEF. This is less than half of the normal level, similar to the difference between a percentage ATK stat and a number ATK stat.

So [PEN] is a attribute can be used early in the game, only in the enemies low level, there is a certain benefit, can not be used when enemies level up.

2.4 Base Attribute Yield Curve in ZZZ

As can be seen, the CRIT Zone curve is smoother over a longer period of time and can be used as a graduation attribute. Referring to the CRIT Zone curve, a straight line with y=0.0067 is selected as a reference line, and when the gain is greater than this value, the current attribute is considered to have a higher gain: Thus it is concluded that the out-of-bounds ATK/DMG Booster threshold is 90%, and stack 90% ATK bonus has a higher gain. PEN Ratio has a higher gain when stacked above 50%, either stack past 50%. Either that or don't stack.

ADDITIONAL: PEN Ratio is reduced from 50% to 20% threshold when combat across 10 levels before level 60. However, since enemy DEF no longer grows after level 60, this basically doesn't happen at later levels.

PART 03. Attribute Anomaly Return Curve & Interpretation in ZZZ

The main difference between the Attribute Anomaly Multiplier Zone and the Regular DMG Multiplier Zone is the absence of the CRIT Zone and the addition of the Anomaly Proficiency Zone.

Anomaly Proficiency Zone = Anomaly Proficiency/100

In the following, the initial Anomaly Proficiency is assumed to be 100.

3.1 Anomaly Proficiency Zone in ZZZ

After assuming the initial anomaly proficiency to be 100, this multiplier zone is of the form (1+n%), so the curve shape is also similar to the ATK/DMG Booster zone.

3.2 Attribute Anomaly Return Curve in ZZZ

Examining the benefit curves related to attribute anomaly damage by putting them into the same coordinate system: you can see that Anomaly Proficiency is a very large boost to attribute anomaly damage, about three times as large as ATK/DMG Booster.

Since the Anomaly Proficiency main stat has a value of 92, which equates to +92% Attribute Anomaly Damage, compared to ATK which is 30%, the manifestation of the 1 Anomaly Proficiency stat benefit is also triple that of the 1 ATK stat.

This design would make it more characteristic to have a significantly higher percentage of Anomaly DMG than Straight DMG in an Anomaly team.

PART 04. Resistance reduction/vulnerability gain curve & interpretation

Since RES and Vulnerability do not exist in the main/sub stats, and there is no equivalent relationship such as "1 unit = 1% CRIT Rate = 1.25% ATK", they cannot be analyzed in the same coordinate system as the attack and wounding multiplier zones. However, it is still possible to make a curve and observe the trend of increase/decrease of its gain.

4.1. Resistance reduction/Vulnerability (RES/VA) in ZZZ

1 unit = 1% resistance reduction/vulnerability has been chosen for plotting. 0 resistance reduction in the Zenless Zone Zero still doesn't halve the gain after getting to 0 resistance like the Genshin Impact, so it's also a higher gain attribute. Currently there are more sources of resistance reduction than vulnerability, such as Lycaon's Core Skill.

PART 05. Survival Attribute Yield Curve & Interpretation in ZZZ

Survival attributes are Health Point (HP) as well as Defense (DEF).It can be analyzed using Equivalent HP, i.e.:

Equivalent HP = HP/DEF Zone

HP = Base HP * (1 + HP bonus)

DEF Zone = 794/[Base DEF * (1+DEF Bonus) + 794]

The base DEF of S-rank agents is usually around 600, and this article takes 600 for calculation.

5.1 Health Point(HP) in ZZZ

The HP gain curve has the exact same shape as ATK.

5.2 Defence(DEF) in ZZZ

DEF is an inversely proportional function of the same type as the HP curve. The shape of the DEF gain curve also validates the previous statement that "stacking DEF is a positive dilution for agents".

5.3 Survival Attribute Benefit Curve in ZZZ

The relationship between HP and DEF is similar to that between ATK and DMG Booster, with the difference being that HP units are more profitable. However, if you stack too much HP, there will be a similar "dilution" phenomenon, so you can balance it out as described in the chart. For the time being, ZZZ does not have any agents that can provide healing, except for the "Avocaboo". Until there is a agent that can provide healing, the above conclusion still applies, and if there is a agent that can provide healing, then the value of DEF in long term combat will increase dramatically.

The above is the entire content of this article, at present the attribute anomaly accumulation, impact (Daze-related) can not quantify its benefits, we are also very happy to see proxies and the strategy group together to explore the study of the benefits of such attributes~