ZZZ DEF Ignore/Reduction and Penetration? Diminishing and Reverse Diminishing?

Welcome to a new article exploring Zenless Zone Zero's underlying mechanics - Penetration Defense Theory. Recently, many players around me have been asking about the relationship between Ignore Defense and Diminishing Returns - topics that are frequently debated. While we've briefly mentioned these concepts in previous discussions about Disorder, a systematic explanation has been lacking. Let's start from the basics today!

1. Introduction to the Zenless Zone Zero Defense Zone Formula

General Damage Multipliers = ATK × DMG Scaling × DMG Bonus Zone × Defense Zone × Resistance Zone × Vulnerability Reduction Zone × Stagger Vulnerability Zone

Direct Damage = General Multipliers × CRIT Zone × (Distance Attenuation Zone)

Anomaly Damage = General Multipliers × Anomaly Proficiency Zone × Level Zone × (Anomaly DMG Bonus Zone × Anomaly CRIT Zone)

We briefly introduced the overall damage formula in our previous article about Anomaly. This article will focus on the Defense Zone, the most complex multiplier in the formula.

The first layer of the formula introduces two new variables:

Let's explain Attacker Level Base: This value increases with the attacker's level, which is why Agent level matters. Currently, the base value at Level 60+ is 794.

Some Proxies might ask: "Aren't Agents capped at Level 60? Why use '60+'?" This is because the formula applies both when Agents attack enemies and when enemies attack Agents. Since enemy levels exceed 60, we use '60+' notation."

Next is Target Effective Defense:

Target Effective Defense = Target Defense × (1 - Penetration Rate) - Penetration Value

Target Effective Defense ≥ 0

We can see that the game's Penetration Rate and Penetration Value are applied here. The minimum Effective Defense is 0, meaning the Defense Zone multiplier cannot exceed 1.

Then, Target Defense:

Target Defense = Base Defense × (1 + Defense% Modifiers) + Flat Defense Modifiers

Current in-game Defense modifiers are percentage-based (e.g., -20% Defense). Flat modifiers (e.g., +100 Defense) do not exist yet, but the formula is prepared for potential future implementation, similar to Astra Yao's flat ATK buff mechanics.

Finally, Base Defense:

Agents Base Defense = Out-of-field Defense = Base Defense Stat × (1 + Out-of-field Defense%) + Flat Out-of-field Defense

Enemies Base Defense = (Lv.1 Base Defense ÷ 50) × Enemy Level Base

If the target is an Agent, this refers to their Out-of-field Defense, calculated in the same way as Out-of-field ATK and Out-of-field Impact. The term 'Target Defense' here corresponds to the character's in-combat Defense stat. For enemies, their initial Defense at different levels can be derived from their Level 1 base Defense.

For specific enemies: Currently, the lowest Level 1 base Defense belongs to small enemies like Tyrfing (36), while most bosses have a Level 1 base Defense of 60. The use of 'currently' acknowledges that future updates may introduce enemies with specialized Defense values exceeding 60. These formulas for the Defense Zone are indeed complex. Let's now simplify them for common scenarios.

First, in the most common scenario where we calculate the damage Agents (max Lv.60) deal to enemies (max Lv.70), substituting the corresponding values and formulas gives:

Further analysis shows that if characters have no PEN (Penetration Value), the 794 in the numerator and denominator can be canceled out:

If there are no Defense% Modifiers and no PEN Ratio, the formula further simplifies to:

Substituting the standard boss base Defense value of 60 at Level 1, the Defense Zone against bosses becomes 0.4545.

2. Diminishing Returns and Reverse Diminishing Returns in Zenless Zone Zero

What we commonly refer to as 'diminishing returns' is more accurately described as marginal utility reduction. Suppose a character's current DMG Bonus is x. When gaining an additional 10% DMG Bonus, the actual benefit is calculated as:

Actual Benefit (y) = [(Current DMG Bonus x + 10%) / Current DMG Bonus x] - 100%

Substituting specific values:

When x = 100%, y = 10%When x = 200%, y = 5%

As x increases, y progressively decreases.

Considering two multiplicative zones (e.g., ATK and DMG Bonus both at 1) with 60% improvement to allocate:

1. Distributing 30% to each zone maximizes the result: 1.3 × 1.3 = 1.69
2. Allocating 60% to one zone yields 1.6

For the Defense Zone formula (ignoring PEN and assuming max enemy level):

Defense Zone = 1 / [60/50 × (1 - Defense Reduction%) × (1 - PEN Ratio) + 1]

Analyzing Defense Reduction or PEN Ratio individually:

1. At x = 0% (no PEN), 10% PEN provides 5.77% actual benefit
2. At x = 90% PEN, 10% PEN provides 12% benefit

This demonstrates reverse diminishing returns: higher values yield greater returns. Allocating 60% to either Defense Reduction or PEN Ratio alone maximizes gains, while splitting 30% to both yields the lowest efficiency - hence their mutual diminishing effect.

3. Main Stat Selection for Drive Disc Slot 5 in Zenless Zone Zero

Penetration Rate (PEN Ratio) is generally only available in the fifth slot of Driver Disks or through the two-piece set effect of Driver Disk sets. Let's first examine its selection as the fifth slot. Aside from PEN Ratio, the other common choice for the fifth slot is the corresponding Elemental DMG Bonus. To determine which is more effective, we can use the formula to calculate that PEN Ratio becomes superior when the DMG Bonus exceeds 199.16% (including the base 100% DMG Bonus).

In simple terms, in a team composed of Lighter, Astra Yao, and Evelyn:

1. Lighter provides 75% DMG Bonus
2. Astra Yao provides 20% DMG Bonus
3. Astra Yao's passive adds an additional 8% DMG Bonus

This totals 103% additional DMG Bonus, surpassing the 99% threshold. Thus, Evelyn's fifth Driver Disk slot should prioritize PEN Ratio.

If a team cannot exceed 99% DMG Bonus, the fifth slot should use Elemental DMG Bonus. For teams where some skills exceed 99% DMG Bonus while others do not, prioritize based on the damage contribution of each skill in the rotation.When factoring in Defense Reduction, the threshold for DMG Bonus increases dynamically. For example: At 25% Defense Reduction, PEN Ratio only outperforms Elemental DMG Bonus if the total DMG Bonus reaches 233.8%.

4. Ignore Defense and Ignore Resistance in Zenless Zone Zero

In Zenless Zone Zero, Ignore Defense and Defense Reduction are functionally equivalent in formula calculations. For example:

15% Ignore Defense + 20% Defense Reduction = 35% total Defense Reduction

Since Defense Reduction itself exhibits increasing returns (the more you stack, the higher the marginal gain), Ignore Defense behaves similarly. The efficiency of Ignore Defense becomes greater when paired with existing Defense Reduction.

The key difference lies in their application targets: Defense Reduction is a debuff applied to enemies. Once active, all attacks from the party benefit from the reduced Defense value. Ignore Defense typically applies to specific skills. For instance, if a skill "ignores 15% Defense," this effect only applies to damage from that skill. The same logic applies to Ignore Resistance and Resistance Reduction.

5. Damage Resistance Zone Caps in Zenless Zone Zero

Speaking of Resistance, let's revisit the Damage Resistance Zone. We know its base value is 1. If the enemy has a weakness, it becomes 1.2; if they have Resistance, it drops to 0.8. However, since a certain update, resistant enemies can now have higher values. For example:

Hugo has 40% Ice Resistance → Damage Resistance Zone = 0.6

Pompey has 40% Electric Resistance → Damage Resistance Zone = 0.6

Do Resistance Reduction and Resistance Penetration have caps like Defense Reduction? The answer is yes. After repeated calculations, the maximum value for the Damage Resistance Zone is 2.

6. ZZZ Penetration Defense Theory Summary

1. Ignore Defense is functionally equivalent to Defense Reduction in damage calculations. The same applies to Ignore Resistance and Resistance Reduction.
2. Defense Reduction and Resistance Reduction have effectiveness caps.
3. The more you stack Defense Reduction/PEN Ratio, the higher their marginal returns.
4. Defense Reduction and PEN Ratio exhibit mutual diminishing returns.
5. PEN (Penetration Value) generally offers low returns unless paired with high Defense Reduction/PEN Ratio.
6. The critical threshold for Slot 5 Drive Disc selection is 99% DMG Bonus under standard conditions (varies by scenario).
7. Agent level directly impacts Defense Zone values - prioritize maxing levels.
8. For Anomaly DMG builds, equip same-element characters with DMG Bonus or PEN Ratio.

That concludes our guide on Penetration Defense Theory in Zenless Zone Zero. Stay tuned for more in-depth analyses! Don't forget to check out merchandises - you might find your favorite Agent's merchandises!