Damage Formula — Additive, Multiplicative & Lone Ranger

Two items both say "+20% damage." One adds to a pool with four other bonuses. The other is a separate multiplier applied after everything else. Those aren't the same thing — not even close. This guide breaks down exactly how Tangy TD calculates damage, which bucket each bonus lands in, and why the Lone Ranger multiplier makes it so much stronger than comparable items.

The Full Formula

Four distinct stages, applied sequentially. The separation between stages is what creates counter-intuitive interactions — a small bonus in the Lone Ranger stage is worth more than a large bonus in the Additive stage once you have other additive bonuses stacked.

Additive Bucket — What Goes Here

The additive bucket pools all percentage bonuses that sum together before being applied to base damage. The more bonuses you have here, the less each additional one matters (diminishing relative returns).

The key insight: if you have +100% damage from additive sources (a reasonable mid-game total), an additional +20% additive item only adds 10% more final damage. But the Lone Ranger multiplier adds its 20% after the doubled base — it's always a true 20% bonus regardless of how many additive items you have.

Multiplicative Bucket

Multiplicative bonuses apply as separate multipliers — they don't pool with each other, which means two multiplicative bonuses compound rather than add.

Example: Ban Hammer +25% amp AND a phase vulnerability of +30% applies as:
Damage × 1.25 × 1.30 = Damage × 1.625

Not: Damage × (1.25 + 1.30) = Damage × 2.55

This means multiplicative bonuses are less powerful than they appear when stacked, but still significantly stronger than an equivalent additive bonus. The compounding nature means 2× multiplicative bonuses give 1.625×, not 2.55×.

Lone Ranger — Its Own Multiplier

The Lone Ranger node (+20% damage when no other Archer is within X tiles) is calculated as its own separate multiplier applied last in the chain — after both the additive bucket and the main multiplicative bucket.

The 4-tile isolation rule

An Archer is considered "isolated" for Lone Ranger if no other friendly unit (Archer, Defender, or Healer) is within 4 tiles. This includes Tangy herself — walking too close to your Lone Ranger Archer removes the bonus until Tangy moves away.

Lone Ranger with Pumpkin King

The Pumpkin King item spawns a puppet companion that counts as a friendly unit for Lone Ranger purposes. A Pumpkin King Archer is never isolated — the puppet is always adjacent. The Lone Ranger build and Pumpkin King builds are mutually exclusive. See the Lone Ranger build guide for positioning alternatives.

Armour & Penetration

Armour reduces incoming damage using a proportional reduction formula, not a flat subtraction:

Examples:

• 0 armour (no reduction): 100 / (100 + 0) = 1.0 → full damage
• 100 armour: 100 / (100 + 100) = 0.5 → 50% damage taken
• 400 armour (Iron Warden): 100 / (100 + 400) = 0.2 → 80% reduction

Armour Penetration

Armour penetration reduces the effective armour value before the formula is applied. Ultra Vision Goggles grants +15 armour pen. Against the Iron Warden (400 armour):

• Without pen: 100 / 500 = 0.2 (80% reduction)
• With +15 pen: 100 / 485 = 0.206 — a marginal improvement on ultra-high armour

This reveals an important truth: flat armour pen is most valuable at low-to-medium armour values. Against common enemies with 50 armour, +15 pen changes 66% of damage taken to 69% — a meaningful 4.5% DPS increase. Against the Iron Warden's 400 armour, the same +15 pen is almost irrelevant.

Critical Hits

Critical hits are a separate multiplicative layer applied between the main multiplicative bucket and the Lone Ranger multiplier:

The default crit multiplier is 1.5× (50% bonus damage). Skill tree nodes can increase this to 2× or 2.5×. Crit chance items add percentage proc probability.

Important: Crit chance does not benefit from the Lone Ranger bonus — the 1.5× crit multiplier is applied before Lone Ranger in the chain. This makes crit scaling marginally less efficient on Lone Ranger builds than on multi-tower builds where the crit lands earlier in a shorter chain.

Worked Examples

Example 1 — Standard Archer, mid-game

Example 2 — Lone Ranger Archer, same setup

Example 3 — Ban Hammer on stunned target

Build Implications

First additive bonus is worth the most

Going from 0% to +20% additive is a true 20% DPS increase. Going from +80% to +100% additive is only a ~11% DPS increase (100 vs 180 base multiplier). Early-game additive items are more impactful than late-game additive items — this is why the skill tree recommendation is to take the Lone Ranger node early rather than stacking more stat nodes.

Multiplicative stacking has compounding costs

Two multiplicative bonuses (× 1.25 and × 1.30) gives a combined 1.625× multiplier, not 1.55×. This seems like a bonus, but it means each multiplicative bonus is weaker in practice than its tooltip suggests — the compounding cuts both ways.

Armour pen is an early-game item

The armour formula means pen is most effective at low armour values (waves 1–10) and nearly irrelevant against boss-tier armour. Prioritise pen items for wave clearing, not boss prep. Against bosses, invest in DoT items (bleed bypasses armour) or armour-bypassing Legendaries.