Item Stacking

Tougher Times, a frequently misunderstood and miscalculated item

As players run through stages, they will collect Items from Chests and other sources. As their run continues deeper and deeper, they will eventually find copies of items they've previously encountered. Picking up additional copies of an item is called "stacking" the item, and the total number of items of the same item is often called "stacks" of that item.

All items in the game (with the exception of equipment and one special itemBeads of Fealty
Seems to do nothing... but...) have amplified effects when stacking. The more stacks an item has, the more that effect is noticeable. For example, a single Paul's Goat HoofPaul's Goat Hoof
Increase movement speed.
Increases movement speed by 14% (+14% per stack). raises movement speed by 14%, which is good, but stacking 30 of them would result in a 420% increase, which is extremely good for both mobility and evasion.

Most items have one or two numerical variables that increase with stacking, but that variable is not always the same between items. For example, AtG Missile Mk. 1AtG Missile Mk. 1
Chance to fire a missile.
10% chance to fire a missile that deals 300% (+300% per stack) TOTAL damage. and Tri-Tip DaggerTri-Tip Dagger
Gain +10% chance to bleed enemies on hit.
10% (+10% per stack) chance to bleed an enemy for 240% base damage. both have a chance to apply an effect that deal damage on a hit, but while stacking the former increases the damage of its effect, stacking the latter increases the chance of its effect occurring. The effect of stacking on each item is described in the item's page.

Note that for some items, the listed value in-game does not reflect the actual effect of the item. This is especially true for items that stack non-linearly, as explained below.

Stacking types

While many items stack in the way you would expect (having 3 items triples the effect, for example), many others do not. However, most items stack consistently in one of 3 ways, and scale in a similar manner to items of the same category. These categories are outlined below.

Note that a few items have several scaling statistics that use different scaling methods. The Fuel CellFuel Cell
Hold an additional equipment charge. Reduce equipment cooldown.
Hold an additional equipment charge (+1 per stack). Reduce equipment cooldown by 15% (+15% per stack)., for example, has both a linearly stacking (+1 charge per stack) and an exponentially stacking one (x0.85 equipment cooldown per stack).

Linear Stacking

Most items in the game stack linearly - meaning that for every added item, the bonus keeps increasing by the same base amount. For example, the Soldier's SyringeSoldier's Syringe
Increase attack speed.
Increases attack speed by 15% (+15% per stack). increases attack speed by +15%. Having 2 of this item raises the bonus to +30%, and 10 of them to +150%.

The formula for an affected statistic is the linear function : $f(x)=1+a\bullet x$, where a is the effect of one item and x is the number of items.

Hyperbolic Stacking

Tougher times block rate per item

For many items using percentages (mostly for control items or items that reduce a statistic), using linear stacking would eventually lead to a 100% chance of triggering or 100% reduction. This would lead to wild imbalance from a design standpoint, such as becoming totally invulnerable or always stunning hit enemies. For this reason, these items use a different type of scaling called hyperbolic stacking.

The formula for an affected statistic is the reciprocal function : $f(x)=1-1/(1+a\bullet x)$, where a is the effect of one item and x is the number of items.

Tougher TimesTougher Times
Chance to block incoming damage.
15% (+15% per stack) chance to block incoming damage. Unaffected by luck. for example, uses a = 0.15 -- so with 10 Tougher Times, the block chance would be equal to $1-1/(1+0.15\bullet 10)=1.5/2.5$, or a 60% chance. Once 223,696,201 Tougher Times are stacked, the block chance becomes 100% due to the limitations of floating point numbers. Interestingly, using this equation when x is 1, you end up with a 13.04% block chance as opposed to a 15% block chance.

An intuitive way of visualizing hyperbolic stacking is through a lottery example. Imagine that there are green and red tickets in a box. Every time the chance is tested, one ticket is drawn at random. If it is green, the effect triggers, otherwise it does not. By default, the box has 100 red tickets and no green tickets. Every time an item is acquired, some green tickets are added to the box. One Tougher Times would add 15 in this case. The more green tickets are added, the more they are likely to be picked over red tickets. However, there is always a chance that a red ticket is picked instead, no matter how small.

Reciprocal Stacking

This is similar in nature to the hyperbolic stacking and in fact mathematically they belong in the same category. The conceptual difference is that the formula in this case is $f(x)=a/x$. Because it does not calculate the complement of the reciprocal, the graph looks mirrored along the x-axis compared to the hyperbolic stacking, i.e., the effect is at its strongest with 1 stack and it decreases as more stacks are accumulated but it never reaches 0%.

Due to the fact that with two stacks the strength of the effect is halved, a lot of items' descriptions that use this stacking behavior incorrectly state that the effect is reduced by 50%. However, this is only true for going from 1 to 2 stacks, as at 3 stacks the effect is at 33.33% strength instead of the expected 25%.

If an item with this stacking behavior affects something unrelated to the item's effect, e.g., not its own cooldown duration, the formula may instead be $f(x)=a/(x+1)$ just so even the first stack can have an effect.

Exponential Stacking

Some items in the game are very powerful because of their exponential stacking, meaning that their stacking effects compound each other. For instance, one stack of Shaped GlassShaped Glass
Double your damage... BUT halve your health.
Increase base damage by 100% (+100% per stack). Reduce maximum health by 50% (+50% per stack). doubles damage and halves health. Another stack will double the new damage and halve the new health, meaning that final damage would be quadrupled instead of tripled, and final health would be divided by 4 instead of 2.

The formula for an affected statistic is the exponential function : $f(x)=a^x$, where a is the effect of one item and x is the number of items.

The power of exponential stacking is that unlike both other types of stacking, they do not suffer from diminishing returns since their effects are compounding. For example, going from 50 to 51 stacks of Soldier's SyringeSoldier's Syringe
Increase attack speed.
Increases attack speed by 15% (+15% per stack). would raise the attack speed bonus from +750% to +765%, which is a negligible increase. Similarly, going from 50 to 51 Tougher TimesTougher Times
Chance to block incoming damage.
15% (+15% per stack) chance to block incoming damage. Unaffected by luck. would increase block chance from 88.2% to 88.4%, which leads to an increase in expected survivability of about 1.7%. By contrast, going from 50 to 51 Shaped GlassShaped Glass
Double your damage... BUT halve your health.
Increase base damage by 100% (+100% per stack). Reduce maximum health by 50% (+50% per stack). would still double an already monstrously high damage and halve a ridiculously low health.

Special Stacking

The BandolierBandolier
Chance on kill to drop an ammo pack that resets all skill cooldowns.
18% (+10% per stack) chance on kill to drop an ammo pack that resets all skill cooldowns. uses its own special stacking formula close to hyperbolic stacking - the drop rate of an ammo box is equal to $1-1/(1+x{)}^{0.33}$, where x is equal to the number of Bandoliers. While this looks like other hyperbolic stacking items, the nature of the 0.33 exponent means that the first couple Bandoliers will have a huge impact on drop chance, while the following ones will rise very slowly. More detailed information on the Bandolier's stacking can be found on its dedicated page.

Stacking limits

All items can be stacked infinitely (at least up to the signed 32-bit integer cap of 2,147,483,647), and their effects will also stack accordingly. However, some items have an effective limit after which further stacks have no effect.

Items with a chance to trigger that increases linearly will not gain any additional effects once the chance reaches 100%. For example, once 100% critical chance is reached through 10 Lens-Maker's GlassesLens-Maker's Glasses
Gain 10% chance for hits to 'Critically Strike', dealing double damage.
Your attacks have a 10% (+10% per stack) chance to 'Critically Strike', dealing double damage., picking up additional stacks of this item will not grant stronger critical hits (unless you're playing as Railgunner, whose passive will convert it to crit damage), or have any other effect. Note that for some items linked to the proc coefficient (on-hit items), the effective limit differs depending on the character.

Focused ConvergenceFocused Convergence
Increase the speed of Teleporter charging... BUT reduce the size of the zone.
Teleporters charge 30% (+30% per stack) faster, but the size of the Teleporter zone is 50% (-50% per stack) smaller. has a hardcoded cap of 3, after which neither of its effects increases.

Fuel CellFuel Cell
Hold an additional equipment charge. Reduce equipment cooldown.
Hold an additional equipment charge (+1 per stack). Reduce equipment cooldown by 15% (+15% per stack). is limited to 255 equipment charges.

Shaped GlassShaped Glass
Double your damage... BUT halve your health.
Increase base damage by 100% (+100% per stack). Reduce maximum health by 50% (+50% per stack). can only be stacked up to 127 copies, as the player will die upon collecting a 128th due to the limits of floating point numbers.

Trivia

• All items can be stacked to a maximum value of 2,147,483,647 - the 32 bit integer cap - before overflowing. Upon overflow, the item will reset back to 1, and act as if there is only 1 item in the player's inventory (a notable exception is the Personal Shield GeneratorPersonal Shield Generator
  Gain a recharging shield.
  Gain a shield equal to 8% (+8% per stack) of your maximum health. Recharges outside of danger., which has a visual bug upon overflow). Due to this, the hard cap for all items is 2,147,483,647, with no items having a hard coded cap before this value, other than the three items listed in the previous category above.
• Prior to the rework it received in the Anniversary update, the Rusted KeyRusted Key
  Gain access to a Rusty Lockbox that contains a powerful item.
  A hidden cache containing an item (80%/20%) will appear in a random location on each stage. Opening the cache consumes this item. used a formula similar to the lottery tickets analogy for its item rarity chances by using (metaphorical) White, Green and Red tickets - which grow at different rates - to determine the rarity of the dropped item. The new version always has the same yields as a large chest and does not stack, but consumes 1 key when opened.