Item Stacking

''This page is incomplete - frequent updates occur. Many items have not been yet fully documented or researched. This page will be updated with the most accurate information currently known as frequent as available.''

Item Stacking is a mechanic that allows for items to have increased effects based on how many of a specific item a player is holding. Most items stack, however not all stacking is equal, and certain items may stack additive or multiplicative, based on the effect. In most cases, the stack value can be found in the log book entry for a particular item, though whether it is additive or multiplicative is not stated.

This page discusses how item stacking functions when one item is stacked, and does not discuss how it interacts with other items.Note that for some items, the listed value in-game is not reflected in the item code. This page strives to give the true values, rather than stated ones. This is why there may be discrepancies with certain items, because the code says different than the item description.

In addition, the only numbers listed here are ones that stack, as full item descriptions are not listed to prevent confusion.

Additive Stacking
Additive Stacking is when an item linearly increases based on number of items. An example would be if the player is holding a Lens-Maker's glasses, they would have 10% critical strike chance, 2 pairs of Lens-Maker's glasses 20% critical strike chance, up until 10 Lens-Maker's glasses with 100% critical strike chance.

Effective Maximum Count is not a hard limit, but any more than the listed value would have no effect on gameplay whatsoever (typically due to surpassing a maximum value in-game).

If a numerical value present in the item does not show up in the table, it does not stack with number of items.

Multiplicative Stacking
Multiplicative stacking is when an item does not increase linearly based on the stack value, it scales based on the current percentage instead of the base one. Another word for multiplicative stacking is "Diminishing Returns". An example would be that if an item grants 10% damage resistance multiplicative, you won't get 100% damage resistance after 10 items, you would get 0.90^10 damage taken, or ~65% damage resistance, where 0.90 is 1 minus the reduction value, and 10 is the exponent representing the number of said item the player has.

Multiplicative stacking is limited to items that reduce values (lower cooldown, health), because if the value was additive, it would allow players to go to zero with these items.

Effective Maximum Count is not a hard limit, but any more than the listed value would have no effect on gameplay whatsoever (typically due to surpassing a minimum value in-game). In positive cases (cooldown reduction), the value is as close to 0.12 as possible, as 0.12 is the fastest ever recorded human reaction speed. In negative effect cases, as with lunar items, the maximum effective count shows simply at what point it becomes obsolete add new ones - these values should never be sought after in-game due to potentially game ruining effects through stacking. Thus, it is recommended not to stack many lunar items.

If a numerical value present in the item does not show up in the table, it does not stack with number of items.

Logarithmic Stacking
At first glance, a player may see an item and believe that obtaining a few of them would grant 100% to the stat of the item, or that they may stack multiplicative and that around 30 would give a near 100% chance.

For some items however, the scaling is different and is dubbed Logarithmic Stacking based on the signature diminishing returns and resultant shape of the graph, which resembles that of a logarithmic one. Although these formulas do not include logarithms, this method of stacking will be referred to as "Logarithmic" for convenience sake.

An item which has logarithmic stacking is typically modeled by a unique equation involving 1/x at some point within it. The intention behind logarithmic stacking is to provide balanced gameplay for specific items which have very powerful effects if stacked high enough, such as block chance or cooldown reset.

''TO DO BANDOLIER, REMOVE TEXT ON TOUGHER TIMES. BL4NK WILL DO IT, PLEASE DON'T MODIFY THIS SECTION UNTIL I'M DONE WITH IT''

Tougher Times
Tl;dr at the bottom of Survivability vs Block chance, but a strong recommendation to read the entire discussion in order to gain a better understanding of the item.

Tougher Times does not stack like any other item, likely due to the extremely powerful potential of stacking the effect. At first glance, a player may see the item and believe that obtaining 7 of them would grant 100% block chance, but this is not true. Even then, if a player believes that it stacks multiplicative, and that obtaining 30 would grant 99% block chance, this too is not true.

Block Chance
The actual formula for the tough times stacking calculation goes as follows

100% - 100% / (15% * item count + 100%), where the final value received is your chance to block.

For example, with one Tougher Times, the player would have a 13.04% chance to block, and with two Tougher Times, the player would have a 23% chance to block. Knowing this, in order to obtain 99% block chance with the Tougher Times, the player would need to acquire about 700 of the item to get a value this high.

Tougher Times Visualized
Here is a graph which demonstrates how the chance to block increases as number of tough times increases (and never actually hits 100):

http://m.wolframalpha.com/input/?i=100*%281-%281%2F%281%2B%280.15*x%29%29%29%29+from+0+to+1000

Tougher Times suffers from extreme diminishing returns due to the code that actually calculates block chance scaling logarithmically. While the block chance initially increases fairly rapidly, it falls off quickly, resulting in requiring a substantially larger amount of tougher times in order to have a noticeable effect on block chance. As can be seen in the table, increasing the player’s tough times from 500 to 1000, an astonishing increases of 500 items, only increases the block chance by 0.56%.

Survivability
Another statistic that should be taken into account is survivability. Survivability models how many hits on average it would take to land a successful hit from an enemy.

The equation is as follows:

1/(1-B) = S

Where B is the block chance (from 0-1), and S is the number of hits it would take, on average, to land a successful hit.

if this equation is graphed, substituting the B value with the tougher times equation, the graph is actually linear. This means that, in terms of a scale of survivability, each consecutive tougher times gives the same amount as the last, with 1 giving an S value of 0.15, and 1000 giving an S value of 150, meaning it would take on average 150 hits for a successul hit to be landed.

A simplification of the survivability equation, is thus:

0.15*T = S

Where T is the number of tougher times the player has, and S is the survivability.

Because of this, this may lead a player to believe that tougher times will thus scale linearly, blocking a linear amount of hits per Tougher Times. However, this is a hasty conclusion that does not take into account game stats.

Tougher Times Conclusion - Survivability vs. Block chance
Because of the survivability, a player may believe that they can block more hits. But, the important thing to note about Tougher Times, is what damage it is blocking. Later in the game, when such a number of tougher times are available, the player typically does not have enough health to survive a single hit alone, and the major factor keeping players alive is block chance.

When looking at a single hit, the survivability does not matter because a range of attacks is not considered. A single hit, therefore, is only affected by block chance, not survivability.

If a player wishes to increase their ability to survive (differentiated from survivability for convenience), and has gotten to the point in a game in which it takes only one or two hits to kill the player, the most important factor is thus block chance and not survivability.

Because of this, the player should take into account how often they are hit by attacks, and scale their tougher times accordingly, not increasing the number of tougher times too high, as they would be missing out on a large amount of damage from other items they could have taken instead that scale linearly.

The highest values of tougher times, generally anything above 50, is not realistic in a typical game. Players do not have item choices, and must constantly stack damage to keep up with their enemies ever increasing health. Because of this, survivability vs. block chance is not a meaningful discussion unless the game is truly long (>6 hours). However, the discussion is not entirely useless - the logarithmic scaling of Tougher Times is still important to players not in the higher numbers of tougher times.

To conclude (tl;dr), the Tougher Times is an Item that grants a logarithmic-scale block chance, diminishing returns, and a linear survivability stat, number of misses per hit on average. Block Chance is the more important than survivability when the enemies deal a large percentage of the players health per hit with a lower fire rate, while survivability is more important when the enemy deals a small percentage of the players health rapidly, with a high attack speed or effect speed. Prioritizing Block chance and Survivability is not wise before starting a game, as the scenarios, enemy scaling, types of enemies, etc. all contribute to factors which determine whether block chance or survivability would give the player a higher chance to stay alive, and thus priority should be on a case-by-case basis, rather than generally.

Sources:
 * 1) Game Code