Mathematics of Facets and Scaling

[Konrad]

The Maths of Facets

Preface(t)

In this post, I will calculate optimal weapon facets for expected damage values (that is, average over many hits, not maximum possible).

TL;DR: Use something with a 20% overall damage boost, not a crit facet.

For damage, there is a clear best group of facets and it’s the same in all circumstances. My analysis ignores benefits not related to damage, though I comment on them in a dedicated section.

Further discussion in this topic explores why current ‘multiplicative’ stat scaling can lead to systemic balancing issues, especially in view of numbers-based and skill-based difficulty (broadly, ARPG vs Souls).

Before Enchantments

Weapons fall in these categories:

10% chance to deal 200% damage
20% chance to deal 150% damage
25% chance to deal 140% damage
40% chance to deal 125% damage

These are balanced and all have an expected damage increase of 10%, which pleases me greatly. Choose a reliable or a swingy weapon!

Facets can provide, amongst others, the following choices related to damage:

40% increased crit damage for 10% less overall damage
10% increased crit chance for ‘free’^[1]
20% increased overall damage for ‘free’^[2]

I will select optimal cases for the first two to show why they are subpar choices.

The 40% crit damage boost should be optimal for weapons with a 40% base crit rate. It is not. Expected damage is 116% (60% chance for 90% + 40% chance for 90% of 165%). The facet with 10% bonus crit chance is worse at 112.5%. The 20% increased overall damage is best (expected damage of 130%).

Boosting crit rate should be suited to weapons with 200% crit damage. It is not. With 10% bonus crit chance, expected damage is 120%. With 20% increased overall damage, expected damage is 130%. Bonus crit damage is last with 114%.

For all weapon types, the 20% increased overall damage is best. It will remain so as enchantments are added.

Enchantments: Crit Chance vs Crit Damage

Optimal combinations of crit chance and crit damage are achieved if crit damage is two times crit chance, until the cap of 100% crit chance. At this point crit damage takes over.

Enchantments provide five mutually exclusive bonuses of 10% crit chance or 20% crit damage, doubled with enchantment power (a pox on it). To keep maths manageable, I will not consider exalting for now.

(Armour penetration of 100% can be achieved without exalting, as can a bonus of 100% to any one element. I will not consider these for my calculations, but for anyone curious, it is marginally better to invest in crits over damage-type bonuses.)

The optimal number of crit chance enchantments for weapons with a facet granting a 20% overall damage boost is thus:

10% chance to deal 200% damage: 3 or 4, for 271.2% total
20% chance to deal 150% damage: 3, for 244.8% total
25% chance to deal 140% damage: 2, for 244.8% total
40% chance to deal 125% damage: 2, for 259.2% total

The optimal number of crit chance enchantments for weapons with a facet granting a 10% crit chance boost is thus:

10% chance to deal 200% damage: 3, for 244% total
20% chance to deal 150% damage: 2, for 219% total
25% chance to deal 140% damage: 2, for 220% total
40% chance to deal 125% damage: 2, for 230.5% total

The optimal number of crit chance enchantments for weapons with a facet granting a 40% crit damage boost at the cost of 10% overall damage is thus:

10% chance to deal 200% damage: 4, for 235.8% total
20% chance to deal 150% damage: 3, for 212.4% total
25% chance to deal 140% damage: 3, for 212.4% total
40% chance to deal 125% damage: 2, for 223.2% total

In this build completely focused on crits, the crit facets still do not catch up. The winner continues to be 20% increased overall damage.

The difference between facets’ expected damage is around 16%. I am unsure whether to consider this a sign of good design or of a lack of choice.

Facets Not Directly Related to Damage

I am not considering the relative merits of focus-generating facets over damage-boosting facets (Ritualistic vs Heavy). I believe this to be a moot point: The optimal strategy consists of a focus-generator (or life-as-focus set-up) paired with a damage-optimised focus-spender (like a one-handed hammer).

I cannot quantify the undoubtedly enormous non-numerical ‘benefits’ of Quick. Though the facet represents a meaningful choice compared to pure damage, I believe attack speed modifications should not exist in their current form, or at least be less accessible. There is no ‘commitment combat’ with the current 35% attack speed boost.

The elemental facets are flexible for build-crafting, but should never be considered for damage: A flaming sword with a mottled ruby is outdamaged by an unfaceted sword with a chipped ruby, even in its own element, but especially on all other elements if accessed by runes. Less damage at the cost of a facet is a poor deal. Use oils.

Poise facets are hard to quantify. I personally think they’re great and the damage trade-off is well worth it. Reliable is an early-game powerhouse.

On Armour Facets

Rapid should be removed.

Facets that trade focus for stamina regeneration (or vice-reversa) feel good, though due to additive stacking with certain Echos can lead to situations where characters cannot recover stamina over time.

The Dense facet is a rough deal, especially compared to its elemental brethren. 5 armour at level 30 is a rounding error, even if 5 weight can often fit into a loadout. The stacking of elemental resistance enchantments, elemental resistance facets, and rings is intransparent.

Durable makes me wonder what its design goal is. Weapon and armour durability, as a system, was intended to provide a delayed punishment and nudge players away from difficulty walls. Currently durability provides no real forced break if one dies repeatedly, a pathway to excessive power (deserved nerf incoming), and no trade-off to hardcore players. Bonus durability could be attractive on tools, but currently resources-per-harvest enchantments would be superior since repairs are a non-issue.

Afterword: Systems and Scaling

Wicked feels initially great to play. The early balance is excellent.

In the long term, however, some systems feel like they scale too strongly. This is not a matter of ‘optimising the fun out of the game’, but of simply playing the game beyond the ten-hour-mark without deliberately ignoring its systems. I do not need the above maths to be too strong.

A build with 100% damage to a beneficial element and 100% armour penetration will do, against an enemy with 50% damage resistance, 1084.8% of the damage of a plain weapon. Exalts can roughly double this (1948.8%).

I personally would prefer a game less focused on numerical power and more focused on player skill; a narrower band, ‘bounded accuracy’. This could begin with the removal of attack speed, armour pen, and percentile boosts to damage of one element.

In addition to the multiplicative damage scaling, some other systems that feel mathematically unsound are armour weight classes, damage reduction enchantments, stamina/focus management past the early game, and bonuses to overall speed.

---

1. Reduced durability is practically free. ↩︎

2. Stamina cost and focus gain can be massively overcompensated for by other systems; stamina consumption is irrelevant if using the weapon as a rune stick. ↩︎

[Hubristic]

this is an excellent summary of the current state of stats scaling, thank you

it’s useful to see everything in one place

i too think that scaling shouldn’t blow out of proportion

i think the “easiest” solution is to use linear scaling instead of multiplicative

i also think that stats shouldn’t determine everything and leave room for skill

so i would like stats to be like this:

+10 weapon damage after dodge for 5s
+5 physical damage

instead of

+15% damage after dodge
5% physical damage

and stats that change how you play instead of just “number go up”, for example:

• after parry, charged attack is 0.5s faster
• after taking more than 30% hp in one hit, add 15 poise damage for 5s
• your dash deals 20 damage to enemies it passes through
• stuff like that

P.S. just to contradict myself:

there are plenty of players who simply like the stats based power curve and it feels good to obliterate the game with a press of a button. they should be able to enjoy the game too. i think Wicked is more flexible in that way.

by flexible i mean, like we talked about SSF regarding self-imposed difficulty: you can always play the harder version if you want it harder.

for example: my latest run is using full cloth, no runes, one sword, no shield, and no durability based resists. the game is hard because everything one-shots me. but i enjoy that kinda thing. it’s even fun to farm pestilence when you can die from a sting of a bee. if i add the dura/resists, all of the sudden the game is irrelevant and i’m just mowing down pixels.

[Konrad]

Thank you! I feel that we have similar wishes for game feeling.

---

Re: Wicked Is Flexible

Power-fantasy players can have their cake without warping the game. What this means is that challenges should not require big numbers. Reducing skill-based difficulty with stats is fine. That’s when we can gimp ourselves and have fun.

I am warning against designs warped by big numbers at the expense of skill, or copy-and-paste content with bigger numbers. I’ve seen this happen before (Elden Ring, many ARPGs). Pestilence and Cyvion debatably already fall into this trap.

Thomas said the current power levels might feel excessive, but upcoming challenges will require them. At other times, he has emphasised that Wicked will absolutely fall on the side of commitment and deliberate combat. I suppose we can trust, wait, and see.

Tangent

On a personal note: I desire a fair, skill-based game that disallows cheating and cheesing. This may be a form of escapism into a just world. I do not think there is an inherent contradiction between minmaxing, stats giving strong benefits, and skill-based gameplay. And yet.

Wicked is between Souls and ARPG, where two cultures clash. It’s a discussion oft poisoned by mutual derision and name-calling: Challenge-seekers are derided as gate-keepers, elitists, or no-lifers. Power-fantasy players are accused of ruining games, seeking instant-gratification, called meta slaves, netdeckers …

Since I think the conflict is completely avoidable, I am always searching for designs that harmonise both sides’ wishes. Wicked has the chance to straddle the line, and so I have high hopes.

---

Re: Enchantment Design

I agree that good enchantments change playstyle, not just numbers.

Numbers bonuses should mostly be tied to risk or opportunity cost. If they have no trade-off or the trade-off is meaningless, you have uninteresting minmaxing (Indestructible and Unrepairable, though that at least has a minuscule opportunity cost).

Conditional bonuses should be skill-based, not just synergistic. Positive examples:

• Your parry-into-charged-attack suggestion is great because parrying is inherently risky and the bonus only allows a slightly safer retaliation.
• Attacks on heavy weapons with ‘guard points’ that permit aggression to act as a timing-based defence.
• Allowing a longer-distance dodge if you roll into a wall to push yourself off.
• The Dawn Never Comes is brilliant in my eyes.

Negative examples:

• MonHan has weapons that are stronger after gathering. If implemented poorly, weapons like this make tedious buff cycles optimal.
• Lifesteal vs. bleeding. Trivial to build around and set up, just free bonuses.

---

Stacking Models

Additive and multiplicative scaling have their place. For instance, you can’t stack elemental resistance (or dodge %) additively without a sensible cap or drop-off. It makes sense that every piece reduces the remaining elemental damage multiplicatively.

Nor am I sure that linear growth should be the goal. On the plus side, it is a very predictable option for players. For instance, if every point of armour represents a consistent increase in effective HP, armour never becomes worthless. The danger is that multiple linear boosts can become multiplicative of each other (like hp–dodge–armour, or base damage and crits).

Solutions in action and pen-and-paper RPGs tend towards diminishing returns or caps. Breath of the Wild simply has a maximum number of attack boosts. In early drafts of D&D 5E, you were relatively strictly limited to ability + proficiency.

These solutions can make buildcrafting too simplistic (or solved, often with hard or soft caps), make items that merely boost stats redundant, or make it too easy to reach a cap if player retention through gear grinding is the goal.

I wonder if the current loot system is compatible with non-incremental gear, and if there is enough design space for conditional boosts.

[Hubristic]

if i understand correctly:

you’re saying that optimally the design should avoid runaway numbers and allow the challenge to be “how well you read the enemy” vs “how high my weapon level is”

i think that is already the situation though. being able to clear some of the challenges (the savage pair, weeping sisters, even the mad fire witch prisoners quest) requires an upgraded weapon, otherwise you’re in for a battle that lasts 15 minutes and needs you to parry the same attack 50+ times. (fighting the Torn Colossus from the side quest with a level 1 weapon comes to mind)

the game ceases to be fun when you do 0.5% damage to an enemy per hit. you can still do it. but the message is very clear: get the blacksmith the things he needs to upgrade your weapon or farm a higher lvl one.

@Konrad

I am warning against designs warped by big numbers at the expense of skill, or copy-and-paste content with bigger numbers. I’ve seen this happen before (Elden Ring, many ARPGs). Pestilence and Cyvion debatably already fall into this trap.

in order to avoid that - and stick to lower numbers - much scaling would have to go extinct already.

@Konrad

Thomas said the current power levels might feel excessive, but upcoming challenges will require them.

this would mean that exponential scaling of stats is intended and future challenges will indeed have badass numbers. i don’t think there is a way of avoiding that !

the ARPG element of this game is the number scaling i guess. the power creep as you get/craft better equipment that allows you to go into harder zones and clear more easily, to get better equipment, to clear harder zones, etc, etc, etc.

and as a long-time ARPG player i gotta say i identify fully with “crafting the awesome perfect relic for my build” it’s a great moment.

can that exist without massive number ramp-up? i don’t know, struggling to see it.

look forward to hear your thoughts on this, particularly about how the game already expects you to have an adequate weapon to solve most of the serious challenges

[Konrad]

Wicked should require both skill and item progression. I do not want one to make the other superfluous. (Great question, thank you. My answer went through a few iterations.)

---

Game Too Hard? Use Numbers!

I’ll use a ‘bad’ challenge as an example. The current difficulty of Cyvion is so high (in numbers and skill: damage, stagger, combo speed, run-back time) that players are pushed into ARPG-style one-shot ‘builds’ or unintended enchantment combinations.

The margin for error is too low without them, especially on leather/cloth. It is natural to see the fight as an unfair challenge. This pushes players not towards upgrading and preparing, but towards minmaxing (immortality or oneshots).

While the high difficulty can be excused to a point (rogue-like environments can generate ridiculous perk combinations; it’s a placeholder wall in Early Access), there are two counters: Cyvion can be one-tapped without any echoes and the difficulty on the path to the boss is rather low. As a consequence, a Cyvion-killing build will trivialise the run to him.

Use Numbers? Game Too Easy!

Player power level is currently degenerately high. This is not, as Thomas claims, because we do not have the later challenges yet, but a systemic flaw. If I upgrade carelessly right now, I destroy my fun.

If I only use what the game lays at my feet – common, store-bought weapons, random blues – the game provides a pleasant challenge during progression. If I upgrade regularly, get mats, pick sensible enchantments, or use oils and vials, the game becomes faceroll-tier easy. That’s not hardcore grinding or minmaxing, that’s engaging with the basic systems.

Shields, groups, enemy mages and such should make me pause to think. Right now, depending on your weapon and armour weight choice, players can button-mash and heal-spam even through bounty targets and bosses.

I am not nudged to upgrade, I am forced to be careful not to do it too much.

Born to Skill, Forced to Number?

Upgrading should be mostly necessary. It funnels me into other systems that I also enjoy.

The 15-minute parrypaloozas are a nice example. I think everyone would agree they’re not fun, but it’s no harm at all if they’re possible, right? While I do some of the bounties you mentioned at level 3 on HC with starter gear, I know that’s not a healthy difficulty and even I wouldn’t want the entire game to be like this.

With infinite time and perfect play, the game can be beaten without any numbers progression. I contend that this should always be the case in this genre.^[1] At the same time, it is perfectly desirable that challenges eventually outscale me: otherwise, to exaggerate, the entire upgrade system is pointless.

But a good upgrade should play catch-up, not make the game easier. Otherwise, as I play more, my skill increases and my relative numerical power increases. The game gets doubly easier.

I don’t mind being nudged to upgrade, I just want challenges to remain relevant when I do.

Progression and Solutions

Current Wicked scaling design is that you become strong within the current difficulty, then it is replaced with a new one as you progress. This works well enough if you don’t make optimisation-like choices (or can’t, pre-Sacrament).

To preserve this good challenge past the five-hour mark there are several routes:

• increase enemy stats as you progress, make optimisation choices necessary
• keep enemy stats (or even reduce for Cyvion), make optimisation increase player power by a smaller amount

Is it contentious if I suggest that many ARPG moments of the ‘perfect build-defining relic’ are degenerate in terms of challenge? Is the joy of a really good item not diminished if it trivialises the game? I think it’s nice to have a short-term burst of power, but would items not need to lose effectiveness in the next zone?^[2]

Curious what you feel about these as an ARPG player. I feel that upgrading should remain relevant and items should never be perfect; or that progression should be horizontal instead (appearance, options over strength).

The class system might end up re-balancing this automatically. From some of the comments, it seems to tie incoming/outgoing damage to character level to an extent. I’m just unclear what the desired experience is. If I want to play Souls instead of Path of Screenwipe, I have to skip a part of the game I’d want to enjoy too.

---

1. But not every genre – numbers should be strictly required in, say, Fire Emblem. ↩︎

2. It has never sat right with me that the legendary Excalibur is vendor trash two zones over. ↩︎

[Hubristic]

getting/crafting a perfect relic in this game feels good because the item isn’t necessarily breaking the game.

it’s so hard to get the correct affixes on the right looking piece, with the right facet, that it’s a cause of celebration “YES THIS IS PERFECT FOR MY BUILD” - does not have to mean “YES I NOW DELETE THE BOSS IN 0.2s”.

i feel like the game does that well, up to a point. it also doesn’t end the journey. the journey is “what’s gonna be the next cool encounter?” not “when do i get the super-mega-legendary win-sword?”. that’s why i sing praise for the current deterministic crafting system - it’s “play and the RNG will eventually give you what your build needs”. amazing. we are pretty much completely off the endless treadmill of chasing stats. getting embers doesn’t seem like such a big deal. fine

that’s why i say that the problem comes in the shape of multiplicative scaling.

if i have 4 pieces of armor with +10% physical damage, those do not multiply with each other — they sit in an additive bucket. so that becomes:

40% physical damage → 1.40x multiplier

NOTE: phys% from gear stacks additively, not multiplicatively.

if i have 50% CSC and 200% CSD that’s another (averaged) ~1.5x multiplier.

that in total is a ~2.1x multiplier for my base weapon damage.

now add that to an expensive rune attack, which has a 10x multiplier, we’re at ~21x.

if my base damage was 150 with no facet this rune now hits for ~3.1k

let’s say i’ve just got a plain 10% damage affix on a weapon, that ~21x becomes ~23.1x.

new rune hit: ~3.4k

then i grab the rune ring with a low 20% roll (1.2x) — my total multiplier for that rune attack is now ~27.7x

new rune hit: ~4.1k

… and so on. we are now miles away from the 150 base damage. your naked starter warrior has no business in this game anymore if killing the boss requires 10 of those attacks.

(if i had one of the aforementioned superior facets that avg hit would now be ~5k+ — massive difference)

NOTE: facets increase base weapon damage, which is particularly powerful because base damage is multiplied by all later systems (phys bucket, crit, rune multipliers, etc.)

now every tiny increment will add a huge chunk to my damage. at the game’s current state, that invalidates all the content.

if future content is expected to match that level of power, it completely leaves in the dust the player who didn’t optimise.

however, optimising requires:

• player knowing that there are 3-4 “king stats” which are the ones that get you to the max damage, everything else is… flavour

so, optimising is inherently a boring thing to do when the game is so full of awesome animations, rich environments, interesting enemies. but it’s a part that’s fun for a lot of people, and it’s nice when you can feel like you have ownership of your build “clicking” into place. it should IMO enable you to clear content faster. i think we are fully in agreement here, the system is fun and we want to take part in it.

my point: if the scaling wasn’t multiplicative, but instead linear, we wouldn’t be so far away from the base 150. optimisers could still optimise without orbit-nuking the content and hardcore naked warriors could still go into each fight with a cathedral-sized eldritch bear holding a butter knife for a weapon and a dinner plate for a shield, wet napkin for armor, and do their thing.

linear scaling will still multiply by parallel stats, but it be subject to runaway damage much less than it is now. linear can be end-game too: 5,10,15,30,60. much easier to control than increasingly big multipliers

at the moment this game multiplies most stats which is guaranteed to blow the system with massive numbers under current construct. a few more increments, another multiplier, and we are in millions.

p.s. i realise this goes back to your initial post about facets - that extra base damage due to a facet is HUUGE in a system that multiplies most stats

but is this system guaranteed to blow out of proportion ? what if they never scale base damage numbers up any more ? they said that currently we are playing ~1/3 of the content. if i’m doing 10 damage on the starting beach and 5k by the time i’m schooling 10x curses pestilence cyvion, that suggests an exponential rate of exactly… too much.

so the concern here is real and in my eyes the design choices for future content become:

• add more power
• or make future loot/progression feel flat
• or hard-reset/rebalance systems

those are real non trivial forks

so i guess my next question is: what design tools could possibly control the unbounded power curve ?

completely off rails & in parallel universe: it would be refreshing to have 6 hearts like Link in a game like Wicked

p.p.s. forgot to answer:

having a perfectly crafted piece of gear doesn’t mean it has to be obsolete in the next level because we have incremental upgrades. i think those work well. they increase base value by a small amount.

but yes the good item only works if it doesn’t delete the game you looked forward to playing so much. another good example next to Dawn Never comes: Wind of Death. tradeoffs are great

edited to reflect the additive bucket reality of elemental damage from multiple sources - thanks Konrad

[Konrad]

Assuming your preferences can be generalised, I am now convinced that a stat scaling model exists which can harmonise ARPG and Souls preferences. I also agree that it’s great to be off the treadmill.

I’m not sure we are completely in agreement as to the meaning of linear and multiplicative scaling, so for the sake of argument I will present several mathematical models. This might become a two-parter.

Tangents, Caveats, Miscellanea

Apologies in advance if I’m preaching to the choir, or if it seems like I’m splitting hairs on what isn’t linear. I just want to define terms to avoid misunderstandings.

I’m really unsure what maths level I should pick for this discussion.

Hearts in Zorldo are a great system! It’s fractional as all damage is by nature (all the floor and ceiling functions below), so despite the abstraction from the numbers the math is much more readable.

I also think minmaxing can be fun in itself, completely separately from gameplay, but that’s a separate discussion.

---

The Maths of Scaling

In the current state of the game, not everything is multiplicative. Four pieces of armour with +10% physical damage currently stack additively as far as I’ve tested, so not 1.4641 but 1.4. Elemental resistance from rings is additive. I will show that additive isn’t linear.

What stacks multiplicatively is % dmg from facets, % dmg per element from enchantments, rune dmg modifiers, and crit calculation. We are in complete agreement that this will escalate.

Before I even get into multiplicative stat interactions, I want to clearly define terms for the various scaling effects of increases to single stats. I’ll also show good and bad scaling models.

Part 1: Linear Stat Increase, Non-Linear Scaling

Example 1.

For this example, I will assume a (mathematically pleasant) baseline enemy with 100 effective hp, against whom a player deals 20 damage per strike, so five hits to kill. As damage increases linearly, the number of hits to kill drops in steps.

The following diagram shows the number ‘f’ of hits to kill as a function of the damage ‘x’ done per swing. Green (or f) is the number of hits to kill, while pale yellow (or g) is a continuous function so the qualitative scaling becomes more apparent.

As damage increases linearly (on the x-axis), the benefits per point diminish hyperbolically. In other words, more points are needed to achieve the next reduction in number of hits-to-kill (the green lines become wider; after the first 5 points you need 8 to make a difference, then 17).

I’d argue that this is good behaviour in the sense that the effect of linear stat-increases actually diminishes with increases stats.

I’ll call this (hyperbolically) diminishing returns.

Does this lead to healthy scaling? – At the extreme end of this, oneshots become possible, which I consider unhealthy (or degenerate). The correct number of hits if you overpower a region is up to personal taste, but could be two to four attacks, depending on the speed of your weapon.

Note also how small increases are much less worthwhile for weapons with large base damage, so this may be unfair towards two-handed weapons, which already receive fewer stats in total.

Example 2.

Now I will assume the enemy hits back. My sample player has 50% damage resistance initially, which I will increase linearly again. Let’s say an incoming hit has 70 raw damage. I’ll now show the reduced damage ‘d’ for this hit as a function of resistance ‘x’.

This is a linear decrease in damage, but I’d argue it doesn’t lead to linear scaling in actual survivability – instead, a better question would be how many hits of this strength the player can survive.

I’ll assume the player also has 100 hp. Initially, the player dies on the third hit of 35 damage. In now familiar fashion, I’ll show the number ‘n’ of survivable hits as a function of resistance ‘x’. In pale yellow, I’ll show what this looks like continuously.

A small initial boost allows surviving the third hit, then the stat has a long interval of limited benefit, and as we approach 100% the number of survivable hits grows (to infinity at full damage immunity).

Unfortunately, this is also hyperbolic in mathematical terms; but effectively the function is defined by the asymptote at the 100% mark. If I had to name this form of scaling, I’d term it asymptotic for that reason.

Obviously this is ridiculous and no defensive stat should scale like this. Small numbers are irrelevant and stacking is extremely worth it. Multiplicative effects like enchantment-power gems are especially horrifying.

Example 3.

Let’s return to the enemy of the first example. His 100 effective hp were secretly 50 hp with 50% damage reduction. Let’s hit him again with 40 raw damage.

This time, we’ll add linearly increasing armour penetration. Armour penetration is a percentile reduction to his percentile damage reduction. This is a linear decrease, and resistances can’t be reduced below zero.

In the following diagram, as a function of our armour penetration ‘x’, we have the real damage (‘dmg’) and the remaining damage reduction (‘dr’).

Not very enlightening – I’m mostly including it so we’re on the same page for armour penetration maths.

The next diagram will display the number ‘n’ of hits to kill the enemy as a function of armour penetration ‘x’, again with a pale-yellow continuous version ‘f’ to compare:

What does this mean? – Armour penetration scales hyperbolically, even if rather gently here. What’s new is the hard cap: No armour penetration lets me kill in fewer than three hits.

This is an extremely useful property if you are with me in the idea that enemies should have a ‘minimum’ number of hits required to defeat them. In this case, this number is the enemy’s raw health divided by our raw damage, rounded up. No amount of overgearing gets us under that.

Maybe this type of scaling could be called hyperbolically diminishing returns with a hard scaling cap. Hardly rolls off the tongue, so I’m open to suggestions. This is a good scaling model, but I’d like to explore its properties more in …

Example 3a.

Same as ex. 3 with one small change: We’ll increase the enemy’s damage reduction to 80%. This has drastic consequences for the number of hits to kill:

Now if I am still wielding my trusty 40-raw-attack weapon and have no armour penetration, I need 13 hits. Yet no matter how much I stack armour pen, I still never need fewer than three.

Benefits: The existence of this minimum achieves our desired goals of strongly rewarding gear, yet never trivialising fights. Low values have sharp benefits, so there’s a built-in catch-up mechanic. There is a sizeable interval near the top in which ‘complete’ builds can fall without requiring absolutely perfect rolls.

Flaws: This scaling still has an (exclusively human) issue in that enemy DR and player armour pen could conceivably approach 100%. Once numbers get high, increasing armour pen by 0.5% will seem to players like a small increase, but actually have huge effects. It just won’t look that way when comparing two items, which is why fake ‘linear’ stats like the current armour are used instead.

As an afterthought, I’d consider an item tooltip complete if it provided the following information (ideally more succinctly; while the ‘linear’ numbers below are asspulled, the percentages make sense):

Currently equipped item: 400 linear AP, equivalent to 90% reduction in enemy defences.

Item being previewed: 450 linear AP, equivalent to 92% reduction in enemy defences; defences reduced by a further 20% compared to the equipped item; equivalent to a damage increase of 1.74% of against foes with 80% damage reduction.

Systemic concerns: If there are many stats where small increases have strong benefits, the correct buildcrafting solution is to spread stats widely rather than specialise. I believe specialising should be encouraged; or in other words, the 50% model is great for early game and the 80% for mid-late game.

I’ll continue in a second post once I find time.