A Better ‘Damage Attenuation’ Formula

The typical damage formula for a video game is the following: $$\frac{Damage Starting}{Damage end}$$ This formula is usually derived by subtracting the value of one death in a game from the value of one death. One death is the point where the player loses all of the health he/she has, and from there, the player dies. This formula is often very confusing for the average gamer, because it seems to imply that the more damage you take, the less damage you take. It is true that the formula above is derived from the assumption that the player takes damage at the same rate regardless of whether he/she is taking damage from a single bullet or a single enemy attack. However, it is widely understood that

When creating a game, the first step is to provide a solid core experience that can be completed by anyone. This provides the foundation for a game that appeals to a broad audience, and is a critical step for any successful game. The second step is to provide a viable means for the player to advance their experience further. This means that the player should have a number of options for how they can progress through the game, and the means for progressing should be sufficient to keep the player interested in the game. The third step is to provide a means for the player to complete the game, with a finish line that allows the player to progress as far as they’d like.

In a previous post, I discussed a formula that is commonly used in game design, specifically in first-person shooters and role-playing video games, to calculate how much damage is inflicted on a players health bar when they take damage. This formula is sometimes called the “damage attenuation” formula, which is a rather misleading moniker. Some readers have expressed confusion, so I decided to describe  the formula in more detail.

TL;DR: Use a single formula to eliminate the diminishing returns of Damage Attenuation, ensuring that a higher DPS always, reliably, and consistently translates to a higher real DPS.

I understand that most people are uninterested in arithmetic, but the current debate about the DR of Liches is a great illustration of how math has a significant effect on gaming. Math is important; if you want excellent gaming, you must have strong math. I’m going to break this up into a few parts, each one delving further into the well.

What is Damage Attenuation, and how does it work?

When battling Liches/Sisters, damage attenuation is that irritating damage reduction (DR) feature that presently makes poor weapons better than excellent ones. This, according to what I’ve learned, is done using your weapon stats. The game calculates a DPS based on your weapon’s current attributes, including Warframe ability buffs, and then adds a DR based on that DPS. This seems to be an easy task, but my lord DE has done everything possible to make it difficult.

• For example, Eidolons’ fancy DR is calculated in four distinct ways. The calculation utilized depends on whether it’s a critical hit or not, as well as the DPS to firing rate ratio.
• Deimos Saxum is another example. These men have six distinct DR values, each with its own equation that scales with DPS, and the choice of which one to utilize is likewise determined by DPS. Crits aren’t taken into consideration in any of these calculations (thus crits essentially disregard the DR), and damage from status effects has its own DR.
• Demolishers (the Disruption opponents) utilize a method identical to Deimos Saxum, although they have the added advantage of six equations for status effect damage.
• There’s no word on what formulae Liches and Sisters employ, but I wouldn’t be shocked if it’s just as complicated as the above.

If the above sounds like cow poop to you, I am pleased to report that you are correct. Before we go on to remedies, I can hear some people shouting, “Kill it with fire, get rid of all the fancy DR.” Before we get there, I believe it’s essential to understand why DE has implemented these measures.

What is the purpose of Damage Attenuation?

Its most basic concept is that it is a damage reduction scaling mechanism that allows for a more constant time to kill an opponent at all levels of development. It’s intended to make it such that players with lower gear may complete the same content as those with higher gear, and neither side has a ‘poor’ experience.

With the absurd levels of power scaling, it’s nearly impossible for DE to choose a static EHP value for an enemy that won’t be trivialized by late-game players but decent for earlier players (star chart bosses), satisfying for late-game players but impossible for mid-game players (Eidolons), or both trivialized by late-game players and annoying for early to mid-game players (Eidolons), or both trivialized by late-game players and annoying for early to mid (Demolysts). The increase in quantity is just too great to make an adversary a danger without simultaneously making them ridiculous.

Damage attenuation isn’t always a negative thing. While it’s just a band-aid for a dysfunctional system, it’s better than nothing if you want bosses that feel like bosses. Personally, I wish it didn’t exist, but DE requires it, so we’re stuck with it.

The issue with DE’s present approach can be boiled down to a single major flaw: it is inconsistent. The damage reduction is based on calculations done at the arsenal level, which do not accurately reflect in-game performance. It doesn’t make opponents more resistant to our attacks; instead, it reduces the damage our weapons deliver depending on their attributes. The result of those events would be the same in an ideal world, but Warframe isn’t perfect. A DPS estimate does not provide a complete view of the situation.

My strategy was to start by shifting the frame of reference from theoretical to real harm. Your weapons continue to pump out the same quantity of raw numbers while opponents suffer damage and grow more resistant to it. Then, to make the DR outcomes seem predictable to the player, I utilized a smooth Damage Reduction algorithm. Finally, the DR decreases with time, so this isn’t simply a case of an opponent becoming endlessly tanky until they die; the DR number stops rising as you fire.

Damage Attenuation is Improved

Enemies get Damage Reduction in proportion to the amount of health they lose. This damage decrease is applied to an opponent as a buff, and it affects all players in the same way. Because the quantity of health lost only accounts for the last two seconds, the damage reduction decreases with time.

When you damage an opponent for the first two seconds, their DR increases in proportion to the amount of damage you deal. The DR would completely reset if you stopped shooting the opponent for two seconds. In reality, if you keep shooting an opponent, the DR will ultimately stabilize at a fixed amount (if the damage you are dealing is consistent). The greater your ‘homeostasis DR,’ the bigger your damage output.

This is “fair” to all incoming sources of harm and ways of us boosting our own damage since it is dependent on the amount of damage an opponent has received. When you strike a Sonar area, the opponent takes much more damage and therefore has a greater DR. Because the frame of reference is on the opponent, self boosts, enemy debuffs, and even abilities are all considered the same.

That’s a lot of words without me explaining what I’m talking about. I’ve been putting off this part since math scares me, but here it is… greater damage attenuation.

https://imagegod.b-cdn.net/wp-content/uploads/2021/08/A-Better-Damage-Attenuation-Formula.png

• is the final amount of damage you inflict on the opponent.
• is the amount of damage dealt to the opponent in the previous two seconds.
• is the amount of damage you would have done before damage attenuation, after all other modifiers (abilities, crit, resistances, armor, etc.).

That… seems a little frightening, I understand. I was just complaining about how complex damage attenuation is, and then I show you this. But bear with me, the mechanics are pretty basic and are based on an existing DR formula, armor.

What is the Benefit of Better Damage Attenuation?

If you’re just interested in the results, you may skip this section.

I’ll break down each component of the equation piece by piece to illustrate how it works and my thinking process. The fundamental concepts of how this DR works, though, come from armor, so it’s better to start there. I’ll explain things as though you’re completely unfamiliar with Warframe’s arithmetic. I want to persuade everyone that this formula works, therefore I’m going to pretend that no one knows anything. Please accept my apologies if you are well-versed in Warframe math; I will discuss a lot of stuff you already know.

Armor is often a source of controversy, however the issue should be focused on armor scaling rather than the formula that translates armor to damage reduction. This formula increases an enemy’s effective hit points in a linear manner. An enemy’s health is multiplied by 1 for every 300 armor points. An opponent with 100 health and no armor, for example, has an EHP of 100. (1x). Its EHP is 200 if it has 300 armor (2x). It has a 300 EHP if it has 600 armor (3x). If you needed a deeper grasp of how beneficial it is to construct armor, you could apply the same logic to a Warframe’s armor. The difference between 0 and 1 armor is the same as the difference between 1000 and 1001 armor, which is a significant factor.

https://preview.redd.it/evxkn40fkye71.png?width=480&format=png&auto=webp&s=7fcc5885e1d3e15e9cfc69736f501e1cd25d3163

This is DR’s tried-and-true armor. Hopefully, I described what armor does well enough for me to explain how it works. The armor value is determined by the green bricks, while the damage reduction strength is determined by the orange blocks. You’d need 500 armor to give a 1x multiplier to an enemy’s EHP if the orange value was 500. Increases in the value of the orange block reduce the armor’s potency, and vice versa.

To calculate the exact damage you inflict to an armored opponent, utilize the formula below.

https://preview.redd.it/z33zq6hgkye71.png?width=550&format=png&auto=webp&s=d3de79d88bd4612966e97fc256339dcdf85a4ea0

The armor formula’s damage reduction must first be translated into damage received. You don’t just multiply DR by damage since it would result in greater damage… that’s not correct. To convert damage reduction (the yellow block) to “damage taken,” just remove 1 from it (the purple block). Then increase it by the amount of damage (blue block).

I’m hoping that this strange color-coding thing I’m doing will assist. Now let’s look at how this applies to the damage attenuation formula.

https://preview.redd.it/p6ztmijhkye71.png?width=624&format=png&auto=webp&s=03e0e812aa8d513f5d11c69e4eb741d98b282500

Damage attenuation is a damage reduction, thus it must be translated into ‘damage received’ exactly like armor. I’m concentrating on the yellow area….

https://preview.redd.it/pw9m7omikye71.png?width=624&format=png&auto=webp&s=72088d3d2b5b49469738d920e5493fff2e30adbf

And see… it’s the armor DR formula all over again (but a little bit bigger). The reason I selected the armor formula as the core is because it raises an enemy’s EHP in a linear manner. The impact on fighting won’t get out of hand if you add large amounts of “armor.” Because this DR is supposed to reign in the very high end of our damage, I went with a much larger number in the orange block.

https://preview.redd.it/pdozvwzkkye71.png?width=624&format=png&auto=webp&s=70dab134e2cd35993d436b02c9af2411cbf1cbbb

The “armor” value is made up of two parts: the overall amount of damage received in the previous two seconds and… something else. The majority of this formula is based on. Over time, this number becomes larger, increasing the “armor” value and hence the DR. What about the pink portion…

https://preview.redd.it/1l28gtjmkye71.png?width=624&format=png&auto=webp&s=fcfd22a645c1625b43cbe814fb54729ca7f67a63

This section was absent from the original draft of this document. The issue this presented was that this DR would have no effect on the initial hit. If you utilized a sniper rifle, all of the fancy DR and one-shot opponents would be ignored as if they didn’t exist. And if you didn’t manage to one-shot the opponent, you’d have a significant advantage in terms of killing them.

I tried making this section simply be to fix it. As a consequence, guns that struck slowly and forcefully were significantly hampered in the long run when compared to weapons with a fast firing rate.

I wanted to find a medium ground; the value was a solid start, but it needed to be lowered to avoid becoming oppressive. So, like any sane person, I included an armor formula inside my armor formula. Another armor DR formula is yellow, which is converted to damage absorbed by the purple. This DR must strike a compromise between lowering the impact of heavy striking, slow rate of fire items while not jeopardizing their long-term effectiveness.

https://preview.redd.it/kiscsuinkye71.png?width=624&format=png&auto=webp&s=4ae4af47ee5b1bdaa2a9a7e894f897fab032d13e

The optimum balance in testing was achieved by combining both and. Everything from only in the armor DR to squaring the calculation to having static values was attempted. When evaluating hit rates, this provided the greatest balance of short term alpha damage and long term sustained damage. There isn’t much more to say about this.

The 100,000 and two-second timeframe I used to write this isn’t the be-all and end-all of this formula. Based on in-game testing, the two-second timeframe would most likely be changed. The 100,000 is intriguing since it has the ability to change depending on the opponent. The greater this value is, the less significant the damage attenuation is. On one hand, in end-game circumstances when it is assumed that you are highly equipped, this might be ramped up. On the other hand, it may be decreased if the material is set in an environment where the majority of players haven’t advanced very far. DE may use this extra knob to fine-tune battle encounters. For example, as a new player boss, Jackal could utilize a value of 10,000. On the other hand, an Eidolon may utilize a value of 500,000. Although the window duration should presumably be constant across all opponents, there is some room to adjust it to suit the battle scenario.

A far-fetched suggestion would be to make this value scale scale with opponent level, say *10,000. Higher level opponents will be less resistant to harm since you will be better equipped to deal with them (keep in mind they also have exponentially increasing EHP as they level up). Theoretically, it improves the consistency of the experience of battling an opponent with damage attenuation as you advance through the game. It’s much too complex to predict how this would function in reality using spreadsheets alone, but it’d be a fun experiment to attempt. With the present state of damage attenuation, anything like this is difficult to conceive; this is the advantage of creating a single equation that scales off just one variable.

I hope that makes sense… Now we’ll look at a bunch of numbers as examples.

Better Damage Attenuation Results

The “homeostasis DR” mentioned before is the first item to show off.

https://imagegod.b-cdn.net/wp-content/uploads/2021/08/1627936825 136 A-Better-Damage-Attenuation-Formula.png

This graph depicts how damage reduction improves and deteriorates with time. After the first two seconds, the drop off is that sharp tip. After that, you can see how it slightly overcorrects before rising and stabilizing at the conclusion of the time period. DPS is exactly proportional to the DR at which it becomes stable.

The figure above has a flaw in that it uses “perfect” DPS, in which every hit does the same amount of damage. This makes for beautiful charts, but it’s not practical; in reality, our damage is much more uneven due to crits, missed shots, melee attacks, and so on.

https://preview.redd.it/10iald3pkye71.png?width=480&format=png&auto=webp&s=9e7e695ae084447cc9177f8ea6db1af3074463c0

Each of the lines above have an average DPS of 200,000 over the course of eight seconds, and they both inflict damage 10 times per second. The blue line’s damage values vary from 17 to 52,728 at random. Even when the damage per hit fluctuates dramatically, the algorithm produces a constant homeostatic DR relative to the average DPS, which I think is a huge gain. For the following several instances, I’ll use “ideal” DPS to make the patterns as apparent as possible; this is to demonstrate that everything holds up even under imperfect circumstances.

https://preview.redd.it/yrza3eupkye71.png?width=480&format=png&auto=webp&s=fbde3875a608ed24115af0100a8654e01014e6e9

This chart shows a DPS range of 5,000 to 1,000,000. Personally, I’m not a huge supporter of using DR to demonstrate how this all works since DR is just a number; what counts is how much harm you do.

https://preview.redd.it/8xezujrqkye71.png?width=602&format=png&auto=webp&s=af6cf7376ccee17119b41a7af3819381249839bb

This, I believe, paints a much clearer picture of what’s going on. If the attenuated DR didn’t exist, the grey line represents how much damage you’d be doing. After the fancy DR, the orange line represents your DPS. Everything comes together here, and you can see how the effect of this DR is much smaller at lower DPS. It also demonstrates that a greater DPS will always be higher, no matter what. The solution will always be to do more damage if you want to cause more harm. This also shows that, despite the substantial decrease in damage, there is still a clear benefit to trying to do more harm. Although the gap between 600,000 and 1,000,000 DPS is no longer the 66 percent boost that it formerly was, it is still 36 percent higher.

The disparity between low and high hit rate items is a not-insignificant problem in this method. I’ll use two instances for this example: one that hits 40 times per second and the other that hits 2.5 times per second. The first hit DR is the DR given to the initial occurrence of damage, while the Homeostasis DR is the DR after 8 seconds. The number of hits required for the 40 case to inflict the same amount of damage as the 2.5 case’s initial hit is known as “Hits to Recover From.” The greater the number, the longer it will take for these two instances to equalize.

This obviously demonstrates that the formula isn’t flawless. Everything works pretty effectively at “normal” DPS levels. Things start to come apart after a million DPS. I don’t see much of an issue with this; it performs well where it counts, and even if it does break, it’s not the end of the world. After eight seconds, the overall damage done in all of the preceding instances is about equal, with a small advantage to the high hit rate. Yes, high alpha damage items will disregard a part of the DR, but the amount of min/maxing required to get to those points leads me to believe that those who go through it deserve it.

One additional graph to illustrate that I don’t believe this is a big deal.

https://preview.redd.it/z31vjv3jlye71.png?width=480&format=png&auto=webp&s=b6cc79ee695fc76561496954b66fdf1696189251

This graphic depicts the total damage done to an opponent over the course of four seconds. This is based on a 10 per second hit rate and a 1,000,000 DPS average. The range of randomized hit values is 0.1 to 416,000. This is the worst difference between perfect and imperfect I could achieve after a few hundred randomizations. The theoretical issue of significant single-instance harm isn’t borne out in practice.

That last graph was the one that persuaded me that this might work. Theoretical work is wonderful, but beautiful equations have a propensity to shatter when subjected to randomness. It’s not perfect; there’s a technical imbalance, but no damage mitigation formula can be. To be really fair, you’d have to be able to look into the future and alter the DR retrospectively. To my knowledge, DE does not have a time machine, thus “perfect” is not an option. In my view, the formula I’ve rambled on about is as near to being fair as we can go.

Should Attenuation be harmed?

I’m placing this at the end because I don’t want my opinions to take precedence.

Progress is essential. We invest hundreds of hours to thousands of hours in order to become more powerful. Adding a mechanism that essentially equalizes the playing field by discounting our time spent charging up is simply plain bad. While I like this formula more than any person should enjoy numbers, slapping it on everything is the last thing I believe is good for the game.

However, the more I consider the many settings in which it has been utilized, the more I agree with DE. It would make sense if the damage attenuation wasn’t caused by an uneven amount of horse feces.

• Demolishers: Disruption is a standard game style that should be accessible to all players. Unfortunately, the mechanics of Disruption may result in a vastly different experience for well-equipped players vs everyone else. Demolishers aren’t simply a threat; they’re also a mechanical goal. Applying attenuated damage to Demolishers ensures a more uniform experience; everyone must take a time to blast down these targets, and everyone is capable of doing so. This is precisely the purpose of damage attenuation. This is, however, just a basic mission kind. The ability for well-equipped players with wacky builds to one-shot Demolishers doesn’t harm much. Going from battling with these men to being able to wipe them out like a Lancer is an amazing feeling. I’m not sure whether these opponents should have damage attenuation in any case.
• Nox: Yes, Nox’s body has a light form of Damage Attenuation on it right now. Adding damage attenuation to common spawn units is, in my opinion, a huge error. Nox has made me rethink. Using Damage Attenuation to promote an enemy mechanic is an excellent approach to avoid making all other methods of destroying an opponent impossible. Damage attenuation offers a lot of potential to establish a more frequent niche for single target items, especially in the present situation where single target has no home in the face of AoE. More adversaries with damage attenuation while you aren’t using their technique, like Nox, may add a lot of variety to the game.
• Deimos Saxum (the two-legged hanging ball sack things): Deimos Saxum feature a mechanism where you can hard DPS their body by popping their two leg things. Damage attenuation now applies to Saxum even after their mechanism has been dealt with, which is… terrible. Saxum already has a flat 80 percent DR on their body before their mechanism is dealt with, on top of complete body damage attenuation. Replace the 80 percent DR with damage attenuation, then remove damage attenuation from Saxum’s whole body by breaking both legs.
• I may be alright with Deimos Jugulus (the towering phallic mortar things) if they weren’t continually produced and were handled as a minor boss on the Cambion Drift. As it stands, these soldiers are just a standard spawn unit with no way to get around damage attenuation. Jugulus should only be used as EHP tanks that can be easily one-shot by well-equipped gamers. Alternatively, their spawn rate may be significantly reduced while their danger level is significantly raised. Alternatively, they may utilize a simple mechanism similar to Nox’s, where just their body uses damage attenuation.
• Deimos Therid (Infested Ancient 2.0): Why… they are basically a standard Infested unit, mechanically speaking, and they don’t offer anything distinctive to the battlefield other than a change of scenery. Unless they’ve been substantially modified to do anything other than spew goo at you, they shouldn’t have any damage attenuation.
• Get rid of the eidolons. It doesn’t make sense for Eidolons to cripple your gear since they’re a gear-check boss. It’s particularly perplexing that this DR is applied to our amplifiers.
• Lephantis: As far as I’m aware, this is the first case for damage attenuation. It’s unclear how it works right now, but most signs point to it being a soft limit on per-hit damage. While one-shotting Lephanits is amusing, it detracts from one of the game’s most intimidating (and better-designed) enemies. At all player levels, I can see it taking a few bullets to take down Lephantis.
• Orphix: I believe this is one of the finest applications for damage attenuation. While it is essential to reward well-equipped players with a large number, this adversary symbolizes a whole phase of a combat engagement. The mission is set up to be a moment in time when everyone concentrates on the big baddie, rather than a fast one-shot that the bulk of the group overlooks.
• Liches and Sisters: The catalyst for all of this… Yes, I believe that damage attenuation should be retained for certain opponents. A battle with your nemesis shouldn’t be finished in a single shot since these adversaries represent a narrative thread as much as they do a combat engagement.

As previously stated, the strength of damage attenuation may be changed by simply changing the 100,000 to anything else. The Orphix, for example, should have a greater rating since you are supposed to go all out in DPSing them. An opponent like Nox, on the other hand, should have a lower rating to make it more difficult to overlook their unique mechanism.

DE has a better tool in their toolkit by making Damage Attenuation a consistent and easy-to-adjust mechanism that can be used to create more fascinating battle by making it a consistent and easy-to-adjust mechanic. And it provides for a fair squeeze for us gamers while yet rewarding our success. It’s a win-win situation.

Thank you for attending my TED presentation.

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