How to develop RPG Damage Formulas?

Question

I'm developing a classical 2d RPG (in a similar vein to final fantasy) and I was wondering if anyone had some advice on how to do damage formulas/links to resources/examples? I'll explain my current setup. Hopefully I'm not overdoing it with this question, and I apologize if my questions is too large/broad

My Characters stats are composed of the following:

enum Stat
{
    HP = 0,
    MP = 1,
    SP = 2,
    Strength = 3,
    Vitality = 4,
    Magic = 5,
    Spirit = 6,
    Skill = 7,
    Speed = 8, //Speed/Agility are the same thing
    Agility = 8,
    Evasion = 9,
    MgEvasion = 10,
    Accuracy = 11,
    Luck = 12,
};

Vitality is basically defense to physical attacks and spirit is defense to magic attacks.

All stats have fixed maximums (9999 for HP, 999 for MP/SP and 255 for the rest). With abilities, the maximums can be increased (99999 for HP, 9999 for HP/SP, 999 for the rest) with typical values (at level 100) before/after abilities+equipment+etc will be 8000/20,000 for HP, 800/2000 for SP/MP, 180/350 for other stats

Late game Enemy HP will typically be in the lower millions (with a super boss having the maximum of ~12 million).

I was wondering how do people actually develop proper damage formulas that scale correctly? For instance, based on this data, using the damage formulas for Final Fantasy X as a base looked very promising. A full reference here http://www.gamefaqs.com/ps2/197344-final-fantasy-x/faqs/31381 but as a quick example: Str = 127, 'Attack' command used, enemy Def = 34.

1. Physical Damage Calculation:
Step 1 ------------------------------------- [{(Stat^3 ÷ 32) + 32} x DmCon ÷16]
Step 2 ---------------------------------------- [{(127^3 ÷ 32) + 32} x 16 ÷ 16]
Step 3 -------------------------------------- [{(2048383 ÷ 32) + 32} x 16 ÷ 16]
Step 4 --------------------------------------------------- [{(64011) + 32} x 1]
Step 5 -------------------------------------------------------- [{(64043 x 1)}]
Step 6 ---------------------------------------------------- Base Damage = 64043
Step 7 ----------------------------------------- [{(Def - 280.4)^2} ÷ 110] + 16
Step 8 ------------------------------------------ [{(34 - 280.4)^2} ÷ 110] + 16
Step 9 ------------------------------------------------- [(-246)^2) ÷ 110] + 16
Step 10 ---------------------------------------------------- [60516 ÷ 110] + 16
Step 11 ------------------------------------------------------------ [550] + 16
Step 12 ---------------------------------------------------------- DefNum = 566
Step 13 ---------------------------------------------- [BaseDmg * DefNum ÷ 730]
Step 14 --------------------------------------------------- [64043 * 566 ÷ 730]
Step 15 ------------------------------------------------------ [36248338 ÷ 730]
Step 16 ------------------------------------------------- Base Damage 2 = 49655
Step 17 ------------ Base Damage 2 * {730 - (Def * 51 - Def^2 ÷ 11) ÷ 10} ÷ 730
Step 18 ---------------------- 49655 * {730 - (34 * 51 - 34^2 ÷ 11) ÷ 10} ÷ 730
Step 19 ------------------------- 49655 * {730 - (1734 - 1156 ÷ 11) ÷ 10} ÷ 730
Step 20 ------------------------------- 49655 * {730 - (1734 - 105) ÷ 10} ÷ 730
Step 21 ------------------------------------- 49655 * {730 - (1629) ÷ 10} ÷ 730
Step 22 --------------------------------------------- 49655 * {730 - 162} ÷ 730
Step 23 ----------------------------------------------------- 49655 * 568 ÷ 730
Step 24 -------------------------------------------------- Final Damage = 38635

I simply modified the dividers to include the attack rating of weapons and the armor rating of armor.

Magic Damage is calculated as follows: Mag = 255, Ultima is used, enemy MDef = 1

Step 1 ----------------------------------- [DmCon * ([Stat^2 ÷ 6] + DmCon) ÷ 4]
Step 2 ------------------------------------------ [70 * ([255^2 ÷ 6] + 70) ÷ 4]
Step 3 ------------------------------------------ [70 * ([65025 ÷ 6] + 70) ÷ 4]
Step 4 ------------------------------------------------ [70 * (10837 + 70) ÷ 4]
Step 5 ----------------------------------------------------- [70 * (10907) ÷ 4]
Step 6 ------------------------------------ Base Damage = 190872 [cut to 99999]
Step 7 ---------------------------------------- [{(MDef - 280.4)^2} ÷ 110] + 16
Step 8 ------------------------------------------- [{(1 - 280.4)^2} ÷ 110] + 16
Step 9 ---------------------------------------------- [{(-279.4)^2} ÷ 110] + 16
Step 10 -------------------------------------------------- [(78064) ÷ 110] + 16
Step 11 ------------------------------------------------------------ [709] + 16
Step 12 --------------------------------------------------------- MDefNum = 725
Step 13 --------------------------------------------- [BaseDmg * MDefNum ÷ 730]
Step 14 --------------------------------------------------- [99999 * 725 ÷ 730]
Step 15 ------------------------------------------------- Base Damage 2 = 99314
Step 16 ---------- Base Damage 2 * {730 - (MDef * 51 - MDef^2 ÷ 11) ÷ 10} ÷ 730
Step 17 ------------------------ 99314 * {730 - (1 * 51 - 1^2 ÷ 11) ÷ 10} ÷ 730
Step 18 ------------------------------ 99314 * {730 - (51 - 1 ÷ 11) ÷ 10} ÷ 730
Step 19 --------------------------------------- 99314 * {730 - (49) ÷ 10} ÷ 730
Step 20 ----------------------------------------------------- 99314 * 725 ÷ 730
Step 21 -------------------------------------------------- Final Damage = 98633

The problem is that the formulas completely fall apart once stats start going above 255. In particular Defense values over 300 or so start generating really strange behavior. High Strength + Defense stats lead to massive negative values for instance. While I might be able to modify the formulas to work correctly for my use case, it'd probably be easier just to use a completely new formula. How do people actually develop damage formulas? I was considering opening excel and trying to build the formula that way (mapping Attack Stats vs. Defense Stats for instance) but I was wondering if there's an easier way? While I can't convey the full game mechanics of my game here, might someone be able to suggest a good starting place for building a damage formula?

Thanks

Answer 1

Creating formulas like this requires knowledge of elementary mathematical functions - the things you learned about in Algebra and Pre-calculus class.

Once you have those mastered, just ask yourself (replace "value" with "damage," or "health," or "speed" or whatever):

• Do you want the value to grow at a constant rate? Use a linear function.
• Do you want the value to grow slow at first, but fast later? Use a polynomial or exponential function.
• Do you want the value to grow fast at first, and slow down later? Use an nth-root or logarithmic function.
• Do you want the value to grow slowly at the start/end but fast in the middle? Or do you want it to grow quickly but still have an upper-bound? Use a sigmoid curve, such as atan or the logistic function.
• Do you want the value to oscillate? Use a sin or some other wave.

Then just tweak it (add/multiply stuff, change the base-value, etc) until it feels right. A graphing calculator will help you visualize how changes to the parameters will affect the function.

---

By the way, the problems you are experiencing are due to integer overflows.

Use variable-types that are large enough to hold the numbers you're working with. Sizes differ by platform in C++, but using the 32-bit Visual Studio compiler, unsigned int is 32-bit, while unsigned __int64 (MS-specific) is 64-bit. Also consider using a double.

Additionally, try to reorganize your operations so that you don't encounter such large numbers in the first place (for example, rather than MDef * MDef / 110, do (int)((float)MDef / 110 * MDef)).

Answer 2

My Characters stats are composed of the following:

There's your real problem: you defined your stats before defining what those stats actually mean. You're putting the cart before the horse.

Look at how D&D (tabletop) works. "Strength" doesn't mean anything by itself; it only means something because there's a rule that says, "Add your strength bonus to your melee attack." That rule is part of D&D's combat rules. Without the combat rules, "Strength" is generally a meaningless quantity.

The first question you need to ask yourself is this: how much differentiation do I want between characters? Again, look at D&D. There, they have 6 basic stats. These stats define different dimensions of play for characters. A character with a high Dexterity will have different options from a character with low Dexterity.

But the reason for even that difference all comes back to rules. A high Dexterity means bonuses to ranged attacks; you can hit more often with ranged attacks. So just between Strength and Dexterity, you have two dimensions of play: ranged vs. melee.

Intelligence and Wisdom also form something of a pairing, but these interact more with specific classes. Int makes Wizards and other arcane spellcasters better (or possible under some rulesets), Wisdom is vital for Clerics and other divine spellcasters. Because divine and arcane spells have different spell lists, these two stats are involved in different dimensions of play.

You need to define the basic rules around stats before you can define growth progression functions and the like. You don't need specifics; you don't need to say that "each point of strength is added into the random roll to determine if a melee attack hits." What you need are meta-rules like "dexterity makes ranged attackers better." You can figure out exactly how it makes them better later on.

There are different ways to progress characters. A common old-school Final Fantasy trick was to simply use the character's level as part of their damage computations. This could be simply multiplying the level by the appropriate stat, or it could mean applying a function to the character's level. Say, a quadratic progression, so that the rate of a character's damage would increase per level.

However you want your combat functions to work, they need to take into account progression. Your functions need hooks for progression.

D&D has a funny way of progression. It is part class-based; every time you go up in level, you get new class features and a flat bonus to your to-hit, based on your character class. However, some class features got better by themselves. Spells in D&D would have progression built into them. A spell might do 1d4 damage per 2 levels of a spellcaster above the first. So every other wizard level makes that spell better.

D&D also used item-based progression heavily. Until 4th edition, item-based progression was mainly for fighting characters, but even in older editions, spellcasters had items that gave them stat buffs or other adjustments (or flat out gave them spells).

So items are another thing your combat functions need to take into account. Do items just buff one or more stats while equipped, or do they do other things as well? D&D was a bit odd, in that stats rarely changed; weapons simply did XdY damage, possibly with a bonus based on one of your stats. And that was that. So your only way to do more damage in battle was to find a better weapon. In many videogame RPGs, they take level into account in addition to a weapon.

Answer 3

Your formulas seem pretty complicated. I'm not sure how professional RPG developers handle this, but I'd recommend on focusing on simplicity. Try to find the simplest possible formula you can that still incorporates the range of stats you want to use. For instance, could you have stats modify each other prior to damage calculation, rather than modifying the damage during the calculation? Once you've got a formula in mind, I'd try graphing it for a wide range of possible values to see how it will behave as players level up. Obviously the fewer variables you have, them more feasible this will be.

Additionally, BlueRaja provided an important explanation of why you might be seeing unexpected values at higher stat levels. Using unsigned types and checking for overflows will be important.

Answer 4

Late game Enemy HP will typically be in the lower millions (with a super boss having the maximum of ~12 million).

This I have a problem with. You should build your bosses around what you think your players should be able to handle. You're building your players and combat formula around what you want your bosses to be like.

Once you get your combat mechanics and roles built, then you can decide on how you want to design your bosses as it should be a good balance between the damage the players can deal/absorb vs what the boss can deal/absorb.

Answer 5

Those numbers you quoted are likely derived from a just tweaking a simulation, over thousands of runs.

So you should write a simulation create a level 20 character, and see how he fares through 200 battles or so against the enemy types he's supposed to be fighting at that point. Do it again for a level 30 character (in presumably the next section of the world).

It'll take a while to get right, but writing automated simulations will definitely make this much easier than just guessing and manually playing it out.

Answer 6

I think you make the mistake that you want to create a formula without having a proper design in mind.

First start with a design, then start thinking about representing the design in formulas. The clearer your design the easier it should be to find simple and/or precise formulas.

Try to implement "types" of enemies, e.g. "armored" => a player attack if it is of type physical is reduced by 50%. Do not make a battle flow to abstract, think about what is relevant and what is not.

If your design says "armored enemies" are weak against magic but strong against physical damage, represent that in code. But remember that you have do a lot of testing, because the values won't work magically the first time you write the code. Try to create a design, put the logic into code, always check whether this the technical representation of what you had in mind and if it isn't change the values until it is.

Answer 7

Although my design have not left Spreadsheet phase, I have came to one conclusion about designing math for RPG:

Keep them as simple as possible. In design that I'm working on, I used very simplistic formulas, that are adequate to semi class-less system. Ie. Fireball Spell have damage of 30. The forumal is:

BaseSpellDamage * Attribute (5 * 6)

We can also add modifiers like this:

Result = (BaseSpellDamage * Attribute) (5*6)
Result = Result + (Result * 50%) (30 = 30 + (30 * 50%))

So end result would be 45 damage. I found that using precentage multipliers is very easy, and very scalable solution to work with. Instead of coming with some odd numbers with complex math to figure out the result we want.

Now we can calulcate damage reseistance, that will just calcucate defense against set type of damage. There are two approaches and I honestly haven't decided which would suit better, but they both are simple and viable:

DamageResult = Resistance * Damage ( 50% * 45)

So the end result will be 22damage taken (I just cut the partial result).

Other formula:

DamageResult = Damage - Resistnace (45 - 22).

So end result will be 23. If it happen that resistance is bigger than damage taken then character just doesn't recieve any damage. Of course it's up to you to make sure that such situation have no place, except for when you want it be the case.

Although I must admit that precentage scaling is somewhat easier to balance and scale. But this also depends on your base numbers. Precentage scaling will work best if you start from 100 and up. If you are operating on small numbers (anything below 100 to be honest), it can get awkard as you will start getting floating point results, that will be hard to balance and actually do anything interesting with them.

Probably optimal solution in that case is to use both approaches when they see to fit. Or if you are fan of big numbers start from 1000.

And at the closing ends. I haven't come to this conclusions completly on my own. I have actually spend quite some time reading various RPG manuals (Hero, DnD). Especially DnD was helpful, as it operates on similiar principles but instead of attributes it's using levels for it's formulas, they can be sometime more complex. Than what I presented here.

In anycase, the best advice is: try to keep them as simple as possible. Do not use any advanced math, or long equations, as they are prone to errors, that are hard to spot, when you have to deal with 87234 other things at the same time.

Answer 8

As others have mentioned, the Final Fantasy X formula is pretty complex. Generally for that series the later the game, the more complicated the formula. It's probably easier to base your damage formula on another game entirely. But in general, I think it's worth discussing from a very general level what kind of damage formulas you can find in the world, and how you can make a game based around them. The first thing you need to decide is how much damage do you want to be able to do at the end of the game, and what kind of stats do you want the player to be able to have? Once you have that, you can pick a formula system, and then optimize the formula and weapon values to reflect those ranges over time.

Purely Stat-Based

This is a good idea if you want your characters to be flexible in terms of what levels of enemies they can challenge. A formula like this is only going to depend on the player's stats, their equipment, and the stats of the enemies. These formulas are usually fairly simple. Final Fantasy Legend II (See http://www.shenafu.com/ffl2/weapon.php for instance, has weapons that do damage based on the simple formula:

(Stat+StatBonus)*WeaponStrength - Defense*4

A formula like this is good if you want a very simple method of estimating damage, or a quick jumping off point for modifying damage based on other factors like skills and elemental weaknesses.

To show how broad this kind of formula can truly go, consider the damage formula for Inflation RPG, an Android and IOS game (See http://inflation-rpg.wikia.com/wiki/Calculations). The formula is heavily both stat and equipment dependent. Each piece of equipment has two stats - a bonus to the ATK stat, and a multiplier value. Some pieces of equipment have low multipliers, but high bonuses, others have low bonuses but high multipliers. For a character with only 10 ATK, the Battle Axe with it's 5000 ATK Bonus but low 145% multiplier is a great choice. The total damage is (10+5000)*1.45 = 7264, but the Estoc, with 0 bonus and a multiplier of 300% is a poor choice - the damage is (10+0)*3 = 30. Later in the game, a character with 5000 attack would prefer switching weapons.

Stat- and Level-Based:

A good example of this is Final Fantasy V, VI, and Final Fantasy XII (See http://www.gamefaqs.com/ps2/459841-final-fantasy-xii/faqs/45900, for example). The formula for swords in FFXII is:

DMG = [ATK x RANDOM(1~1.125) - DEF] x [1 + STR x (Lv+STR)/256]

and the damage formula for staves is:

DMG = [ATK x RANDOM(1~1.125) - DEF] x [1 + STR x (Lv+MAG)/256]

They're very similar, but notice that the sword formula only depends on the strength and level, while the staff formula depends on strength, magic and level. The advantage to this kind of formula is that it allows the player two avenues of growth - building their stats, or building their levels. The downside is that it also penalizes characters both ways as well. What this really ends up doing is allowing the player to level up to increase their damage output (for FFXII this amounts increasing their damage output by ~4% per level around level 50 when you factor in stat gains) to help customize the difficulty to their comfort level.

Fixed Damage:

Fixed-damage formulas do not depend on the character's stats or level, they depend only on the internal damage formula of the weapon itself. They can still vary over a range, but they deal the same damage regardless of the user (barring any other special effects or character traits). They're best used if the weapon is going to do fixed-damage and the ability to equip the weapon depends on either stats and/or level. Diablo 2, for example, does this, as do many roguelikes which have weapons that depend on die rolls. That being said, 'fixed damage' does not imply "non-random" - and in fact there's usually some element of randomness to the damage done.

This is a good methodology if you want to have weapons that are easy to transfer between characters or to carefully control the damage output that characters at certain points in the game can do if you know what equipment they have access to (via drop tables, chests, and steal tables).

Another place you'll run into this is with certain types of equipment or items in Final Fantasy. 1000 Needles, for instance, always deals 1000 damage. In Final Fantasy Legend II, martial arts deal damage based on the formula:

Damage = WeaponStrength*(90-UsesLeft) - 4*Defense

Final Fantasy XII has somewhat fixed damage for guns as well, dealing damage according to the formula:

DMG = [ATK x RANDOM(1~1.125)]^2

Although the damage is somewhat random, it only varies by 26.5 percent over the total range, so you're guaranteed to do a certain level of damage on average over time. These types of attacks are useful for characters who have both low stats and low levels in games that normally account for those factors in dealing damage. Plus, they ignore the defense of the target (although the formula could be easily reworked to fit in defense if you so desired).

Answer 9

all games in the final fantasy series have had a stat cap of 255 because of the problem you are encountering. at level 100 stats would be 255.

you say about increasing stats with abilities and equipment and I remember seeing this in the games but the way this is done is in the formula. there is an extra step that checks for ability and equipment modifiers and applies them after the stats have been used.

in your case it would be step 21: apply ability modifiers step 22: apply equipment modifiers step23: final damage.

if you are interested google for final fantasy formulas, they are out there. I have copys of the actual battle mechanics including AI for final fantasy 4, 6, 7 and 9. people cracked them from the original games when they were creating roms for emulators. There not that hard to find if you look hard enough.

the biggest thing for creating formulas is testing. set up a script to run your battle wit ai on both side and run several hundred battles. vary the monsters and the stats and see if it works or if lv 40 kills everything, it is entirely possible that a boss is actually impossible to kill lol. a tip would be to turn off all animation as it is impressive how fast AI's can battle when no one is watching.

Answer 10

I was wondering how do people actually develop proper damage formulas that scale correctly?

The first 2 things are:

• decide what you mean by 'correctly' - what is your idea of 'correct damage'?
• decide what you mean by 'scale' - what values are going to change, and what effect do you want those changes to have?

Once you know that, you have enough information to use the mathematical formulas that BlueRaja mentioned in his answer. Just remember there is no such thing as a 'proper' damage formula - just one that matches your design for the type of experience you want your players to have.

Answer 11

In essence, you need to figure out two things.

1. How to properly preform computation with large numbers.
2. How you want and expect attack damage to behave against weaker and strong opponents.

1

You can either use a double data-type or a big number library. Doing computations with large numbers (like in clicking / idle games often relies on such big number libraries). In your case, the numbers are relatively small so using 64-bit floating point will allow the needed flexibility.

2

How do you want the game to behave? some examples:

• One approach is to use ATK to decide if you hit or not and roll a die with a known damage rating depending on the weapon: for instance long sword 1d8, two handed sword 1d10, dagger 1d4. This will not scale amazingly well if the hp can grow on and on. If you use ATK to determine if the characters hit or miss, you can do another throw after a successful hit with ATK * X% to determine crits. This would mean game characters get more crits against weaker opponents but far less against stronger ones.
• If you want to use a damage formula, the damage formula can be as simple as (ATK - DEF) +/- 20% as it's done in World's End (an RPG strategy game). This will mean that attacks against an opponent with strong DEF could do no damage at all. For instace ATK = 10, ENEMY DEF = 12, (ATK - DEF) = -2 and it also means that high attack vs slightly lower def, could do little damage, potentially making battles longer with increasing HP. For instance with ATK = 1010, ENEMY DEF = 1005, the damage would be 5. The damage will also increase dramatically now if the ATK is increased by a small percentage. For instance ATK = 1055, will do 900% more damage with only a %5 increase in ATK value.
• To avoid this, you can do something like ATK / DEF * WEAPON_DAMAGE This will scale more gradually when ATK or DEF are increased and allow a weak attacker to damage a creature with strong DEF.

tl;dr You need to treat your numbers responsibly and figure out how you want the game to behave in different situations. Some questions to ask yourself:

1. How many attacks (turns), should it take to defeat that boss / minion?
2. How should different skills, buffs, debuffs affect x battle? Should it be hopeless, should winning be y% less / more likely?

Answer 12

alternativly, if your looking for a very simple formula, Do the following:

(Note that this is my idea for a formula,)

First, for a simple attack formula, you will only need a few stats. THe player's Attack stat (this could be Strength, or Magic, depending on the type of move.)

Then, create a variable, called MovePower. This will depend on the move, and better moves would have a better MovePower. For my formula's, general "Attack" commands have a MovePower of 5.

Then, make the defence a percent based stat, (and, of course, make it inpossible to achieve 100% defence on damage)

Then, during the attack phase, simply multiply the attack stat by the move power, and remove damage based on the foes defence! simple!

You can also opt to have other modifyers, like if the move will hit (Accuracy), if the move has another added effect (Bio, inflicting poison,) and opt for that to hit, (Accuracy again,) have damage upgrading/downgrading moves that change stats, ect. Have fun with it!

Answer 13

that physical damage formula is the EXACT SAME as the one used in Final Fantasy 10. Those formulas were meant for stats UP TO 255, especially the defense stat. Here is the page on GameFAQs that explains the formulas: https://gamefaqs.gamespot.com/ps2/197344-final-fantasy-x/faqs/31381 Scroll down to the full damage calculation and you will see that those were pretty much copy and pasted. Tweak the formulas so that the defense will actually absorb damage like it is supposed to and so the damage does not increment so fast, because a 3 point strength increase will result in a massive increase of damage from my perspective from my playthrough of FF10 when I increased a party member 3 strength. Those formulas are meant for small increases of stats like how the sphere grid works. They are NOT meant for level ups where your stats increase quicker. Here is why you should cap your stats at 255 only if you are using those formulas: In FF10 once you are at 255 strength with the celestials, you will deal at least 99,999 damage, if not more since the cap is 99,999. Now try to find out how much damage it will be at 999 strength.