Basic Math and Games

Note: This article is mostly about action-rpg that have game mechanics rooted in heavy numbers.

Well, I believe it's time to take a little tangent here to talk about a thing that has plagued me for years: Mathematics.

It comes from the fact that I love action games with rpg elements or rpg games with heavy action elements and such. (Examples: Castlevania: SoTN, Secret of Mana, Diablo.. Etc.) They are typically rich games with a good mix of customization from the RPG side and fast-paced gaming from the action side.

...And then math rears it's ugly head. Suppose that you have a game where Damage is Calculated by (Attack - Defense) = Damage. Simple. Perfect. Understandable. Of course, simple is better, right?

Slime hits you for 3 damage!
You hit Slime for 7 damage!
Slime hits you for 3 damage!
You hit Slime for 7 damage!
You have defeated Slime!

What's wrong with this picture? For starters, it's utterly boring. You know exactly what will happen next time you face the dreaded slime. You'll take 6 damage. If you don't have 6 hp, well, then you'll die.

What happens if you get +3 defense? Well, then.. even if you face a million slimes, you won't have to fear a thing as they all bounce uselessly off your armor. What if you had 3 less defense? Then you'd take twice as much damage from all slimes. In this 6 point margin, you can go from 'Slime is a terrifying force' to 'Slimes are trivial.'

In 6 points. That's a thin margin.

Now, think about what happens to the next mob? Say, the Blue Slime. If he hits for 3 more attack than a Green slime and you take 0 damage from Green Slimes, then the Blue Slime will be as hard as a Green Slime before. The monster isn't getting harder. It's running hard to stay in place. However, if you didn't have you armor... then the Blue Slime would be more than 50% harder than the Green Slime. Such a jump in difficulty would be hard for most slow paced rpg's to stomach.

The problem with using straight attack - defense = damage formulas is the thin margin of balance you have.

For example:

If my attack is less than the monster's defense, I essentially do nothing to the monster and it is impossible.

Alternatively, if my defense is less than the monster's attack, the monster is challenging for a while, until he decimates me by having just a few extra points of attack. (The difference between doing 5 and say.. doing 9.)

If my defense is greater than the monster's attack, then the monster is essentially irrelevant. It does nothing to me.

And the difference between impossible and ridiculously easy is only a few points. A few pieces of equipment for defense. A level or 2. Fall a little behind, everything is impossible and you're forced to stay behind to catch up. Fall a little ahead, everything is ridiculously easy.

This problem becomes compounded if you want to have a character that does repeated small strikes. Suppose we have a 'Thief' type character who hits twice.

Thief hits for 0 damage. (Attack less than their defense.)
Thief hits for 2 damage. (Attack one more than their defense.)
Thief hits for 4 damage. (Attack two more than their defense.)

This character will do very little or start doing twice as much as his compatriots. How are you going to balance this when it all comes down to a razor thin margin of points? If you give a character 50 attack but 2 attacks and a character with 100 attack, these characters are essentially ALWAYS unbalanced no matter what the defense of the monster is. The thief character will always be affected twice by the monster's defense and the 100 attack character will always out-damage the thief. But it seems balanced, at first glance, and that is the difficulty.

Typically, games that use this formula follow a distinct pattern: "This game is too hard." "Wait, now it's too easy." "Wait, now it's too hard again, did I miss something?" "Wait, got that piece of armor. This game is too easy." "Aagh. Can't afford that next armor. This game is too hard."

What was missing? "This game is just right." or.. "Hm. Well, I don't need that armor, I could just play a little better."

But you might say.. "Well, I can perfectly pace my game such that it curves out perfectly. At level 1, they'll fight green slimes and at level 2, they'll fight blue slimes with such and such bonuses and they'll have such and such equipment..."

Which is all fine and good, until you realize that you are not flawless, that players will either miss the equipment/skill/level or player's will innovate and achieve far greater equipment/skill/level and make all your careful balancing irrelevant. Putting the entire difficulty of the game on a razor thin edge by using simple math? A game that's too hard is no fun. A game that's too easy is no fun.

In a large complex game featuring complex interplay between monsters and players... Having the entire difficulty of monsters come down to whether or not people have the stats to EXACTLY put them on this razor thin edge of 'just right' is insane. If the players have control over their stats (Attack/Defense) then it's essentially hopeless. There is no margin of error if you use this kind of simple math.

And if you screw up.. well..

Slime attacks you! It does 0 damage!
Slime attacks you! It does 0 damage!
Slime attacks you! It does 0 damage!
You attack Slime! You do 7 damage!

*Yawn*

Big Slime attacks you! It does 13 damage!
You die...

*Sigh*