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View Full Version : RPG math and scaling Or "Fighters are already Quadratic! Wizards are even weirder!"



Vitruviansquid
2017-06-23, 07:20 PM
Fun observation:

In a fight, an RPG character's power can be thought of at its most basic as the amount of change he can bring about. When I say "change," I generally mean the change from alive monsters to dead monsters.

So if you look at it that way, a character's power is equal to the character's ability to deal damage.

A character who can deal 10 damage is twice as powerful as a character who can deal 5 damage.

But that's only half of the puzzle, because we are not considering that as the character is trying to make changes, there are also NPC baddies who are trying to change them. When I say "change" here, I mean the change from an alive adventurer to a dead adventurer. So health also matters.

A character who has 100 hp is actually twice as powerful as a character who has 50 hp.

But now that we realize damage is actually a value to mean "change over time" and hp is actually a value to mean "amount of time to be changed" we realize that a character's power can be found with the formula:

Power = hp * damage

Basically, a character with 100hp and 10 damage is AS powerful as a character with 50hp and 20 damage which is AS powerful as a character with 200hp and 5 damage. You see that all three of them have the same power (which in this case would be 1000 power)

This also gives rise to the understanding of these hp and damage is only as good as how much of the other you have.

The character with 20 damage is dealing twice as much damage as the character with 10 in the above example, but only gets to apply that damage for half the time.

So if you look at the D&D fighter who generally improves in both defense and offense as he levels up, he is actually growing quadratically. That's why you don't have enemies who are way over or under levelled to fight your PCs' party. (Before anyone asks, since the fighter scales quadratically it does not make him equal to the wizard. The wizard actually does not scale quadratically and has even more extreme scaling.)


Another fun observation:

Many dice systems break down as differences between modifiers and target numbers get too large. Why is this?

Each addition to your chance of success gets more less valuable as your chance of success goes up.

If I had a 5% chance of succeeding something, adding another 5% doubles my likelihood of success. I now have 200% efficacy.
If I had a 50% chance of succeeding something, adding another 5% only adds a tenth to my likelihood of succeed, so it only brings me to 110% efficacy.

That is if I am rolling 1 die. If I roll more dice to add up the values, the bell curve that forms exacerbates this problem.



In any case, I wanted to start a thread about the underlying math of RPGs because I, and I know a number of others, on this forum are writing RPG systems.

What are virtues and what are follies in an RPG's math system?
What do RPGs do to mitigate those?
Which systems have you seen have really good underlying math?

Koo Rehtorb
2017-06-23, 07:40 PM
In a fight, an RPG character's power can be thought of at its most basic as the amount of change he can bring about. When I say "change," I generally mean the change from alive monsters to dead monsters.

I don't know why you added the second sentence there, the thought works much better without it.

obryn
2017-06-23, 07:46 PM
Your initial statement about fighters doesn't quite work. In fact, early 4e made some mistakes with monster roles using exactly that philosophy.

Assuming you're fighting opponents who also have damage and hit points, the ability to deal massive damage all at once is more valuable than the ability to deal the same amount over time. Someone with a 100 damage alpha strike is much better off facing off against a 100 hp monster than someone who deals 10 in each of 10 rounds.

This was the "soldier/brute" problem before 4e monster math got fixed.

Soldiers dealt less damage, but have higher defense
Brutes dealt more damage and had more hp, but were less accurate and easier to hit.
Skirmishers were average in all categories.

Mathematically - using something like that "power" calculation - these were all even. But in practice...

Soldiers were unthreatening and annoying to hit, making combat drag.
Brutes couldn't hit and were big hit point sinks, making combat drag while they hoped for a critical.
Skirmishers were pretty good, though.

The combination of these in early 4e, like in Keep on the Shadowfell, made those combats drag, and contributed to the (then-accurate) perception that 4e combats were overlong and draggy.

Vitruviansquid
2017-06-23, 07:53 PM
Your initial statement about fighters doesn't quite work. In fact, early 4e made some mistakes with monster roles using exactly that philosophy.

Assuming you're fighting opponents who also have damage and hit points, the ability to deal massive damage all at once is more valuable than the ability to deal the same amount over time. Someone with a 100 damage alpha strike is much better off facing off against a 100 hp monster than someone who deals 10 in each of 10 rounds.

This was the "soldier/brute" problem before 4e monster math got fixed.

Soldiers dealt less damage, but have higher defense
Brutes dealt more damage and had more hp, but were less accurate and easier to hit.
Skirmishers were average in all categories.

Mathematically - using something like that "power" calculation - these were all even. But in practice...

Soldiers were unthreatening and annoying to hit, making combat drag.
Brutes couldn't hit and were big hit point sinks, making combat drag while they hoped for a critical.
Skirmishers were pretty good, though.

The combination of these in early 4e, like in Keep on the Shadowfell, made those combats drag, and contributed to the (then-accurate) perception that 4e combats were overlong and draggy.

My point was about how to evaluate a character's power, not how to set up a good game flow.

Though you are right that there is such a thing as good balances of offense and defense and bad balances of offense and defense.

But since we are going there and acknowledging that, everything else equal, higher damage over time is better than higher defense because having higher offense makes you a greater threat to the enemy... the exception to this rule is when you combine high defense characters with high damage over time characters. If you can rig it so the enemy can only attack the more defensive characters, you are effectively giving the higher offense character extra survivability, which adds to your power.

We might also mention that if you duplicate a character, you are actually tripling, not doubling their combined power.

goto124
2017-06-23, 07:56 PM
We might also mention that if you duplicate a character, you are actually tripling, not doubling their combined power.

Why? Action economy?

Vitruviansquid
2017-06-23, 07:59 PM
I have a hero with 10 health and 1 damage fighting an infinite supply of orcs with 10 health and 1 damage.

I kill 1 orc because on the 10th turn, I have dealt a total of 10 damage and been hit for a total of 10 damage.



I have 2 heroes with 20 health and 1 damage each.

I kill 1 orc on the 5th turn.

I kill my 2nd orc on the 10th turn, but one of my heroes has died.

I kill my 3rd orc on the 20th turn as my second hero also dies.

goto124
2017-06-23, 08:07 PM
If someone deals a high amount of damage in one strike, it downs individual enemies a lot faster and makes battles easier. Say we have 3 enemies, each has 3 HP and each deals 2 damage. We also have a PC who deals 3 damage per attack.

Round 1: PC1 takes 6 damage from 3 enemies. PC kills 1 enemy.
Round 2: PC1 takes 4 damage from 2 enemies. PC kills 1 enemy.
Round 3: PC1 takes 2 damage from 1 enemy. PC kills last enemy.

Now, let's swap out for a PC who deals 1 damage to each enemy.

Round 1: PC2 takes 6 damage from 3 enemies. PC kills no one.
Round 2: PC2 takes 6 damage from 3 enemies. PC kills no one.
Round 3: PC2 takes 6 damage from 3 enemy. PC kills all enemies, but took more damage than PC1.

Every time an enemy is killed, the battle gets significantly easier. I haven't factored in AoOs. And what if one of the enemies was spewing poison gas, or blocking your attacks, or some other form of debuff? Even more damage or disadvantage mitgated there.

Damage means nothing when you don't actually kill your enemies, since they're just as effective whether they're at 1 HP or full health.

HisHighestMinio
2017-06-23, 08:46 PM
Linear fighters/quadratic wizards wasn't a mathematical description, it was an analogy. It says that Wizards don't just get better stuff than a Fighter, they're better in multiple ways all at once. Quadratic equations end up bigger than linear equations even if they start smaller because they don't only grow faster, they grow in multiple dimensions/measures/whatever at once. So the Wizard can do as much as or more damage than the Fighter through buffs or summoning or BFC, AND they can teleport and mind control and do all sorts of other useful stuff. So it wasn't at all based on math, it was an analogy.

Defense and offense don't scale linearly, so fighters aren't exactly quadratic anyway.

Your power stat tends to work better in combat systems with less depth. It's exactly the formula I use when my FTL crew gets in a fight and I'm trying to judge whether I can win.

Vitruviansquid
2017-06-23, 08:53 PM
The point is to create a framework for people to discuss in theory how different RPGs' mathematical foundations lead to consequences in play.

I am aware of the discrepancies between the theoretical "power" value and an actual game of D&D.

Spore
2017-06-23, 11:03 PM
For these exact reasons I feel dice pool systems (e.g. Vampire) or bounded accuracy (D&D 5e) work so much better than D&D 3.5 for equality. But people have grown to love 3.5/PF to be able to simulate a vast power difference while keeping Bob the Fighter and Agamagar the Wizard in the same level range.

And some higher level spells are out of whack while I feel in 5e some high level spell options feel "neutered", especially controll spells such as Dominate Person. As a Wizard I don't necessarily need to be able to solve ALL my problems with magic. I shouldn't be able to secure my fortress against any and all magical and mundane intrusion via Sanctuary, Consecration or similar. I shouldn't be able to warp reality to a degree as did Wish in 3.5. But on the same notion it is weird that a spell that is called "Power Word KILL" only works on people below xyz health.

Also there shouldn't be spells to be able to deal with ANY problem. Plagues sweeping the land? Na, we got tons of clerics able to cure those. A tricky assassin successfully poisoned you? Remove Poison! YOU MANAGED TO KILL SOMEONE?! Na, Raise Dead.

Imho, spells that deny ways of engaging with the story OR flatout cancel other effects are necessary but sometimes just plain boring. A game like Skyrim isn't perfect but it's my example for now. Got sick? An Alchemist or praying at a shrine would help you. Poisoned? You can heal the damage it does but you need an antidote. Being of a certain race helps you too. Killed that guy with your sword? Too bad, he is dead and stays that way. You want a magical item? You have to first slay a powerful monster with a soul trapping device. Or you know, ritual murder of humans with black soulstones? What? You don't need another flaming sword, I guess.

icefractal
2017-06-24, 04:21 PM
Technically speaking, monsters in D&D are exponential, not quadratic or linear, and PCs theoretically should be as well. It's right in the CR system: x2 creatures = +2 CR. 1024x CR 1 = CR 20.

In practice, this tends to be done in a patchwork way that works out 'close enough' for the lower levels and falls apart but is patchable through items and scenario choice at the higher levels.