Reading the threads about Resonance and CLW Wands has been very enlightening, is a very complex issue with no easy answer. But one of the things that calls me more is that HP and healing increases linearly while overall power level and treasure increases quadratically, that brings me the questions:
1- Why somethings increase linearly and while others quadratically?
2- Is just for legacy or are other considerations too?
3- It would not be better that everything increased quadratically?
HP for example, in a quadratic progression would be Con mod * Class bonus * Levels, so a 10 lv Fighter with Con 16 would have 3*10*10 = 300 HP. This way high level healing magic items could heal enough to justify the high price.
4- Would be a bad idea that HP and healing increased quadratically?
I think having quadratic progression for everything would make easier to balance the game than having somethings increasing linearly while others quadratically.
Why quadratic should be the default? Because Pathfinder answer NO to Bounded accuracy, a high level character should not have a problem wiping low level ones.
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The problem comes that if currency is to increase as PCs fight tougher and tougher enemies, then costs must be quadratic. The problem being that it is underwhelming if a 17th level fighter, fighting a Great Wyrm gained the same as a 1st level fighter, fighting a few goblins. And it is equally problematic if HP or other limited resources are quadratic, as that means that a 17th level fighter means that 16th level enemies are obsoleted.
As such, I can agree with the Resonance solution of managing linear resources at quadratic costs with a separate linear cost. But I don't necessarily agree with the way of doing it. I think Resonance works better than most alternative systems, but I think rather than resonance as the whole of the system, it should be but a part. Now, as much as I'm not a fan of 4e or 5e, I think a system of non-magical, non-resonance healing could work as to allow resonance or other resources to be spent only if the necessity of such is warranted either by virtue of misfortune or poor tactics.
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I have said it before, most recently back on June 8, at its core Pathfinder is not linear nor quadratic. It is exponential.
Early Dungeons & Dragons was governed by tables of numbers, so the mathematical trends where difficult to see. In 3rd Edition, Wizards of the Coast made many things additive.
In 3rd Edition, reaching 2nd level took 1,000 xp. Reaching 3rd Edition took 2,000 more xp for a total of 3,000 xp. Reaching 4th level took 3,000 more xp for a total of 6,000. Each level being harder to reach represented the tougher challenges, which gave more xp, the party faced. The series 1, 3, 6, 10, 15, etc. is a quadratic series called the triangular numbers. The exact formula is the accumulated experience to reach level n is 500 × n^2 - n)
The longer gap between levels also gave more treasure. If treasure is proportional to experience points, then treasure increases linearly with each level and quadratically with total levels. 3rd Edition had its own wealth by level table: 900 gp at 2nd level, 2,700 gp at 3rd, 5,400 gp at 4th, etc. The exact formula is the accumulated wealth at level n is 450 gp × (n^2 - n).
Since the cost to buy an enchanted weapon or suit of armor appropriate to a level should be a set fraction of the accumulated wealth at the level, Wizards of the Cost made the cost quadratic. The cost of a +X weapon is 2,000 gp × x^2, the cost of a wand of spell level x and caster level 2x - 1 is 1500 gp × (x^2 - (1/2)x). Those two quadratics are not the same formula as x^2 - x, but they are also quadratic. Thus, they roughly tied the cost of weapons, armor, and magic items to a fixed portion of wealth by level.
However, Pathfinder abandoned quadratic xp and moved up to exponential xp. The fast progression in Pathfinder corresponds to the old D&D 3.5 progression but changed a little: 1,300 xp, 3,300 xp, 6,000 xp, 10,000 xp, 15,000 xp, 23,000 xp in Pathfinder rather than 1,000 xp, 3,000 xp, 6,000 xp, 10,000 xp, 15,000 xp, 21,000 xp in D&D 3.5. The exact formula for accumulated xp is 3000xp × (1.43^(n-1) - 1).
Wealth by level converted to exponential, too. 1000 gp, 3000 gp, 6000 gp, 10500 gp, 16000 gp, 23500 gp, etc. The first three numbers are triangular, but the rest are exponential. That does not match a formula well, but the best one I found was (5000×1.3^n - 8000) gp. It gives a negative number for starting gold and only 450 gp for 2nd level, but those number were not on the exponential curve anyway.
By the way, exponential xp shows a much greater understanding of what makes levels and challenges interesting. Paizo designers displayed genius in switching to it.
However, despite changing xp and WBL to exponential, they kept the old quadratic prices. The exponential curve is close to the original quadratic curve at low levels, so the low-level prices did not need adjustment. And for high-level play, Paizo appeared more interested in correcting the classes than the magic items.
Thank you Mathmuse for the maths lesson.
I think there could be scope for alerting the various scaling formulas to achieve better balance.
Pathfinder answer NO to Bounded accuracy, a high level character should not have a problem wiping low level ones.
Just pointing out, from what I have seen this is very much the case even with the more bounded accuracy in PF2. Adding level as well as proficiency to attacks and AC and saves and DCs means a higher level character facing a lower level enemy has a sizable advantage offensively and defensively before factoring in things like having better equipment, more uses of daily resources, and more powerful feats and spells than lower level entities.
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3- It would not be better that everything increased quadratically?
HP for example, in a quadratic progression would be Con mod * Class bonus * Levels, so a 10 lv Fighter with Con 16 would have 3*10*10 = 300 HP. This way high level healing magic items could heal enough to justify the high price.
4- Would be a bad idea that HP and healing increased quadratically?
I performed an analysis of how character power increases with the various numbers in leveling up. I began writing an article titled "41% Better", but gave up when I passed 10 pages and was still explaining basic principles. The key is the CR system. Two identical creatures are twice as powerful as one of those creatures, and the CR goes up by 2. Thus, power doubles every two levels. Since the square root of 2 is 1.41, that means a character must improve by 41% at each level.
Consider a warrior with 1d10 hit dice and Con 10. He has 10 hp at 1st level. He rolls a 5 (I will alternate 5s and 6s to keep close to the average) at 2nd level for 15 hp, a 50% improvement. Going to 21 hp at 3rd level is a 40% improvement. Going to 26 at 4th level is a 24% improvement. Hit points don't keep up with the 41% increase in power. (Yes, going from 1st to 2nd level is a much bigger increase than the CRs call for, around 80% more powerful. That is why 1st-level characters feel so weak. They are weak compared to 2nd level.)
Fortunately, hit points don't stand alone. Overall defense increases with armor class, too. The net effect is quadratic. Alas, since power goes up exponentially, even quadratic is not enough. But quadratic stays close to exponential for several levels.
Also consider the progression in the Cure Spells: 1st level Cure Light Wounds is 1d8+1 hp restored, averaging 5.5 hp per spell. 3rd level (caster level) Cure Moderate Wounds is 2d8+3 hp restored, averaging 12 hp restored. That is more than double the healing and since we jumped up two levels, we want a full doubling in power. 5th level Cure Serious Wounds is 3d8+5 hp restored, averaging 18.5 hp per spell. That is only a 54% improvement, far short of the 100% improvement we needed. 7th level Cure Critical Wounds is 4d8+7 hp restored, averaging 25 hp per spell. That is only a 35% improvement over 5th-level Cure Serious Wounds. The healing spells don't keep up with the power level.
The 1st level Cure Light Wounds applied to a 1st-level warrior with 10 hp heals 55% of his total hp.
The 3nd level Cure Moderate Wounds applied to a 3rd-level warrior with 21 hp heals 57% of his total hp. The 5th level Cure Serious Wounds applied to a 5th-level warrior with 32 hp heals 58% of his total hp. The 7th level Cure Critical Wounds applied to a 7th-level warrior with 43 hp heals 58% of his total hp. The healing spells do keep up with the hit points up to 7th level.
Pathfinder uses a trick I call Moving the Goalposts to make the linear increase in Base Attack Bonus act like an exponential increase. That trick applies only differences between numbers, such as the difference between attack bonus and armor class, and cannot be applied to hit points. Offense grows in power exponentially. The lack of matching exponential growth in defense leads to rocket tag.
Trying to answer the OPs questions:
Because it creates better games, better rule systems. If everything were linear, for instance, you are incentivized to "farm" in MMO terms, low level/trivial mobs for exp to rapidly level. If everything is quadratic, you very quickly get situations were each level is too big a jump in power and you can't effectively have encounters with things more than two away from you. (Mooks will never be challenging even in mass)). Over time as Mathmuse indicates designers have been refining the exact non-linear aspects, but they are superlinear for good (not legacy) reasons.
If HP/healing increased super-linearly, it feeds into the rocket-tag aspect they designers are trying to get away from. If a hit doesn't do enough damage to kill an opponent, they are likely able to heal it all immediately.
1st level Cure Light Wounds is 1d8+1 hp restored, averaging 5.5 hp per spell. 3rd level (caster level) Cure Moderate Wounds is 2d8+3 hp restored, averaging 12 hp restored. That is more than double the healing and since we jumped up two levels, we want a full doubling in power. 5th level Cure Serious Wounds is 3d8+5 hp restored, averaging 18.5 hp per spell. That is only a 54% improvement, far short of the 100% improvement we needed. 7th level Cure Critical Wounds is 4d8+7 hp restored, averaging 25 hp per spell. That is only a 35% improvement over 5th-level Cure Serious Wounds. The healing spells don't keep up with the power level.
If we think of it in terms of the total healing a cleric can put out in a single day, the increase is quadratic.
EG, at level 1 the Fighter has 13hp and the Cleric can heal about 11HP from spell slots. At level 7 the Fighter has, say, 74HP and the Cleric can heal about 200HP from spell slots.
So for out of combat healing, the cleric keeps up with the curve, and the number of combats you can handle per day (even without wands) tends to increase, but as an in-combat action it becomes less efficient... until you get to Heal and then it suddenly gets more efficient again.
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If we think of it in terms of the total healing a cleric can put out in a single day, the increase is quadratic.
Let's number crunch how much a Wis 18, Cha 14 cleric with no healing domain can heal.
1st level: spells (2)(1d8+1) = 11 hp, channel (5)(1d6) = 17.5 hp
2nd level: spells (3)(1d8+2) = 19.5 hp, channel (5)(1d6) = 17.5 hp
3rd level: spells (3)(1d8+3) + (2)(2d8+3) = 46.5 hp, channel (5)(2d6) = 35 hp
4th level: spells (4)(1d8+4) + (3)(2d8+4) = 73 hp, channel (5)(2d6) = 35 hp
5th level: spells (4)(1d8+5) + (3)(2d8+5) + (2)(3d8+5) = 117 hp, channel (5)(3d6) = 52.5
6th level: spells (4)(1d8+5) + (4)(2d8+6) + (3)(3d8+6) = 156.5 hp, channel (5)(3d6) = 52.5
7th level: spells (5)(1d8+5) + (4)(2d8+7) + (3)(3d8+7) + (2)(4d8+7) = 223 hp, channel (5)(4d6) = 70 hp
8th level: spells (5)(1d8+5) + (4)(2d8+8) + (4)(3d8+8) + (3)(4d8+8) = 279.5 hp, channel (5)(4d6) = 70 hp
The series 11, 19.5, 46.5, 73, 117, 156.5, 223, 279.5 is close to the series 11, 20, 47, 73, 118, 163, 226, 289, which is two interleaved, interrelated quadratic series. Hence, the total healing of a cleric is quadratic up to 8th level. The pattern of Cure spells breaks past 8th level, switching to Mass Cure spells and the Heal spell, so the quadratic curve changes, too. However, it will remain approximately quadratic.
EG, at level 1 the Fighter has 13hp and the Cleric can heal about 11HP from spell slots. At level 7 the Fighter has, say, 74HP and the Cleric can heal about 200HP from spell slots.
Wait, why compare the total healing to the total hit points? It is not as if the CR system scales the monsters' attacks to always hurt the same percentage of a character's hit points. Hit points goes up linearly but challenge goes up exponentially.
Instead, the proper comparison is total healing to total damage. A true modeling of total damage is beyond me, but I can make a quick approximation via the CR system, because two creatures deal twice as much damage as one creature. Thus, total damage doubles every 2 levels, or a 41& increase in damage every level.
Let's set 11 as our 1st-level baseline for damage. The 2nd level challenges would deal (1.41)(11) = 16 damage. The damage curve by level would be 11 hp, 16 hp , 22 hp, 32 hp, 44 hp, 64 hp, 88 hp, 128 hp, etc. That is below the 11 hp, 19.5 hp, 46.5 hp, 73 hp, 117 hp, 156.5 hp, 223 hp, 279.5 hp that a cleric heals and gets further below at each level. Not only can the cleric keep up, but he needs a smaller percentage of his spells to do so at higher levels.
So for out of combat healing, the cleric keeps up with the curve, and the number of combats you can handle per day (even without wands) tends to increase, but as an in-combat action it becomes less efficient... until you get to Heal and then it suddenly gets more efficient again.
Okay, the cleric can handle the healing and more. What about wands? A lot of people argue about healing wands in this Playest subforum. The Cure Light Wounds, Cure Moderate Wounds, Cure Serious Wounds, and Cure Critical Wounds increase linearly, so wands of them also increase in effectiveness linearly. Since the damage goes up exponentially, which is also close to a quadratic curve, the number of wands needed would have to go up linearly to handle the quadratic load. Um, isn't this a problem that Paizo wants to correct in Pathfinder 2nd Edition? They can't change the math without changing the spells or the wands.
On the other hands, if wands switched to operating purely on resonance, with no internal charges that limit the lifetime of a wand, then the load that healing wands could handle would be the product of the spells, which go up linearly, and the resonance, which also goes up linearly. The product of two linear curves is quadratic. Resonance-only wands could handle a quadratic healing load.
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Let's set 11 as our 1st-level baseline for damage. The 2nd level challenges would deal (1.41)(11) = 16 damage. The damage curve by level would be 11 hp, 16 hp , 22 hp, 32 hp, 44 hp, 64 hp, 88 hp, 128 hp, etc. That is below the 11 hp, 19.5 hp, 46.5 hp, 73 hp, 117 hp, 156.5 hp, 223 hp, 279.5 hp that a cleric heals and gets further below at each level. Not only can the cleric keep up, but he needs a smaller percentage of his spells to do so at higher levels.
I may not be following correctly - but if I'm reading this correctly - yes the cleric can keep up - for one encounter at CR = APL - but that's now how a typical adventure will go right?
@Mathmuse Thank you very much for such a detailed explanation, I certainly haven't cracked the math to such detail and it had help me to have a better understanding of the game.
Although I did not include exponential in my initial post I think the premise still stand, why have somethings increase linear, others quadratic, and others exponential, it would not be better to just stick to exponential?
However, Pathfinder abandoned quadratic xp and moved up to exponential xp. The fast progression in Pathfinder corresponds to the old D&D 3.5 progression but changed a little: 1,300 xp, 3,300 xp, 6,000 xp, 10,000 xp, 15,000 xp, 23,000 xp in Pathfinder rather than 1,000 xp, 3,000 xp, 6,000...
PF2 looks like the perfect place to fix this and convert prices to exponential, that way you can have exponential HP, healing, and damage.
On the other hands, if wands switched to operating purely on resonance, with no internal charges that limit the lifetime of a wand, then the load that healing wands could handle would be the product of the spells, which go up linearly, and the resonance, which also goes up linearly. The product of two linear curves is quadratic. Resonance-only wands could handle a quadratic healing load.
Having Wands operate only on resonance would still be linear, you are only going to use the highest level Wand, not the whole series of Cure Wounds.
Trying to answer the OPs questions:
Because it creates better games, better rule systems. If everything were linear, for instance, you are incentivized to "farm" in MMO terms, low level/trivial mobs for exp to rapidly level. If everything is quadratic, you very quickly get situations were each level is too big a jump in power and you can't effectively have encounters with things more than two away from you. (Mooks will never be challenging even in mass)). Over time as Mathmuse indicates designers have been refining the exact non-linear aspects, but they are superlinear for good (not legacy) reasons.
But what Mathmuse show us is that the CR increases exponentially, this is already the case in PF1 and I imagine is going to be for PF2, and yes mooks should not be a challenge even in mass, that is what it means answering NO to Bounded Accuracy.
If HP/healing increased super-linearly, it feeds into the rocket-tag aspect they designers are trying to get away from. If a hit doesn't do enough damage to kill an opponent, they are likely able to heal it all immediately.
Sorry but I did not understand this part, why super-linear will generate more rocket tag than linear HP/healing?
From what I understand rocket tag arises when offensive options growth much fastest than defenses causing defensive options to be meaningless, and given that damage is already increasing super-linearly I think HP/increasing super-linearly too would actually decreases rocket tag.
Sorry but I did not understand this part, why super-linear will generate more rocket tag than linear HP/healing?
From what I understand rocket tag arises when offensive options growth much fastest than defenses causing defensive options to be meaningless, and given that damage is already increasing super-linearly I think HP/increasing super-linearly too would actually decreases rocket tag.This is because if healing increases at the same rate as damage and health, 1 of 2 things happen.
1.) You take enough damage in one hit to die.
2.) You don't die in one hit, and you're healed back to full hp.
This results in battles for players that feel like they have little control over, reduces the fun, and increases the rocket tag.
I still don't follow, why would you receive damage to die in one hit, or why you will be healed back to full instantly? And how is this different increasing linearly?
Damage and HP/healing increasing super-linearly doesn't mean that damage and healing are going to be equal to HP, they are going to keep being a fraction of HP.
Edit: misread the post. The rocket tag is a result of numbers, if the numbers are increasing faster, the game seems to be moving faster, till you're going rocket fast. Hence rocket tag at higher numbers.
I haven't hear that meaning of rocket tag, but anyway this is unavoidable if you want to keep answering NO to Bounded Accuracy.
Although I did not include exponential in my initial post I think the premise still stand, why have somethings increase linear, others quadratic, and others exponential, it would not be better to just stick to exponential?
Linear is very convenient, because all the player needs is addition. We don't want to give up that convenience where linear is good enough.
PF2 looks like the perfect place to fix this and convert prices to exponential, that way you can have exponential HP, healing, and damage.
The PF2 prices of healing potions in the Paizo Blog: Potency and Potions go 3 gp, 8 gp, 20 gp, 60 gp, 250 gp, 1,200 gp. That increase is faster than exponential. The hypergeometric sequence 5, 8, 19.2, 61.44, 245.76, 1179.65, 6606.03, ... comes close. The only hypergeometric sequences most people have seen is the factorial sequence 1, 2, 6, 24, 120, 720, 5040, ....
Having Wands operate only on resonance would still be linear, you are only going to use the highest level Wand, not the whole series of Cure Wounds.
Let me illustrate the progression with a character. A PF2 party has no healer, except a bard who would rather use his spell slots for other spells than Cure Luight Wounds. The party agrees that the bard should use his spells as he sees fit, but they do need healing. At 2nd level, they pool their money and buy an unlimited minor wand of healing, which does 1d8 healing for each resonance spent by the bard (numbers based on the Healing Potions from the Paizo Blog: Potency and Potions). The bard has 5 resonance (2 levels + 3 Cha bonus), so he can heal 5d8 hp (average 22.5 hp) per day with the wand.
At 5th level, the party sells the minor wand of healing and buys a lesser wand of healing, which heals 2d8+4 for each resonance spent. The bard has 8 resonance, but uses one for his +1 armor, leaving 7. He can heal 7(2d8+4) hp (average 91 hp) per day with the wand.
At 8th level, the party sells the lesser wand of healing and buys a moderate wand of healing, which heals 3d8+8 for each resonance spent. The bard has 11 resonance, but uses two for his armor and amulet, leaving 9. He can heal 9(3d8+8) hp (average 193.5 hp) per day with the wand.
The healing of the wands goes up linearly, the spare resonance of the bard goes up linearly, but the total healing is the product of the two, and the product of two linear sequences is quadratic.
Sorry but I did not understand this part, why super-linear will generate more rocket tag than linear HP/healing?
From what I understand rocket tag arises when offensive options growth much fastest than defenses causing defensive options to be meaningless, and given that damage is already increasing super-linearly I think HP/increasing super-linearly too would actually decreases rocket tag.This is because if healing increases at the same rate as damage and health, 1 of 2 things happen.
1.) You take enough damage in one hit to die.
2.) You don't die in one hit, and you're healed back to full hp.
This results in battles for players that feel like they have little control over, reduces the fun, and increases the rocket tag.
Edit: misread the post. The rocket tag is a result of numbers, if the numbers are increasing faster, the game seems to be moving faster, till you're going rocket fast. Hence rocket tag at higher numbers.
In willuwontu's case (1), no healing was applied, so I don't see how that relates to healing. It does relate to edduardco' original idea of hit points increasing super-linearly, which would decrease the chance of this happening.
In willuwontu's case (2), I see assumptions about rates of growth that don't necessarily hold. First, just because healing increases super-linearly, it does not have to increase to match damage per round. Damage per round increases super-linearly already, and we have several super-linear curves to chose from. For example, consider a barbarian's damage. His rate of hitting increases linearly due to linear increasses BAB and in weapon enhancement. His damage per hit increases linearly due to linear increases in his bonus to strength from raging and in weapon enhancement. In addition, his number of attacks per turn increase linearly, too, though he cannot necessarily use that all the time unless he has pounce. That is not just quadratic, that is cubic! If healing is quadratic, it will still fall behind the barbarian's damage.
And suppose we set healing so that the strongest healing spell heals 50% of the damage a fighter or barbarian of that level typically deals in one round. And that an enemy fighter needs 2 rounds to kill the typical PC. The fighter attacks on the first turn and the PC is at half max hp, so the healer brings the PC back up to 75% max hp. The fighter attacks on the second turn and the PC is 25% max damage. Then the PC takes down the enemy fighter and combat is over. Or combt is not over, so the healer heals him back up to 50% damage and the PC charges another enemy. The risk still exists, especially for the other two party members who don't have a dedicated healer keeping them on their feet.
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For the record total healing from Wands is likely higher than that because they duplicate the Heal spell and can thus almost certainly be used for lower amounts in an area.
A Level 5 Wand can be expected to heal 2d8+4 HP (or so) in an area, which totals 13 points of healing for each target, or 52 points of healing if the whole party is damaged, all for 1 Resonance.
Based on price I'd expect to see one of those at around 8th level or so rather than actually picking it up at 5th most times, but it's worth noting.
This doesn't change the main point, but it seemed worth noting if we're talking Wand healing.
Although I did not include exponential in my initial post I think the premise still stand, why have somethings increase linear, others quadratic, and others exponential, it would not be better to just stick to exponential?Linear is very convenient, because all the player needs is addition. We don't want to give up that convenience where linear is good enough.
Fair enough I guess, although I'm not sure how well that aligns with answering NO to Bounded Accuracy
Having Wands operate only on resonance would still be linear, you are only going to use the highest level Wand, not the whole series of Cure Wounds.Let me illustrate the progression with a character. A PF2 party has no healer, except a bard who would rather use his spell slots for other spells than Cure Luight Wounds. The party agrees that the bard should use his spells as he sees fit, but they do need healing. At 2nd level, they pool their money and buy an unlimited minor wand of healing, which does 1d8 healing for each resonance spent by the bard (numbers based on the Healing Potions from the Paizo Blog: Potency and Potions). The bard has 5 resonance (2 levels + 3 Cha bonus), so he can heal 5d8 hp (average 22.5 hp) per day with the wand.
At 5th level, the party sells the minor wand of healing and buys a lesser wand of healing, which heals 2d8+4 for each resonance spent. The bard has 8 resonance, but uses one for his +1 armor, leaving 7. He can heal 7(2d8+4) hp (average 91 hp) per day with the wand.
At 8th level, the party sells the lesser wand of healing and buys a moderate wand of healing, which heals 3d8+8 for each resonance spent. The bard has 11 resonance, but uses two for his armor and amulet, leaving 9. He can heal 9(3d8+8) hp (average 193.5 hp) per day with the wand.
The healing of the wands goes up linearly, the spare resonance of the bard goes up linearly, but the total healing is the product of the two, and the product of two linear sequences is quadratic.
OK now I understand what you were meaning, but in that case the Bard with the Wand is still lagging behind your Cleric example and the damage that needs to be healed, is it not?
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Conversation is moving too fast to respond to some of the points, but the key points to me are:
a) as mattmuse has shown the system uses way of combining linear/near-linear systems to produce super-linear growth.
b) We take as a given that the principle behind the CR progression is what we want to base the entire system around.
c) This means that most individual aspects need to scale at a lower rate than the CR system. Otherwise the combined influences of the different systems exceed the power budget given by the CR.
Ie if HP scaled quadratically/exponentially, then AC would need to be fixed and never changing across levels. Otherwise your eHP is growing faster than the CR allows.
Ie if HP scaled quadratically/exponentially, then AC would need to be fixed and never changing across levels. Otherwise your eHP is growing faster than the CR allows.
I don't think so, AC is competing against Attack Bonus not Damage, if Attack Bonus increase so should AC, and needs to be at the same rate also.
I propose that HP/healing should increase super-linearly so that it could keep pace with the already super-linear Damage.
however number of attacks should grow much more slowly in pf2, so it might drop the dpr back to the quadratic. (Ie number of attacks was linear in pf1, effectively constant in pf2 -- 3 + one from haste-like effects kicking in. Doesn't seem like we expect to see that number grow in a linear manner.)
Ie if HP scaled quadratically/exponentially, then AC would need to be fixed and never changing across levels. Otherwise your eHP is growing faster than the CR allows.I don't think so, AC is competing against Attack Bonus not Damage, if Attack Bonus increase so should AC, and needs to be at the same rate also.
I propose that HP/healing should increase super-linearly so that it could keep pace with the already super-linear Damage.
HP increases linearly as does damage, ac, and attack.
The combination of attack and damage cause dpr to scale quadratically, as does the combination of hp and ac, causing your effective HP to scale quadratically.
If healing could keep pace with damage (as in healing = dpr), my earlier example holds true.
Edit: Now technically dpr is a cubic increase, since your number of attacks factors in as well, similary DR causes a cubic increase in effective HP.
This is only true if healing == DPR, you could have healing keeping pace with damage distributed in a longer time frame, in other words, healing would be a fraction of damage and it would take longer to negate the damage, but still should be enough healing available to negate the damage, like Mathmuse last example shows.
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Ie if HP scaled quadratically/exponentially, then AC would need to be fixed and never changing across levels. Otherwise your eHP is growing faster than the CR allows.I don't think so, AC is competing against Attack Bonus not Damage, if Attack Bonus increase so should AC, and needs to be at the same rate also.
I propose that HP/healing should increase super-linearly so that it could keep pace with the already super-linear Damage.
HP increases linearly as does damage, ac, and attack.
The combination of attack and damage cause dpr to scale quadratically, as does the combination of hp and ac, causing your effective HP to scale quadratically.
When we think about scaling in that way, we should think about it in comparison to where we once were in all regards, as the factors really combine to make the whole character:
Imagine a character that only scaled attack bonus / DC and damage, while keeping HP, saves, and AC somehow equivalent, so we're looking only at offense. Damage dealt actually does scale roughly linearly based on level, although piecewise jumps based on things like potency increases, so that's roughly a linear increase. Accuracy increases are linear in the number of successes you get, but they can often produce a pretty large multiplier on your damage vs the same AC. Going up two levels could easily produce around 50% more damage from accuracy increases alone, possibly more if the levels included multiple non-proficiency accuracy boosts. If our level 6 goes up to level 8 and now does roughly 4/3 the damage with 3/2 the hits from accuracy, that's doubled offense.
A doubled offense is worth two characters with normal offense, but in truth, there's also defense scaling in the same way. You'll have higher AC, meaning you might take perhaps 2/3 the damage from hits as before if you fight the same enemies, and you have more HP too, this time superlinearly over enough levels as you gain a set amount per level but your Con stat boosts increase the amount per level. We don't have that happen leveling from 6 to 8, so we get under 4/3 the HP (because of the starting racial HP), giving us slightly less than double the lasting power against those sorts of attacks.
Our true strength is a factor of both our offense and our lasting power, plus we also probably learned some cool new abilities beyond the pure math increases to make our character more powerful, so really we're more powerful than that. In a pure fight alone, supposing we fight a single foe that we could barely defeat, we can use both our offensive boost and our defensive boost to defeat two such opponents (though if we can avoid being flanked, it's probably easier than before if we focus down one of the opponents first, as we then start taking less damage past that point). If we fought the same single opponent alone as before, instead of barely winning, it would only take out about 1/4 of our HP (our HP and AC scale such that we take proportionately half of our HP each turn from the foe, and we deal roughly double damage to it).
So if we consider that foe that we used to be able to fight only one but now could take four one at a time in rapid succession with no healing in between, in some ways we're kind of four times as strong as before. It's not a perfect analysis for real play though, as the defensive benefits only help you if you're the one being attacked. However, when it does apply, you're sort of multiplying four linear factors together, for a quartic increase in math alone; multiply in some factor to represent the increasing power of your special abilities and feats and it's kind of a quintic increase.
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stuff
Posts like this are why I'm glad they hired Mark.
stuffPosts like this are why I'm glad they hired Mark.
Indeed
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The rocket tag is a result of numbers, if the numbers are increasing faster, the game seems to be moving faster, till you're going rocket fast. Hence rocket tag at higher numbers.
It's what the term "rocket tag" originates from. If you consider lower numbers (at level 1) as a walk, as you go up numbers you go into a jog, running, sprint, and then you're strapped to a rocket trying to hit (tag) enemies, who are also strapped to rockets and trying to hit you.
That's...not at all what "rocket tag" means. It's about the lethality of the attacks. If everyone can kill each other with single shots, then the first one to hit wins. It's essentially a game of tag at that point. The "rocket" part comes from rocket launchers, which are likely to kill in a single shot in most games.
This is a common term (or was when I was a kid, anyway) in first person shooters and the like, and was typically a game mode. The one for Halo (called "Rockets") where everyone has rocket launchers was often played in a relatively close arena means that it's a fast-twitch game of rocket tag where everyone dies in one hit, but no one is actually moving faster (although anyone who stopped moving would be killed very quickly).
In Pathfinder and other pen/paper games, it comes from the increasing likelihood that someone goes from "fine" to "super dead" in single attacks, whether through pure damage or save/die spells. Part of this has already been discussed, with hit points not scaling with level as fast as damage does. It's amplified by the death system from 3.x/PF1, where the amount of negative hit points required to actually die did not scale with level, so after a certain point you would almost always jump past "dying" to "dead" because that was only a 15 point window or so.
The death/dying system in PF2 completely eliminates the need for negative hit point scaling (because you can't go below 0 and the dying condition escalates separately). I think it's super elegant so far. With 0 other changes to HP and attacks, the dying rules make the "instant death" aspect of rocket tag gameplay less likely.
If you want a bunch more examples across different games, or even someone else explaining that it's about the lethality of attacks...here's the tvtropes link.
https://tvtropes.org/pmwiki/pmwiki.php/Main/RocketTagGameplay
lotsa good stuffNow that I look the link and back on memories, you're correct (though that makes me wonder when my perception of rocket tag changed). Still it is a factor of the numbers. And despite pf2 lowering the likelihood of it happening, it'll still probably happen in the end game (though the method of reduction is quite elegant as you say).
Yeah, they left some room for deadliness with the danger of critical hit putting someone straight to Dying 2 followed by a crit fail on the fortitude save, which I think takes you to dead. Or in the Glass Cannon podcast, where Jason had the mindless undead keep attacking the nearest "living" thing whether or not it was conscious, where each hit kept escalating the dying condition.
Also, I found this cool thread back from March on resonance nerd math where Mark suggested some damage numbers and Dasrak ran simulations showing how unlikely you were to be knocked below zero without also straight-up dying in PF1.
There is a quote from, I believe, SKR that states that for fighters and the like to approach the flexibility of casters they'd need more flexibility in swapping out feats--daily, for example.
I may be misremembering.
There is a quote from, I believe, SKR that states that for fighters and the like to approach the flexibility of casters they'd need more flexibility in swapping out feats--daily, for example.
I may be misremembering.
The brawler class from the Advanced Class Guide had Martial Flexibility, gain any combat feat the character qualified for at the cost of a move action.
But combat feats are oriented toward combat, which the fighter already excels at. What I like about the flexibility of casters is their non-combat utility. And wizards are famous for their mastery of battlefield control. Battlefield control is possible with a martial; for example, my wife managed it with a gunslinger.
Flexibility is hard to see on the power scale of linear vs quadratic vs exponential. Swapping out one option that does not fit the circumstances for another option of equal power that fits the circumstances still counts as equal power.
I think the biggest issue is what the numbers represent.
In 3.x, the numbers meant specific things in the narrative milieu, but paizo and just about everyone else has chased after the mmo craze of numbers meaning absolutely nothjng beyond the mechanics.
Frankly, in my opinion, if you want to ignore what the numbers mean narratively, then every level should be the same relative to the other levels. By this I mean lvl 5 compared to lvl 3 should be identical to lvl 9 compared to lvl 7, with the exception of the number of available choices, a.k.a. the flexibility.
If you do that though, then you can't directly associate lvl with power, otherwise you get demigods that can kill the world with a word. Instead, everything needs to be powered up of down regardless of narrative description to provide appropriate challange. For example, orcs would get powered up to remain roughly similar in power compared to the pcs, losing ground far more slowly than the gaining of lvls.
Some decisions need to be made here though, such as what range of lvls should be viable threats (-+ 3 lvls, or 7?) and how many levels should be available to be gained over the career of a character. Both of those are important as they change the needed rate of level growth and power growth per level.
Of course, I really do prefer numbers grounded in the narrative milieu, which means power really needs slowed down by a lot, or even better, separate raw power from the amount of flexibility one has, thus allowing a polymath that is still a natural human being.
In 3.x, the numbers meant specific things in the narrative milieu, but paizo and just about everyone else has chased after the mmo craze of numbers meaning absolutely nothjng beyond the mechanics.
Do you have an example? I remember that in 1st Edition D&D, experience points were tied to treasure amount, 1 gp equalled 1 xp, but that link was broken before 3rd Edition.
Frankly, in my opinion, if you want to ignore what the numbers mean narratively, then every level should be the same relative to the other levels. By this I mean lvl 5 compared to lvl 3 should be identical to lvl 9 compared to lvl 7, with the exception of the number of available choices, a.k.a. the flexibility.
Pathfinder does that. That is exponential progression.
If you do that though, then you can't directly associate lvl with power, otherwise you get demigods that can kill the world with a word. Instead, everything needs to be powered up of down regardless of narrative description to provide appropriate challange. For example, orcs would get powered up to remain roughly similar in power compared to the pcs, losing ground far more slowly than the gaining of lvls.
It depends on your definition of demigod. By Pathfinder's CR system, a 20th-level character would be as powerful as 724 1st-level characters. That is an impossible one-man army, but not someone who can kill the world with a word.
GM DarkLightHitomi, what are your narrative descriptions? To me, 1st level feels like a beginner, called fighter or wizard or cleric mostly due to potential rather than current abilities. 2nd level feels average, a common townsfolk. 3rd level feels capable. 4th and 5th level are veterans, people experienced in adventuring that common townsfolk turn to for rescue. 6th level are masters of their craft--if their profession had a guild, they would be asked to serve as guildmaster. 7th level is extraordinary. 8th level is an Olympic medal winner, and at 9th level that medal is definitely gold. At 10th level, the PCs are people of legend, though more like Paul Bunyan and Pecos Bill and Johnny Appleseed (United States folktales, though John "Johnny Appleseed" Chapman was real) than demigods. Each one can do the work of 16 common 2nd-level people.
The advantage of exponential progression in character abilities with level is that every level feels fresh. The player has the fun of mastering new abilities at each level without being overwhelmed by too many new abilities. Going up a level makes a character 41% more powerful, which is room for a significant new ability.
Contrast that to a linear progression. Imagine that the level numbers, 1, 2, 3, etc., were proportional to the power of the characters. Going from 1st to 2nd level would be a massive 100% increase in strength. From 2nd to 3rd is a significant 50% increase. From 3rd to 4th would feel a little short, only 33%, but still would gain new abilities. But when we reach 10th level? Advancement from 9th to 10th is an 11% increase. That would be covered by maybe a +1 to BAB and one more Hit Die of hit points. The only way to fit a new ability in there would be by depriving the character of advancement for a few levels to make room.
Dungeons & Dragons 3rd Edition had a quadratic progression, with more room for improvement at high levels than a linear progression. But we still had dead levels. Take a look at the D&D 3.5 Barbarian. At 9th level, his special ability is that his trap sense goes up to +3. His saves don't even improve. Boring, though he does receive his long-awaited 4th feat then. For the D&D 3.5 Rogue 14th level is a true dead level: +1 to BAB, +1 to Reflex save, the usual hit points and skill points, no new ability, no new feat.
Only exponential progression has room to make each advancement interesting.
Read the Alexandrian's Calibrating your expectations.
Take the Hobbit/Lotr. Gandalf is level 5 or possibly 6. I can't think of him ever using magic above a 3rd lvl spell.
A lvl 1 is a professional. Nothing special perhaps, but someone who is competant at their job. A college graduate even.
A lvl 2 is someone that is superior, someone that is a cut above the common folks.
A lvl 3 is an extraordinary person, someone who is a star at what they do. These are the folks that make a real difference in their profession. The guys who invent the techniques that everyone else uses.
A lvl 4 is a grandmaster, someone who understands their craft to a degree that lvl 1 only wonder at. These are the mentors of lvl 3s. These are the guys who get their names into history books.
A lvl 5 is someone who changes the world. Someone who goes beyond theif profession, beyond just a group, and becomes a household name. Princess Diana, Einstein, Hitler. For good or ill, these are the people who shape the course of history.
Beyond them are myths and legands. The demigods and demons.
For me though, the going up in level is not about skill nor ability, but in agency.
A lvl 1 is someone who just acts as taught. They don't do much beyond simply doing the job.
A lvl 2 is someone that really investigates what they are doing and tries to achieve a better result, and therefore is generally more confident and competant. These are often the folks others turn to when stuck because these guys are the ones who can handle it when thjngs go "off script." These are the people who read the broken printer and notice that it says "paper jam."
A lvl 3 or 4 is someone that goes chasing after their goals and dreams with confidence and mindset to accomllish something. The explorers of new things. These are the guys that took computers and made them do more than simply calculate numbers. These are the visionaries that see possibilies beyond the intended purpose, beyond the obvious.
A lvl 5 is someone who sees that common sense is flawed, that everyone's worldview is messed up, and then reconciles this and is able to build on it to accimplish things others claim are impossible.
As to dead lvls, that has nothing to do with linear vs quadratic. That has to do with scope, with how quickly or slowly one gains flexibility and breadth rather than height.
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As to dead lvls, that has nothing to do with linear vs quadratic. That has to do with scope, with how quickly or slowly one gains flexibility and breadth rather than height.
Flexibility and breadth are dimensions of power. How quickly or slowly you gain those affects your power curve just as much as increasing numerical bonuses.
There is a quote from, I believe, SKR that states that for fighters and the like to approach the flexibility of casters they'd need more flexibility in swapping out feats--daily, for example.
I may be misremembering.
I believe that you are right. SKR's "upcoming" game Five Moons uses a system like that. Spells and martial feats ("Cronks" he calls them) are more or less the same thing in that system but with thematically different mechanics and benefits.
I think that is certainly an interesting approach, but I am not sure if I find it satisfying and I am not sure that SKR is 100% correct;
you can balance a game where one character has a lot of varied utility limited by finite resources and another character has more focused utility but is not limited by resources.
Wielding a sword = the D&D/PF Fighter in novels...crikey, so obnoxious and stupid, it's embarrassing, just stop.Take the Hobbit/Lotr. Gandalf is level 5 or possibly 6. I can't think of him ever using magic above a 3rd lvl spell.Ha, not this garbage again; Gandalf is a Maia (Order of the Istari), an Angel, Celestial, Demigod servant-type of Manwe. He is not a human wizard, or anything so mundane.
JRR was also not writing with Gary's writing in mind.
No, he is not as mundane as a wizard. He is more mundane than that. Every magical ability we see could be pulled off by anyone using that gear, and what do we see when he's up against the only opponent that gives him a run for his money? He drops the act as soon as it closes, and fights with a sword.
Angel or no, Gandalf's just a Fighter.
Maybe he is a PF2 fighter with lots of skill feats invested in Arcane.
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I first read The Alexandrian's D&D: Calibrating Your Expectations article years ago. The article was written before the Pathfinder Core Rulebook was published. I reread it again today.
The Alexandrian wrote about the minimum levels that heroes and townsfolk would require for their accomplishments. For example, Gandalf from The Hobbit is a powerful wizard, so his D&D level must be in the teens, right? No, he could have managed everything as a 4th-level wizard. (From other sources, we learn Gandalf is a Maia acting below his full capacity, but let's focus on how Gandalf presented himself to others). His spells were pretty low-level, mostly flashbangs and speak with giant eagles without any Fireballs. His most impressive stunt was that he wielded the sword Glamdring with skill, perhaps representing a +3 BAB and a bit of multiclassing for martial weapon proficiency. So 5th level, wizard 4/ranger 1.
Likewise, The Alexandrian demonstrates that a gifted (+4 Int bonus) blacksmith who devoted his life (and feats) to blacksmithing could manage to regularly turn out masterwork exotic weapons as early as 5th level. No-one in town needs to be above 5th level to keep the market stocked for the adventurers' needs.
The Alexandrian even claimed that a village can get by with a 1st-level blacksmith. However, he played fast and loose, because to reach +10 to Craft(weapons) the village blacksmith needed an apprentice who could always make the DC 10 Aid Another rolls. Sounds like the apprentice has a better unaided bonus than his boss.
For an example from real blacksmiths, I regularly watch a televised blacksmithing contest called Forged in Fire. In the first two rounds, four blacksmiths compete to create a simple melee weapon in 6 hours. In Pathfinder, a simple melee weapon has craft DC 12. The judges throw in extra requirements, such as defective materials (rusty lawnmower blades) or exotic techniques (canister damascus), in order to prevent the Take 10 of routine crafting. Pathfinder's rules for the rate of crafting are ridiculously messed up--higher DC items are made more quickly--but we viewers can see that the smiths are working at an exhausting pace. The extra requirements and the pace raise the DC, so let's call the crafting DC 22. The success rate is about 75%, so a d20 roll of 6 or higher is necessary. The blacksmiths, therefore, must have a +16 to Craft(weapons). If we subtract the +3 for Craft(weapons) as a class skill, the +3 for the Skill Focus(Craft(weapons)), and the +2 for masterwork tools, that leaves a +6 bonus for skill ranks and intelligence. So the contestants range from natural genius 2nd-level blacksmiths to average-intelligence 6th-level blacksmiths. (Or I could have pulled too many numbers out of a hat in trying to match Pathfinder skills to real life blacksmithing.)
I can accept The Alexandrian's claim that no-one in the towns and cities of Pathfinder has to be higher than 5th level to be able to perform their jobs. But that does not mean that everyone in the entire world is 5th level and below.
Does this mean you should never throw a 10th level blacksmith into your campaign? Nope. D&D is all about mythic fantasy, after all. But when you do decide to throw a 10th level blacksmith into the mix, consider the fact that this guy will be amazing.
My Iron Gods party excelled as smithing. They forged mithril and adamantine. Any forging of those mythic metals is amazing. I let them do things no-one else in Numeria did, such as convert the nuclear fusion powerplant of an annihilator robot into an adamantine smelter. It was appropriate for 12th level.
A lvl 5 is someone who changes the world. Someone who goes beyond theif profession, beyond just a group, and becomes a household name. Princess Diana, Einstein, Hitler. For good or ill, these are the people who shape the course of history.
Going from "no-one has to be higher than 5th level" to "A lvl 5 is someone who changes the world" is quite a jump. Okay, The Alexandrian modeled Albert Einstein as a 5th-level physicist. "He’ll know the answers to the very hardest questions (DC 30) about 75% of the time," and with aid of scientific journals (+2 circumstance bonus), fellow colleagues (+2 bonus from aid another), and top-of-the-line equipment (+2 circumstance bonus) this 5th-level physicist could sometimes manage DC 40.
The Alexandrian made a big mistake. He created a generic dedicated 5th-level physicist and dubbed this character Albert Einstein. If we measured Albert Einstein's skill as a physics professor--lecturing, grading tests, answering routine questions about physics--maybe he was only 5th level. But that does not imply the reverse, that all 5th-level physicists are Albert Einstein. Would you insist that Enrico Fermi, Neils Bohr, John von Neumann, and Richard Feynman were only 4th level? Despite all the other physicists of equal Knowledge(physics), only Albert Einstein invented the Theory of Relativity.
What is the DC of inventing the Theory of Relativity? The answer is that there isn't one. That is not the kind of thing covered by Knowledge skill checks. Those checks relate to known facts, not new ideas. Changing the world is not accomplished by a skill roll, no matter how high the skill bonus.
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For the record, I actually mostly agree with the Alexandrian article and would cap people on Earth at level 4-6.
However, Golarion is not Earth, and much more powerful people than that clearly exist, making someone who would be extraordinary on Earth merely mediocre on Golarion as compared to the truly extraordinary people in that world. It's much like a superhero universe in that regard. People who are better then the best possible warrior on Earth (say, a 10th level Fighter) are still nowhere near world shaking or the best in the world on Golarion because there's that 20th level Fighter over there who has that honor.
Lame, and none of that has anything to do with a novel written decades before D&D (PF).
What are you calling lame, my dude? Mathmuse is just sharing stuff from a fun little thought experiment w.r.t. how the rules of DnD can or cannot successfully model a full universe.
Or the apprentice could just take 10 and hope the modifier isn't negative.[...]
The Alexandrian even claimed that a village can get by with a 1st-level blacksmith. However, he played fast and loose, because to reach +10 to Craft(weapons) the village blacksmith needed an apprentice who could always make the DC 10 Aid Another rolls. Sounds like the apprentice has a better unaided bonus than his boss.
[...]
Aid Another
You can help someone achieve success on a skill check by making the same kind of skill check in a cooperative effort. If you roll a 10 or higher on your check, the character you're helping gets a +2 bonus on his or her check. (You can't take 10 on a skill check to aid another.) In many cases, a character's help won't be beneficial, or only a limited number of characters can help at once. ...