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This is something that I've been trying to work out for awhile. If others have insight on this issue I'm all ears.
As an example, take a 1st level Paladin and give him a Holy Avenger:
Holy Avenger
Aura strong abjuration; CL 18th Slot none; Price 120,630 gp; Weight 4 lbs.
DescriptionThis +2 cold iron longsword becomes a +5 holy cold iron longsword in the hands of a paladin.
This sacred weapon provides spell resistance of 5 + the paladin's level to the wielder and anyone adjacent to her. It also enables the paladin to use greater dispel magic (once per round as a standard action) at the class level of the paladin. Only the area dispel is possible, not the targeted dispel or counterspell versions of greater dispel magic.
Construction RequirementsCraft Magic Arms and Armor, holy aura, creator must be good; Cost 60,630 gp
With a 1st level Paladin swinging that around, how does it affect the APL of the party?
You're typical 1st level party of Pathfinder characters are going to be able to handle APL+1 just fine. Normally you wouldn't want the party to deal with anything beyond APL+3. However, when you dump that much gold value onto a party, things are a bit different.
A really epic fight for four 1st level characters might be fighting an Owlbear. It has an AC 15 and 47 hitpoints.
The Paladin might have a 16 strength. He could double hand the Holy Avenger and end up with something like +9 to hit (+3 str, +1 BAB, +5 HA) and doing 1d8+9 damage (+4 str, +5 HA). That ends up with a 75% to hit chance and taking away about a 4th of the hit points from the owlbear in one hit, half with a crit.
In the same party you could technically have a Barbarian with a 20 Strength. When he rages it's 24 which gives a greatsword attack of +8 to hit (+1 BAB, +7 Str) and 2d6+10 damage. That's a 70% to hit and 17 average damage which is more than a third and if crit then more than two-thirds of the owlbear's hitpoints.
So this is kind of the mystery I'm trying to sort out, is how wealth value affects the APL of a character? It doesn't seem to have a clear value and instead is so wedded to a particular treadmill evolution of the character within an assumed system progression.
Another way of looking at it would be to get a Manual of Gainful Exercise +1. It's worth 27,000gp and gives you a permanent +1 to Strength. If you had a 1st level character read through it, it ain't all that big of a deal, at best getting their character to a 21 strength, which doesn't have any impact on the game.
Likewise, a Gem of Seeing (75,000gp) handed to a 1st level character just isn't that big of a deal. It's nice to have, but even if you used it in the middle of combat it requires a standard action, so it's only really useful to figure out things when people aren't trying to kill you.
The thing is that the gold values don't seem to correlate very well with the CR system. If you stay on the treadmill then everything fits into place, but the more you deviate from the wealth by level the more chaotic the results can be. It might have no affect at all, or it may throw the math off too far, or perhaps it will fit within it but the wealth value is so off base that it doesn't fit with the economy of the game state.
The major reason I'm wondering about this is when you have a campaign where people end up having wildly different levels. If a bunch of high level characters take on a newly minted 1st level character and tell the guy to duck and but also shower the low level guy in gold, how does this affect the underlying math?
I know one answer is “don't do that” but I'm not concerned with that. I'm trying to see if there is any underlying math that can be used to at least gauge how to adjust encounters when things are totally off the rails.
If you pour 400K in gold pieces onto a 1st level party, how much does their APL change in a way that might be considered somewhat reliable?

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He will live much longer if he was ingesting 115.000 gp in armour and shield and the leftover for a weapon.
Some equipment will imbalance the game other will not do so much. It is not directly related to the GP value of the item but to the utility for a first level character. So there is not a monetary math that will allow you to gage the effect on encounters.
The math you need to check is the the number needed to hit the party members, the damage they do and the survivability they get with the extra equipment.

Hyperion-Sanctum |

If you pour 400K in gold pieces onto a 1st level party, how much does their APL change in a way that might be considered somewhat reliable?
Ummm... depends on what they're buying really....
can 400K buy the Paladin +5 Full Plate with an additional +1 or +2 enchantment on it?
If so his AC will be through the roof at first lvl.
The issue here is the balance of stats:
Giant AC probably means still crap HP, so a monster than CAN hit the character WILL kill the character.
Giant attack basically does the same thing to the villains, to the point where you have to up the CR of the baddies to be hit less often, but then if they hit the PCs... they kill them....
Enforce balance, and you could probably add +5 to the level no problem.

Robb Smith |

A 1st level paladin with a holy avenger can still easily take a dirt nap from a falchion crit from an orc, so my answer is "not by as much as you'd think".
The disparity would probably start to show more in the middle levels I'd really think. I'd say a party might be able to go a little higher in low levels (like, perhaps up to APL+4 instead of +3)), a fair amount higher in mid levels (5-9) (Up to APL+6?), and a depending on if they curve out or keep getting insane sums of money, stay the same in the higher levels (10-15).
You'll note I don't talk about 15-20 really, and that's because as far as I'm concerned, most things numerically start breaking down a bit past this point. When it starts taking players about 5+ minutes to take a turn and they're legitimately going as fast as they really can, the game stops being fun for myself and most of my friends, so we don't play above 15 much.
I agree with those above me - if you do this, you introduce too much variability into the system, and making things more random = making things more dangerous for the PCs.

Are |

If the party gets one single item of a very high value, that probably won't change the challenge they face from monsters by very much.
However, the same total gold value split into multiple items likely will, as the party will be better equipped to deal with multiple weaknesses, and multiple offenses.
I don't think it's possible to find any "hard rules" for how much APL is increased based on increased wealth. The best such rule is "don't give too much wealth" :)
The Holy Avenger's greatest affect on encounters will be against creatures with higher-than-average AC or creatures with DR. The addition of the +5 weapon will make such encounters easier than they would otherwise be.

Preston Poulter |
This reminds me of a story told by a now dead friend of mine who was the first person I ever really role played with.
Back in first edition days, he had a starting level Paladin, that the GM decided to give a Holy Avenger to as a "sword in the stone" kind of plot hook. But, so as to not unbalance the encounters, he role played the character as feeling as though he were unworthy of the sword, so he just carried it around. There was one encounter where they were fighting a devil and losing where he felt this was a challenge worthy of the sword, so he used it in that one encounter.
I miss him.

Ashiel |

Post
First, it depends on a number of factors. You drop 400k on a party (or heck, every party member). Are there any limits for what they can buy with it, or are we just assuming "anything within a total value of 400k"?
If it's the 1st, I'd say it probably won't affect much. This is exceptionally true if you have to craft your own items, as the enhancement bonuses for items are limited by caster level (you must be 15th level to craft a +5 armor, for example), which means you can give them all the gold in the world and they won't overshoot their power level by very far.
If it's the second, then you're getting into a can of worms. The main reason being the fact that some pieces of equipment are more useful at different levels. If you have a +5 suit of mithral full plate, a +5 cloak of resistance, energy resistance 10 vs acid, fire, cold, electricity, and sonic on your armor, and a 5/day casting of stoneskin, you will be near immortal as a 2nd level character (or more).
I'd actually go so far as to say offensive items are less powerful at low levels than defensive items are. Being able to deal an average of 22 damage per hit twice per round isn't that amazing when enemies only have 5 hit points, and can get a 1d12+3 hit in on you while you're cleaving them down.
In the first case, I'd estimate maybe APL +2. I've actually ran games where money was not an issue (face it, by about level 13, PCs have access to infinite wealth if they desire it), but they were basically told they could get what they could craft, and what they could find in adventures. In a way, this actually made the game take a very nice pace. They upgraded their equipment as they leveled (higher +X bonuses as desired), and didn't worry too much about splitting loot. Instead, they focused more on what they wanted to do with the loot to change the world (building towns, starting guilds, etc).
In the second case, I'd have to say it entirely depends on the item. Some items are just not intended to be used by low level characters, just like some spells aren't. Take a level 3 wizard who can cast shapechange and you get similar results.