Treasure and magic item prices -- smooth curve or no?

[mneme]

I killed the "how much treasure do you get going between level 11 and 12" thread with this concept, so I figured I'd give it its own thread instead.

I hadn't looked at the numbers, but an examination of Wizards' magic item prices (and by extention, their treasure guidelines for how much gold players get) reveals some really odd discontinuities.

We all know the theoretical rule -- an item that is 5 levels higher always costs 5x as much. But a cursory examination of the treasure tables reveals a very interesting fact: The intervening values don't actually follow this progression--instead, it looks like Wizards picked some numbers that at the +2 number resemble the actual just tempered curve (more on this later), and then just multiplied them all up the level chart.

This means that in every sub-tier (1-5, 6-10, etc), the numbers/costs are relatively flat (level 2, 6, etc items are overpriced, but there's a sharp dropoff as you approach the end of the subtier) -- and there's a sharp jump as you cross subtiers, when the cost/treasure jumps from less than 3x the base cost of the tier up to 5x that number--nearly doubling.

The problem is, we have a really good method of tempering these numbers so that the curve is even -- in fact, we've had such a method since the 1700s, when it was developed as a method of tuning an insturment (typically, a piano) with even spacings between the notes so it would be equally in tune [or out of tune ] in every key. It's called "equal tempering", it works by making the relationship between the steps multiples of (period) to the power of ({iteration, starting at 0} / (period)) -- so for the D&D example, the a first level item will cost 360 * (5 ** (0 / 5)) = 360, a 6th level item will cost 360 * (5 ** (5/5)) = 1800 -- just like in PH1 -- but using this calculation method, every single cost will have exactly the same relationship to the previous cost (barring corrections for rounding) as the one before it did -- a smooth curve.

Admittedly, there is one big advantage to the PH1 system: it under-prices 5th, 10th, etc level items. As those items typically -are- substantially less powerful than the items just one level higher, there's some argument for doing so. OTOH, given that the characters are getting the same item set over each level as the one before, there's not a great argument for doing the same with player treasure.

As such, if I were to run a game, I'd be tempted to stick with the PH1 numbers for item costs, but give treasure along the true curve. This does give out a bit more treasure than the PH1 version suggests (a party gets an extra 700 by 5th level, an extra 5k by 10th level, an extra 25k by 15th level, etc), but it's just not a huge deal.

FWIW, here's the 6 line perl one liner I used to produce my comparison data (I'm working with the "amount of treasure earned by the party between levels" number, which is 2x the item cost for that level, but the ratio between numbers is identical.

perl -e 'use strict; my $sum; use POSIX("ceil"); my @adds; @adds[1..29] = ((320) x 4,(1600) x 5,(8000) x 5,(40_000) x 5, (200_000) x 5,(1_000_000) x 5 ); for(0..29) { my $tres = ceil(720 * (5)**($_/5)); $sum+=$tres; print "$tres/$sum "; (($_+1) % 5) ==0 and print "\n"; }; print "\nsmooth total: $sum\n"; $sum=0; for my $lvl (0..29) { my $tres = 720; for(my $i=0; $i<=$lvl; $i++) { if($adds[$i]) { $tres+=$adds[$i] } } $sum+=$tres; print "$tres/$sum "; ($lvl+1) % 5 == 0 and print "\n"; } print "\nwotc total: $sum\n" '

And here are my results (first my expected/smooth numbers, then, for comparison, I recalculated Wizards' numbers by the formula MrMyth gave in the how much treasure thread -- the first number in each pair is the treasure (2x the item cost) for that level, the second is the total earned by the party over the entire course of the game:

720/720 994/1714 1371/3085 1892/4977 2610/7587
3600/11187 4968/16155 6854/23009 9456/32465 13047/45512
18000/63512 24836/88348 34266/122614 47278/169892 65231/235123
90000/325123 124176/449299 171329/620628 236388/857016 326151/1183167
450000/1633167 620879/2254046 856645/3110691 1181938/4292629 1630755/5923384
2250000/8173384 3104392/11277776 4283222/15560998 5909688/21470686 8153772/29624458

smooth total: 29624458
720/720 1040/1760 1360/3120 1680/4800 2000/6800
3600/10400 5200/15600 6800/22400 8400/30800 10000/40800
18000/58800 26000/84800 34000/118800 42000/160800 50000/210800
90000/300800 130000/430800 170000/600800 210000/810800 250000/1060800
450000/1510800 650000/2160800 850000/3010800 1050000/4060800 1250000/5310800
2250000/7560800 3250000/10810800 4250000/15060800 5250000/20310800 6250000/26560800

wotc total: 26560800





Any thoughts?

 

[Nolan]

the equation itself

could you give it to us in mathematical terms? in some way that is not so hard to plug x = lvl and get the gold the party should get &/or have at that level? maybe even if you have to plug that into a graphing calculator.

on a related note, has anyone ever created a formula for how much equivalent gp a monster of a given level should drop, considering what fraction of the xp that monster provides toward the next level?

 

[mneme]

Nolan: I actually did do that; look at the perl code above (** = power, / = divided by, * = multiplied by, $_ = level).

The key formula is this (pulling it directly out of the perl above):
ceil(720 * (5)**($_/5)); # for numbers from 0 to 29

In English, that's:
If you're calculating a smooth curve where val(lvl+5) = v*(val(lvl)), then the value of a magic item of (lvl) should be 360 times 5 to the power of (level minus 1 divided by 5), and the amount of treasure a party gets between any given level and the next one should be twice that -- id est 720 times 5 to the power of (level minus 1 divided by 5). Since this is an irrational number except at divisions by 5, you'll have to round slightly, but it won't mess with the math much.
Using this formula, the treasure at level 1 will be 720 * ( 5 ** (0/5) ) = 720, and at level 6: 720 * ( 5 ** (5/5) ) = 720 * 5, with a logarithmically smoth curve between them (you can actually start the formula at any point; I'm just starting it at level 1 to synch with the base system).

The total, btw, is harder; you more or less have to do that iteratively:

At level X, the total gold the party should have (using a smooth curve, not the WOTC's curve), and not incluidng hte starting 100 gp is:
sum( as X goes from 0 to level-1) : 720 * (5 ** (X / 5))

You could consturct a similar formula to get the numbers I calculate for the official, non-smooth numbers, but it's more annoying (as you end up having to add 320 * the half-tier * the distance between level and 5 to the "base" value for that half tier (720 * 5**HT) ) and not as mathematically interesting.

 

[the Jester]

While I admire mathematically beautifully systems, I think you're fixing something that isn't broken, and I'd say that the pricing discrepancies are a feature, not a bug.

Just my opinion, though.

 

[mneme]

As I said above, it -is- to an extent a feature, as items (or, at least, items with plusses) across a half-tier boundary -are- substantially more powerful than even those one level down.

The issue (from my pov) is that while it's worth deflating, 4th, 9th, etc level items, I don't see that deflating the amount of treasure a 4th, 9th, etc level party gets has similar value; they're often fighting monsters of level + 2 or so -- so (particularly at level 8 and 9 and level 18 and 19), they're fighting -harder- monsters than they'd be hitting early in the half-tier -- and getting stronger items (as most items they win will cross the half-tier boundary), but getting a dispurportionatly depressed gp reward.

Thus, were I running a long running campaign, I'd probably shoot for the smooth curve in terms of how much gold the party got between levels -- but keep the prices of items from the book rather than adjusting them to fit.

 

[eamon]

As I said above, it -is- to an extent a feature, as items (or, at least, items with plusses) across a half-tier boundary -are- substantially more powerful than even those one level down.

The issue (from my pov) is that while it's worth deflating, 4th, 9th, etc level items, I don't see that deflating the amount of treasure a 4th, 9th, etc level party gets has similar value; they're often fighting monsters of level + 2 or so -- so (particularly at level 8 and 9 and level 18 and 19), they're fighting -harder- monsters than they'd be hitting early in the half-tier -- and getting stronger items (as most items they win will cross the half-tier boundary), but getting a dispurportionatly depressed gp reward.

Thus, were I running a long running campaign, I'd probably shoot for the smooth curve in terms of how much gold the party got between levels -- but keep the prices of items from the book rather than adjusting them to fit.

I think you're right; I also think it doesn't much matter. Also, have you considered that treasure parcels are actually handed out in the form of items-or-item-value-equivalents per DMG guidelines - and that these items are of various levels? That's a form of rudimentary smoothing.