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May 18, 2013, 9:32am

Original D&D Discussion :: Dungeons & Dragons (1971-1978) :: Monsters & Treasure (1974) :: Analysis of OD&D treasure types

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geoffrey
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Re: Analysis of OD&D treasure types
« Reply #15 on Aug 10, 2012, 9:52pm »

These are fascinating analyses!

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Re: Analysis of OD&D treasure types
« Reply #16 on Aug 10, 2012, 11:04pm »

Quote:

If there's a better way to determine the average worth of a treasure type I'll be happy to adjust my figures accordingly.

edit: I suspect the above is pretty straight forward. What is less obvious is how to get the value of gems and jewellery, and I suspect that this may be where differences have appeared.

Gems are likely to be the biggest difference, but I found a couple of unrelated bugs in my script. I'll report back once I squash them.

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Re: Analysis of OD&D treasure types
« Reply #17 on Aug 11, 2012, 12:21pm »

After fixing a couple of bugs in my treasure type analysis script, my results look more like those supplied by waysoftheearth.

Even averaging several thousand runs, the high degree of randomness in treasure generation gives some differences in values from one run of the script to the next. For copper, the difference is one or two gold pieces. For gold, the difference is one or two hundred gold pieces either way. Still, the rankings of the treasure types relative to each other remains constant.

[image]

This is the chart waysoftheearth provided:

[image]

My copper values are about twice his, so he's probably using 1 gp = 100 cp, while I'm using 1 gp = 50 cp. Our silver and gold values are very close.

The remaining question is the gem and jewelry values. My values are much higher than wayoftheearth's, and change the treasure totals enough to change the relative rankings of treasure types.

Quote:

(Gems are worth a mean of 55 gp each, while a piece of jewellery is worth an average of 1,137 gp).

My average values are 419 gp for gems and 3,410 gp for jewelry. My reading of gem and jewelry generation may be wrong. Let's take a closer look at that. Hopefully, one of you can set me straight.

Here's how I generate jewelry:

roll d100
if d100 <= 20, jewelry value is 3d6 times 100
otherwise, if d100 <= 80, jewelry value is 1d6 times 1,000
if d100 >= 81, jewelry value is 1d10 times 1,000

Doing this by hand a few times, I get: 3000, 4000, 6000, 1000, 2000, 7000, 4000, 6000, 9000, 4000. That's an average of 4,600 gp for each piece of jewelry, so reasonably close to my proposed average of 3,410 gp for only ten rolls.

The average chance for jewelry to be found in a treasure (averaged across all treasure types) is 35%. 35% of my average jewelry value is 1,193 gp. That's close to waysoftheearth's jewelry average of 1,137, but I don't know if that's how he calculated it.

Generation of gems is more complicated than generation of jewelry, with a far greater range of potential values. Here's how I read gem generation:

roll d100
1-10 = 10 gp (but see below)
11-25 = 50 gp (but see below)
26-75 = 100 gp (but see below)
76-90 = 500 gp (but see below)
91-100 = 1,000 gp (but see below)

The value from the d100 roll isn't "fixed" until we make at least one d6 roll. If we roll a one on a d6, the gem gets bumped to the next highest value (e.g.--a 50 gp gem becomes a 100 gp gem).

Additionally, (and this may be the part I'm misreading) once a gem gets bumped to it's new value, that value still isn't fixed. We roll d6 again. A roll of one means the gem gets bumped to the next higher value (e.g.--our 100 gp gem increases again to a 500 gp gem).

Value bumping continues for the gem until we roll a d6 result other than one. At that point, the value of the gem is final. If your string of one's takes you past the 1,000 gp mark, use these tiers: 5,000; 10,000; 25,000; 50,000; 100,000; and 500,000 gp.

I'll roll a few gems by hand: 1000, 500, 500, 100, 100, 10, 100, 100, 10, and 100 gp. That's an average (mean) of 242 gp--less than the 419 gp average I propose, and more than the 55 gp average waysoftheearth provides.

With a mode of 100, and values that only get bumped up, we're unlikely to get a mean of less than 100 gp. Possibly, as I surmise for jewelry, wayoftheearth's gem value of 55 gp averages gems that exist with those that do not.

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Re: Analysis of OD&D treasure types
« Reply #18 on Aug 11, 2012, 9:28pm »

You are spot on Paulg! Thanks for making me double check -- I found an error in my mean gem and jewellery calculations. With this corrected, my revised figures are very close to yours.

(The funny thing is, I had it right originally but though to myself "those figures are WAY too high", and by thinking to fix it I actually broke it).

Gems

I ignored gems that are extremely unlikely to occur. Being so rare, they don't make much difference to the average in any case.

Jewellery

Of course this correction further exaggerates the importance of gems (and especially jewellery) in the treasure tables. Everything else (except magic items) is worth peanuts by comparison.

Updated sorted mean treasure values

Updated percentage values

The value "order" of types B, C, D, E, F is revised to:

C > E > B > D > F

So Type C (ogres, gargoyles, lycanthropes, minotaurs, gnomes) is the least valuable treasure type in the game, and Type D (probably the most frequently encountered type in the early game -- orcs, hobgoblins, gnolls, trolls, etc.) is almost four times as valuable.

Thanks for your diligence Paulg, have an exalt!

« Last Edit: Aug 11, 2012, 9:29pm by waysoftheearth »

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Re: Analysis of OD&D treasure types
« Reply #19 on Aug 12, 2012, 10:15am »

Awsome work gentlemen! Definetly takes out the guesswork. For your analyzing pleasure, if it strikes your fancy, here is the Treasure Table given in Beyond This Point be Dragons:

[image]

The table (originally on the bottom and top of two seperate pages), is either an older version of the one in the 3lbb's or Arneson's revision of an older table. Most likely the latter, since it is somewhat differently organized and the footnote info in the 3lbb's is incorporated in the text.

There are lots of small differences and there is also no type I. There's also some difference in generating gems and jewelry:

Gems: roll a d6. 1 = 1000; 2 = 500; 3-6 = 100
Roll another d6 for each gem. Results of 1 bump up to next highest value. Values above 1000 are 5000, 10,000, 20,000 and 5000.

Jewelry: Roll d6. 1-3 = 100; 4-6 = 1000.
For base 100, roll 3d6 and multiply by 100
For base 1000, roll 1d6 and multiply by 1000.

It's worth noting too that the following monsters have Lair treasure types that differ from the 3lbb's (3lbb in parenthesis):

Orcs - Type A (D)

Ogres - Type B (C)

Wraiths - Type D (E)

Vampires - Type E (F)

Wyvern - Type E (F)

Dragons - Type D (H)

Lycanthropes - Type B (C)

« Last Edit: Aug 13, 2012, 6:32pm by aldarron »

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aldarron
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Re: Analysis of OD&D treasure types
« Reply #20 on Aug 14, 2012, 9:50am »

Okay, I used the method employed by WotE, on the BTPbD tables. Here's what I got:

Gems average value: 386 GP each.

Jewelry average value: 2275 GP each.

Type average values (greatest to least):

A:water – 70,114
H – 59,451
A:land – 28,466
A:Desert – 14,059
F – 4901
D – 4496
E – 2388
G – 2163
C – 1107
B - 363

Observations: There's some really significant revisions here. More than I expected. Gems are about the same in value to 3lbb's but jewelry is significantly less. The value order is also changed in significant ways. In particular Type B (ghouls, wights, ogres, hydras, lycanthropes, Nixies) is now the least. Type A:land is here almost twice as valuable as Type A:desert, whereas in the 3lbb table, desert is more valuable than land by about 20%. Type G (Rocs) is also a lot less due to the reduction in Jewelry value. In fact all the treasure types are reduced significantly from the OD&D amounts.

Overall the differences strikes me as deliberate "corrections" especially the adjustment downward in the amount of treasure present.

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Re: Analysis of OD&D treasure types
« Reply #21 on Feb 7, 2013, 11:13pm »

I wanted to develop an exact result for the mean value of a gem, based on theory, one that kept the rare gems valued from 5,000 gp to 500,000 gp, without resorting to simulations.

The theoretical result I got (417.563 gp) matches the result paulg got (418 gp) by simulation.

Aug 11, 2012, 9:28pm, waysoftheearth wrote:

I ignored gems that are extremely unlikely to occur. Being so rare, they don't make much difference to the average in any case.

WotE is right that the probability of getting a gem greater than 1,000 gp is tiny -- 2.32767%. The probability of getting a 500,000 gp gem is a mere 3 in a million. But these "black swans" do skew the mean. The distribution has a skewness of 120.86 and a kurtosis of 36,821.2, compared to a normal distribution which has skewness 0 and kurtosis 3.

To put these rare gems back into the theoretical account of the distribution, I took the following steps:

In Mathematica, I created a categorical distribution that captures the essence of rolling a d100 and using the table on page 40 of volume 2 in order to get an initial value for the gem. However, instead of assigning the gem a value in gp, I merely assign it a category, 1 through 5. Call this distribution X:

Code:
gemtable = EmpiricalDistribution[{.1, .15, .5, .15, .1} -> {1, 2, 3, 4, 5}]

Next I create a distribution that counts how many 1s I roll on a d6 before I get some other result. In probability theory, this is known as a geometric distribution:

Code:
d6while1 = GeometricDistribution[5/6]

Then I lumped all the probability mass in this distribution to the left of 11 together. Call this distribution Y:

Code:
pdfd6while1max11[x_] =
Which[x < 0, 0, x >= 0 && x <= 10, PDF[d6while1, x], x == 11,
Sum[PDF[d6while1, k], {k, 11, \[Infinity]}], x > 11, 0]

Now we want the distribution Z=X+Y. We do this by "convolving" the probability mass functions of the two distributions whose random variables you're adding. Here is the code to perform the convolution:

Code:
gempdf[k_] := Sum[PDF[gemtable,j] pdfd6while1max11[k-j], {j, 0, 11}]

Again we want a "censored distribution" that lumps everything to the left of 10 together, since there are only 11 categories of gems:

Code:
gempdflist = Append[gempdf[#]&/@Range[1,10], Fold[Plus,0,gempdf[#]&/@Range[11,16]]]

Here is the "lookup table" that maps gem categories into gold piecess:

Code:
gp = {10, 50, 100, 500, 1000, 5000, 10000, 25000, 50000, 100000, 500000}

Finally, here is the probability distribution we are after:

Code:
gems = EmpiricalDistribution[gempdflist -> gp]

Now we can operate on this distribution like any other built-in distribution, in order to find its statistics:

Code:
Mean[gems]
417.563

Median[gems]
100

StandardDeviation[gems]
1504.61

Skewness[gems]
120.86

Kurtosis[gems]
36821.2

Probability[x > 1000, x \[Distributed] gems]
0.0232767

(* Generate 100 random gems *)
RandomVariate[gems,100]
{1000,10,100,100,100,100,1000,100,1000,100,100,100,10,1000,1000,500,100,100,500,100,50,100,50,100,100,1000,100,100,50 ,1000,100,1000,500,100,500,500,500,500,10,500,100,100,100,1000,100,100,100,100,100,100,100,1000,100,1000,100,100,500,100, 5000,10,500,100,1000,100,500,500,50,500,100,100,100,100,10,1000,500,50,500,10,500,50,100,10000,50,100,100,500,1000,1000,1 00,500,100,500,100,50,500,500,100,100,100,100}

(* Count how many of these 100 random gems > 1000 gp. We expect about 2. *)
Select[%,#>1000&]
{5000,10000}

(* Chance of a half-million gp gem is one in how many *)
1/gempdflist[[11]]
334067.

(* Create a table with all the probability values *)
TableForm[Table[{x,PDF[gems,x]},{x,gp}],TableHeadings->{None,{"gp","Probability"}}]

And here is the table of probabilities produced by that last line of code:

gp	Probability
10	0.0833333
50	0.138889
100	0.439815
500	0.198302
1000	0.116384
5000	0.0193973
10000	0.00323288
25000	0.000538814
50000	0.0000898023
100000	0.000014967
500000	2.99341*10^-6

The probabilities from 10 gp to 1,000 gp don't differ all that much from WotE, but those big gems with tiny probabilities on the long tail do skew the results by a noticeable amount. The difference between WotE's mean (378 gp) and mine (417.563 gp) is about 40 gp.

Since I never trust my own calculations, I always try to reproduce the results in some other way. So I did write a monte carlo simulation in Lisp called gems.lsp, and you can see it here in my pastebin. If you have a Lisp interpreter and want to try it yourself, you can use a command-line like so:

Code:
elisha@ubuntu:~/workspace$ clisp gems.lsp 10000000
Number of runs 10000000
Total value of gems (in gp) 4174236350
Mean value of gems 417.4236
Standard deviation 1486.8490

As you can see, I ran this simulation 10,000,000 times (it didn't take that long), because 500,000 gp gems are so rare, and the resulting mean (417.4236 gp) is a good match for the exact result I got in Mathematica (417.563 gp).

« Last Edit: Feb 8, 2013, 6:25am by aher »

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