A Case Study in Probability: Endgame

[Yomo Kimyata]

What if I told you that there was a piece of salvage that could let you convert by rarity?  What if I told you there was one that would let you convert by category?  And what if I told you there was one that lets you convert in set?

 

 

Of course, all these things come from the same wonderful, magical animal that we know and love as the enhancement converter.

 

At this point in our program I decided to try to answer the question that nobody is asking:  how many licks does it take to can I make all of my IOs into Luck of the Gambler +rech and make a profit?

 

To begin, I put in bids on level 25 uncommon recipes at 5 inf each, hoping to be able to pick up a hundred of them within a week.  Why level 25?  Well, I figured that was probably the sweet spot where the number of uncommon/rare defense IOs had the highest ratio when compared to the entire pool, and it was the lowest level so therefore the cheapest crafting cost.  Why uncommon?  Didn't want to pony up for the rare salvage.  Now, I know that there probably aren't many/any recipes just sitting around for sale there, so I put in bids of 5 on blocks of ten in likely areas.  I think it took a day or two to get filled, which was the first lesson:  people continue to dump their stuff on the /AH for the sake of convenience.  I also found a few recipes that converted directly into rares at that level, and I put those aside for other purposes.

 

I bought all my salvage at vendor levels, so 250 inf each for two commons and 1000 for one uncommon.  Tack on the crafting fee of 37,400 and my all-in cost basis of each IO was 38,905.  To be honest, this number isn't really that important.  But it adds up.

 

Next, I wanted to develop a crafting strategy.  I decided to eyeball it instead of actually crunching the numbers because I felt pretty confident that it was a good method that would minimize the number of converters used.  The procedure was as follows:

 

1.  Start with a level 25 uncommon IO.

2.  Convert by rarity (so "uncommon") until either:

  a.  I rolled an uncommon defense IO (Karma, Serendipity, Red Fortune); or

  b.  I rolled an uncommon that was guaranteed to convert to a rare if converted by category.

3.  If a. then convert by category ("defense") until Luck of the Gambler, then if necessary convert in set until the +recharge piece.  End.

4.  If b. then convert once by category to get a rare IO.

5.  Convert by rarity ("rare") until rolling a rare defense IO (Kismet, Gift of the Ancients, Reactive Defenses, Luck of the Gambler).

6.  If not LotG, convert by category ("defense") until Luck of the Gambler, then if necessary convert in set until the +rech.  End.

 

Honestly, I could have made a Monte Carlo process to simulate this, because the probabilities are relatively straightforward, but I'm a glutton for punishment.  So I loaded up with 100 uncommon IOs and went to town.

 

Much clicking ensued.

 

At the end of the day, I ended with a data set that I will share with you here:

 

Spoiler

Trial		Conversions	Conversion	Conversions	Conversion	Conversion		Total
		by uncommon (1)	to rare (2)	by rare (1)	by defense (2)	in set (3)		Converters
1		4	1	20	3	0		32
2		1	1	14	0	8		41
3		1	1	22	6	3		46
4		8	1	2	6	0		24
5		2	1	3	2	6		29
6		7	1	25	5	5		59
7		1	1	1	2	0		8
8		8	1	16	0	17		77
9		7	1	5	6	3		35
10		3	1	2	1	4		21
11		1	0	0	2	0		5
12		2	1	5	3	7		36
13		4	0	0	14	12		68
14		3	1	1	3	0		12
15		3	1	14	1	8		45
16		6	1	11	0	1		22
17		4	0	0	3	9		37
18		3	1	5	0	3		19
19		9	0	0	3	3		24
20		1	1	5	0	5		23
21		2	1	36	1	2		48
22		6	1	2	8	0		26
23		2	1	2	8	1		25
24		2	1	26	2	2		40
25		6	1	21	5	6		57
26		4	0	0	7	9		45
27		8	1	15	5	7		56
28		2	1	95	10	0		119
29		4	1	35	0	6		59
30		5	0	0	3	7		32
31		1	1	68	2	2		81
32		10	1	1	5	7		44
33		1	1	6	6	14		63
34		10	1	12	0	3		33
35		3	1	8	6	1		28
36		1	1	9	1	1		17
37		6	1	1	7	8		47
38		3	1	11	1	0		18
39		2	1	9	0	3		22
40		2	1	15	5	10		59
41		2	1	1	4	5		28
42		1	0	0	7	6		33
43		11	1	1	5	6		42
44		6	1	4	0	2		18
45		1	0	0	7	4		27
46		9	1	12	0	10		53
47		1	1	1	15	5		49
48		8	1	15	2	13		68
49		5	0	0	4	1		16
50		17	1	22	14	1		72
51		3	1	8	9	11		64
52		1	0	0	1	2		9
53		4	1	8	2	2		24
54		1	0	0	2	5		20
55		7	1	9	19	2		62
56		5	1	4	10	1		34
57		1	1	14	0	11		50
58		4	1	43	2	20		113
59		1	1	72	0	3		84
60		3	1	11	1	3		27
61		1	0	0	9	5		34
62		10	1	7	0	2		25
63		1	1	18	17	1		58
64		1	0	0	1	1		6
65		11	1	27	52	11		177
66		16	1	13	1	1		36
67		1	1	47	5	3		69
68		2	1	1	2	3		18
69		3	0	0	1	2		11
70		5	1	47	2	8		82
71		9	1	48	0	2		65
72		3	1	12	0	6		35
73		2	1	3	3	8		37
74		7	1	2	8	8		51
75		2	1	16	3	1		29
76		11	1	6	0	1		22
77		2	1	9	3	1		22
78		2	0	0	9	1		23
79		13	1	24	3	2		51
80		9	1	29	16	0		72
81		1	0	0	3	4		19
82		13	1	9	2	6		46
83		5	1	6	2	2		23
84		2	0	0	6	2		20
85		3	1	3	1	3		19
86		5	0	0	10	1		28
87		2	1	29	4	4		53
88		1	1	5	1	2		16
89		5	0	0	2	10		39
90		2	0	0	22	4		58
91		5	0	0	4	3		22
92		5	0	0	2	0		9
93		4	1	3	5	6		37
94		1	1	37	2	2		50
95		1	1	3	0	0		6
96		5	1	27	2	0		38
97		6	1	6	0	0		14
98		3	1	14	5	2		35
99		4	1	19	6	1		40
100		1	1	6	10	9		56

 

 

But I anticipate that this graph may prove more informative:

 

 

The mean was 40.3, median of 35.0, standard deviation of 26.0.

 

You may draw your own conclusions from this.  I have to go ice my clicking finger.

 

 

[Andreah]

2 hours ago, Yomo Kimyata said:

ice my clicking finger

I feel for you.

[Andreah]

I'd be interesting in knowing how many classical 'equally-likely alternatives' there were at each stage. With that, it would be a simple (tho perhaps tedious) thing to adapt my existing Monte Carlo simulator to replicate your results.

 

[Yomo Kimyata]

Let me see.  If I've counted correctly (and someone certainly should double check me):

 

There are 52 different uncommon IO sets at level 25.

There are 66 different rare IO sets.

There are 3 uncommon defense sets.

There are 4 rare defense sets.

There are 9 categories that will guarantee a rare from an uncommon with one conversion.

 

So, starting with a random non-defense uncommon IO, a conversion by rarity has a probability of 3/51 of rolling to a defense set, a 9/51 chance of rolling to a set that will have a guaranteed rare conversion (none of the recipes I bought were in those categories), and a 39/51 chance of rerolling.

 

Once you have a non-defense rare, you have a 4/65 chance of rolling a defense and a 61/65 chance of rerolling.  That defense has a 1/7 chance of being a LotG (and from there, a 1/6 chance of being a +recharge).

 

Once you have a non-LotG defense, you have a 1/6 chance of rolling a LotG (and a 1/6 chance of being a +recharge) and a 5/6 chance of rerolling.

 

Once you have a non +recharge LotG, you have a 1/5 chance of rolling the proc and a 4/5 chance of rerolling.

 

Based on a post by @UberGuy (all hail UberGuy), the conversion process entails creating a list of possible sets, picking a set randomly, and then picking a member of the set randomly, so this seems pretty straight forward to program.  I just don't feel like doing it!

 

Edited March 6, 2022 by Yomo Kimyata

[Andreah]

I'll see what I can put together and run.

[Andreah]

 

Whew! I've been playing with this and getting statistically very different results -- mean around 26 and not anywhere near 40.

 

I scrubbed my code and found a minor bug, and fixed it .. and it was a trivial difference. 

 

Then I scrubbed the process Yomo described to make sure I was actually simulating that, in detail, exactly, and not some not-good-enough look-alike. I found a few things that weren't quite right, but still only resulted in trivial result differences.

 

Then my brain exploded. I was counting Conversions not Converters. LOL.

 

Fixed! I will post results after my long ten million sims finish up.

[Andreah]

Results from 10 Million repeated trials in simulation.

 

#
# Total Runs = 10000000
#
# Minimum =      3
# Mean =    39.430 Standard Error: +/-  0.007
# Stdv =    23.158
# Maximum =    287
#
# Frequency of Results:
#    Range     Count   Percentage Cumulative
#    0 -   4   40659      0.4066     0.4066
#    5 -   9  326289      3.2629     3.6695
#   10 -  14  654478      6.5448    10.2143
#   15 -  19  893858      8.9386    19.1528
#   20 -  24 1010160     10.1016    29.2544
#   25 -  29 1026673     10.2667    39.5212
#   30 -  34  978519      9.7852    49.3064
#   35 -  39  887945      8.8795    58.1858
#   40 -  44  778525      7.7852    65.9711
#   45 -  49  666114      6.6611    72.6322
#   50 -  54  557765      5.5777    78.2099
#   55 -  59  459508      4.5951    82.8049
#   60 -  64  373540      3.7354    86.5403
#   65 -  69  298680      2.9868    89.5271
#   70 -  74  237109      2.3711    91.8982
#   75 -  79  186212      1.8621    93.7603
#   80 -  84  145445      1.4545    95.2148
#   85 -  89  112962      1.1296    96.3444
#   90 -  94   87437      0.8744    97.2188
#   95 -  99   67480      0.6748    97.8936
#  100 - 104   51635      0.5163    98.4099
#  105 - 109   39298      0.3930    98.8029
#  110 - 114   29741      0.2974    99.1003
#  115 - 119   22746      0.2275    99.3278
#  120 - 124   17027      0.1703    99.4980
#  125 - 129   12808      0.1281    99.6261
#  130 - 134    9478      0.0948    99.7209
#  135 - 139    7152      0.0715    99.7924
#  140 - 144    5304      0.0530    99.8455
#  145 - 149    4041      0.0404    99.8859
#  150 - 154    2963      0.0296    99.9155
#  155 - 159    2244      0.0224    99.9379
#  160 - 164    1642      0.0164    99.9544
#  165 - 169    1226      0.0123    99.9666
#  170 - 174     891      0.0089    99.9755
#  175 - 179     640      0.0064    99.9819
#  180 - 184     474      0.0047    99.9867
#  185 - 189     366      0.0037    99.9903
#  190 - 194     256      0.0026    99.9929
#  195 - 199     195      0.0019    99.9948
#  200 - 204     118      0.0012    99.9960
#  205 - 209     106      0.0011    99.9971
#  210 - 214      69      0.0007    99.9978
#  215 - 219      72      0.0007    99.9985
#  220 - 224      34      0.0003    99.9988
#  225 - 229      30      0.0003    99.9991
#  230 - 234      26      0.0003    99.9994
#  235 - 239      15      0.0001    99.9995
#  240 - 244      15      0.0001    99.9997
#  245 - 249      10      0.0001    99.9998
#  250 - 254       4      0.0000    99.9998
#  255 - 259       7      0.0001    99.9999
#  260 - 264       2      0.0000    99.9999
#  265 - 269       2      0.0000    99.9999
#  270 - 274       3      0.0000   100.0000
#  275 - 279       0      0.0000   100.0000
#  280 - 284       1      0.0000   100.0000
#  285 - 289       1      0.0000   100.0000
#

 

[Andreah]

[Yomo Kimyata]

Yeah, the reason why I target guaranteed conversion from uncommon to rare is that it costs more.  It drives me nuts to convert from one uncommon to another, then back again.  It’s also trivial to demonstrate that a guaranteed path is cheaper.

[Hedgefund]

Good stuff folks.  The MC sims look pretty close, like a smoothed out version of the empirical set, at least at a glance.  I certainly haven't done a fit test (and I'm not going to) but it looks reasonable.

[Andreah]

22 minutes ago, Hedgefund said:

Good stuff folks.  The MC sims look pretty close, like a smoothed out version of the empirical set, at least at a glance.  I certainly haven't done a fit test (and I'm not going to) but it looks reasonable.

The test you would want to run would be the Kolmogorov–Smirnov test. You put the two  cumulative distributions next to each other, and measure the maximum difference, and then look it up in the appropriate table for significance (I don't remember which density function it follows -- school was LONG ago).

 

You could either treat the Monte Carlo result as a theoretical distribution, or treat them both as samples and do a two-way K-S test.

 

However, I can spare you all that; it's easy enough to replicate the Monte Carlo simulation for many runs of 100, just like Yomo's actual real world example, and then you find the mean and standard deviation Yomo found are consistent with what repeated monte carlo experiments produce, and so the simulations are consistent.

 

We don't really ever say we prove something is right, but we do make the claim there's, at most, very weak evidence it is wrong.

 

Here's a chart showing 1000 repeated 100-trial Monte Carlo Mean/Standard Deviations. The heavy circle is Yomo's result. It's not dead center, but that's fine -- lots of other dots are about that far from the cluster center. 

 

 

If the experimental result had been out on the edge, or worse, way out by itself all alone, that would have said "Go back and look at your model again, something's not right here", but we don't see that.

 

[Yomo Kimyata]

Thanks @Andreah for doing the computational work I didn't feel like doing!  Also, I made about 500mm by actually doing it, so I've got that going for me.  Which is nice.

[Andreah]

19 minutes ago, Yomo Kimyata said:

Also, I made about 500mm by actually doing it, so I've got that going for me.  Which is nice.

I estimated the profit around 5 Million per unit, so that sounds spot on! All we need to do now is figure out the profit/minute and profit/keystroke-mouseclick. :D

[iBot]

And would you believe there are still people that ask "Why do I need to learn this?!?? When will I ever need statistical analysis??" 🤪  As for the profit/click should be a straight forward multiplication using the average converters.  But the real question that needs asking is "How much beer did this take?"

[Troo]

On 3/5/2022 at 5:39 PM, Yomo Kimyata said:

all hail UberGuy

🙂

[Ukase]

On 3/6/2022 at 7:56 PM, Yomo Kimyata said:

Thanks @Andreah for doing the computational work I didn't feel like doing!  Also, I made about 500mm by actually doing it, so I've got that going for me.  Which is nice.

<smh> 
In my mind, I'm appreciating the info. But in the back of my brain, I'm saying, "Fools! How much MORE could they have made if they didn't take the time to post these things?!" 

Some time ago, a post mentioned a guess at the total amount of inf. Just tallying it up is an opportunity cost. Can you really be an "ebil marketeer" if you take the time to measure how high the gold pile is? I say, if you have more than 50 billion, you probably don't need to market any more - unless it's just fun/habitual/interesting. Clearly, for some of us, it's quite interesting. 
I have been trying really hard to leave it alone and explore other facets of the game. But old habits do not die easily. 

[Cancrusher]

@Ukase  Hey, man- if you want to live the life of a Scrooge McDuck, you've got to take a little time to go for a swim in your money vault.  😄🤷‍♂️

[Yomo Kimyata]

3 hours ago, Ukase said:

<smh> 
In my mind, I'm appreciating the info. But in the back of my brain, I'm saying, "Fools! How much MORE could they have made if they didn't take the time to post these things?!" 

Some time ago, a post mentioned a guess at the total amount of inf. Just tallying it up is an opportunity cost. Can you really be an "ebil marketeer" if you take the time to measure how high the gold pile is? I say, if you have more than 50 billion, you probably don't need to market any more - unless it's just fun/habitual/interesting. Clearly, for some of us, it's quite interesting. 
I have been trying really hard to leave it alone and explore other facets of the game. But old habits do not die easily. 

 

I doubt this outcome wasn't already a forgone conclusion to anyone, but at this point in the cycle, all I really care about is shaking the jar and seeing what happens.

 

By making it so obvious, maybe more people will try it?  Maybe there will be a run on converters?  Maybe there will be a glut of LotG +rech?  Who knows?!?   Running experiments on the /AH keeps me warm at night.

[Andreah]

4 hours ago, Ukase said:

I say, if you have more than 50 billion, you probably don't need to market any more - unless it's just fun/habitual/interesting. Clearly, for some of us, it's quite interesting. 

That marker long since came and went. 

 

One of the real-world jobs I had was related, and I enjoy it, especially when it's not 'real'.

 

I suppose it is also fun and habitual. But it's also a bit of a race. Eventually someone's going to get to a Trillion, and I'd like to think I was in that race, even if I was never really anywhere close to leading it.

 

[Troyusrex]

On 3/9/2022 at 9:10 AM, Ukase said:

I say, if you have more than 50 billion, you probably don't need to market any more - unless it's just fun/habitual/interesting. Clearly, for some of us, it's quite interesting. 
I have been trying really hard to leave it alone and explore other facets of the game. But old habits do not die easily. 

 

50 Billion? For me it's 15 Billion. I realized a while ago that I am a satisfiser. That is to say, I'm usually not looking for the optimal solution, just a good enough one. I'm highly motivated to play the market and make money up until a point. I find repeatedly that at about 15 Billion my motivation wanes and I'd rather play the game. Then, when funds get a bit lower, I dive back in until funds are replenished.

 

I just find that at 15 Billion I have more than enough to buy anything I want for as many alts as I want. I never lack. I've had a good bit more at times (and probably a huge amount more in inventory in bases that I never use). I certainly have the skills to make a lot more, but I'm quite happy being "Fairly Rich" and can't be bothered to get to "Very Rich" let alone "Uber Rich" or "Plaid". I'm not in the running to be the first to 1 Trillion, simply because having money beyond the amount to buy everything I need seems like a waste of my time.

 

Of course, if you find it interesting or just like trying to get to a trillion, more power to you! It's just not my cup of tea. 

 

 

[Ukase]

I do get it. I think when HC first opened up shop, I wasn't sure which way to go with playing the market. Other than Very Rares and LotG, I wasn't sure what flavors of IO were going to be popular, so I invested heavily into Rare salvage. Easy enough to get via AE tickets and just drops in general, and then the HC devs seeded them, effectively putting a damper on my plan. And if they did that - what might they do next? Seed the lotg 7.5%? Put a ceiling on the cost of every IO? They could still do that! 

Mind you - it would just make my stack ridiculously larger in terms of need. 

But there's one thing I do know - if you step away from the AH, even if you have plenty - there may come a time when certain rules change - like emails being limited to 10 days, or a month. (I believe it was like that on live) 
So what would I do if that happened!? I'd need to make about 300 more characters pretty damn quick! Not to mention, when you step away from a niche, things change! I just checked - PvP recipes are ALL selling for more than 2.5M, up from the usual 1.5-1.8M. And they're barely holding price on crafted IOs at 8m. Someone is being competitive, as @Yomo Kimyata likes to talk about. And he's right. It's good for the community for folks to get more for the recipes and for other folks to pay less for the crafted IOs. But I digress..


I know I can make more, even if I were stripped of all my storage, all my inf. But I much prefer to do these things when I want to, instead of being level 22 and needing/preferring to optimize my character with an LotG 7.5%. I don't team often, but when I do, the last thing I want is to head to a trainer, level up and then determine what/how I'm going to put in the slot. I like to have the IOs already in my tray at the ready. And without "plenty", I'd be struggling and letting any prospective teammates down with a sub-standard character. 

I know a number of folks have no issue leaving things unslotted until they have a chance to take care of these things. But most of those folks are nicer people! Me? I'm a grumpy curmudgeon type. If I don't at least come to the party with a kitted out character, I might not ever get a task force done, unless I do it solo, which is fine, but takes longer. 

[Andreah]

49 minutes ago, Ukase said:

But there's one thing I do know - if you step away from the AH, even if you have plenty - there may come a time when certain rules change - like emails being limited to 10 days, or a month. (I believe it was like that on live) 
So what would I do if that happened!? I'd need to make about 300 more characters pretty damn quick!

I think we're okay, for the moment, but like you say, it could be annoying if suddenly email isn't going to be safe, or certain items in the auction change value suddenly. This lead me to posting this a while back: 

I still like the idea in that thread -- it would remove a lot of dependence off of the email system and off the auction too.

[Bionic_Flea]

58 minutes ago, Ukase said:

It's good for the community for folks to get more for the recipes and for other folks to pay less for the crafted IOs. But I digress..

 

That's just crazy talk.

 

[Ura Hero]

3 hours ago, Troyusrex said:

 

50 Billion? For me it's 15 Billion. I realized a while ago that I am a satisfiser. That is to say, I'm usually not looking for the optimal solution, just a good enough one. I'm highly motivated to play the market and make money up until a point. I find repeatedly that at about 15 Billion my motivation wanes and I'd rather play the game. Then, when funds get a bit lower, I dive back in until funds are replenished.

 

Of course, if you find it interesting or just like trying to get to a trillion, more power to you! It's just not my cup of tea. 

 

 

This is me.  I could be in the running for the trillionaires club if I were so inclined, but after about 10-15b or so I lose interest.  I keep a healthy safety net of funds and pre-crafted IOs, but I don't go out of my way to try to make it into the major leagues.

[lemming]

22 minutes ago, Ura Hero said:

 

This is me.  I could be in the running for the trillionaires club if I were so inclined, but after about 10-15b or so I lose interest.  I keep a healthy safety net of funds and pre-crafted IOs, but I don't go out of my way to try to make it into the major leagues.

I think I have less than a billion in loose influence, but I have a pretty good stock of purples, AT and Winter enh, so to outfit a character usually only takes me a little shuffling of enhancements with a little bit of converting.  Takes me longer to figure out how I'm going to outfit them and do the respec than the actual doing so.