[Mathematics] Why The AH Is The Main Problem

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Many people ask, "Why am I not finding upgrades?" and "Why am I forced to use the AH to progress?"

These are both valid questions, and the answer in both cases is the existence of the AH. Let me explain.

This blue post by Wyatt Cheng should provide a background for what I am about to delve into:
http://us.battle.net/d3/en/blog/6317360/6317360#misc

If you don't want to read through the math and want to skip to the conclusion, scroll down to the part where there are two bulleted lists and start from there.

The Item Scale

First we need a mathematical scale to assign items. By convention, I am going to use a 0 to 100 scale. Let us only consider rare quality (yellow) items.

A 0 is the worst possible rare item you can find in Inferno. E.g., a level 51 chest piece that rolled 10 dex, 46 thorns, 200 health globe healing bonus, and 6 gold pickup radius. You can probably imagine worse, but that item is well under a 1 on the 0 to 100 scale.

A 100 is the absolute best-in-slot item for some slot, with perfect rolls on the stats you want. E.g., Archon Armor chest piece with 200 str, 300 vit, 3 sockets, 80 resist all, and perfect rolls on base armor and bonus armor. This item is a 100.

Everything in between works in percentiles. An item of scale level 99 is better than 99% of Inferno rares. A 10 item is only better than 10% of Inferno rares.

When we talk about "good" items, we only care about items near the 100 end of the scale.

10% More Stats Is NOT 10% More Rare.

I'm going to give two examples of chest pieces to demonstrate relative rarity, which is key to the argument. Say both are ilvl 63:

Chest A:
70 str
70 int
70 vit
70 resist all

Chest B:
77 str
77 int
77 vit
77 resist all

The common but false conclusion to jump to is that Chest B is 10% more rare than Chest A because it has 10% more of every stat. The conclusion is false because several 77's are much harder to roll than several 70s. The math is below. We conclude that Chest B is in fact 5 times rarer than Chest A.

Say the roll range for Str and Int are 20-100. It's possible to roll more due to also rolling an affix that has 2 stats, but let's not worry about that. Say these chest pieces rolled the same types of affixes. Similarly, say that the vit roll is 20-200, and the Resist All roll is 20-80. I know these exact ranges may be off but the fundamental result is the same for my demonstration.

Supposing you rolled a chest that had Str, Int, Vit, and Resist All, what is the chance that it will be strictly better than or equal to Chest A?

The chance is (31/81)*(31/81)*(131/181)*(11/61) = 1.91%

Is the chance to roll strictly better than or equal to Chest B only 10% lower than that? No, it is actually a lot lower!

The chance is (24/81)*(24/81)*(124/181)*(4/61) = 0.39%

Chest B is almost 5 times rarer than Chest A.

And to make it worse, the percentages above are only of the items that rolled Str, Int, Vit, and Resist All, which is a fairly desirable combo despite having only 4 properties. Even having a terrible roll with these 4 properties is still better than any 4-prop chest with Health Globe Healing Bonus, Gold Pickup Radius, Thorns, and say Dex.

Most of Your Items Are In the Top 1%

But wait, you probably don't just have such 4-prop items. You probably have 5- or 6-prop rares that are much better. But what we have shown above is that even Chest B is in the top 0.39% of ilvl 63 chests that roll those relatively desirable stats.

So Chest A might be around 95 on the 0-100 scale, and Chest B, being 5 times rarer, is a 99. But what about 5 and 6 prop gear? Consider Chest C and Chest D below:

Chest C:
77 str
77 int
77 vit
77 resist all
3 sockets

Chest D:
77 str
77 int
77 vit
77 resist all
3 sockets
597 life regeneration

Chest C is clearly much better than Chest B, and Chest D is much better than Chest C!

How much rarer is Chest C than Chest B? There are 3 factors at play: rarity of 5-prop versus 4-prop, rarity of the +sockets affix, and rarity of 3 sockets within the +sockets affix.

With these combined together, Chest C is likely 30 or more times rarer than Chest B, but just to keep numbers simple, say it is only 10 times rarer. Thus Chest C would have the value 99.9.

Chest D adds a 6th affix as well as an extremely high roll of a nice affix. It's probably at least 100 times rarer than C, putting Chest D at the very least 99.999.

The RNG

Now say you are poor and can only afford something of Chest B quality. The middle class probably has gear more like Chest C, while the rich have Chest D quality or better. This might not exactly match, but the relative quality is what I am getting at.

Now, a Random Number Generator (RNG) is generating random decimal numbers between 0 and 100. You play Diablo 3. Every time you find a chest armor piece, you take a number from the RNG.

Say you're poor and you have only bought Chest B. The RNG must give you a number higher than 99 for it to be an upgrade. Statistically, it is very unlikely.

Say you're a bit wealthier and you have Chest C. The RNG would have to give you a number higher than 99.9 for it to be an upgrade. Extremely unlikely. You'd have to farm for a long, long time.

Say you are even wealthier and have Chest D. Well, good luck beating a 99.999.

The Top 1% Is Your Starter Set

The problem is, most people didn't find Chest B themselves. They bought it from the AH. This is why it is extremely rare for people to find upgrades. They already have Chest B, something better than 99% of other chest pieces. The next chest they identify has only a 1% chance to be an upgrade. Those of you with Chest C have only a 0.1% chance of finding an upgrade. And those with Chest D have a measly 0.001%. To find something better than D, the chance would be akin to drawing a royal flush. It doesn't matter if you have 300% MF, it's still a gamble.

The Blizzard Internal Testers Did Not Have Top 1% Gear

They didn't even have top 5% gear, because they did not have an Auction House. Even Chest A, while laughable to many of us, would probably have been considered a great item when they were testing the game.

Of course, to get to Inferno they had to have had at least semi-decent gear, say in the top 50%. That translates to their gear level being around 50.

This is why in the article linked at the beginning of this post, Wyatt Cheng claimed that upon entering Inferno, a player might find an upgrade in 30 minutes. After all, if you start with your gear at 50 on the 0-100 scale, then any rare chest piece you pick up has a 50% chance of being an upgrade. Picking up 2 chest pieces sounds reasonable in 30 minutes.

However, using the AH screws things up because it makes you jump from 50 to 99.

Here is a comparison of what happens:

Blizzard Testers/No-AH Players
* At item scale level 50 upon entering Inferno.
* The first chest armor they find is a 37, so it isn't an upgrade. But the next is a 64, which is an upgrade.
* To upgrade again, they must roll better than a 64.
* They roll a 28, 61 (which would have been an upgrade if they didn't already have the 64), 12, and finally a 73, an upgrade.
* To upgrade again, they must roll better than a 73.
* Etc. Each time they get an upgrade, the amount of time to the next upgrade goes up gradually (at least on average). Nonetheless, they are steadily finding upgrades. This is what it felt like in Diablo 2, because there was no AH.

AH Players
* At item scale level 99 upon entering Inferno.
* The first chest armor they find is a 37, so it isn't an upgrade. Neither is the next piece, which is a 64.
* Neither is the 28, or the 61, or the 12, or the 73.
* Or the 91, 8, 29, 67, 62, 72, 84, etc.
* Or the 95, 93, 22, 60, 71, 89, or 96.
* Since seemingly nothing they find is an upgrade, they are drawn to the conclusion that the only way to find a better item is to scour the AH for a 99.9.
* Thus they feel they cannot upgrade on their own. They feel forced to use the AH, which leads to a vicious cycle. Once you use the AH, you are enslaved to it.

The AH is the problem! Of course, another issue that can be seen from the numbers above is the "Doubled It" fiasco. Before they doubled it, it was probably possible to beat the game with 98/99 gear. So it might take a few days to farm the top 1%/2% gear, but you can do it without much sweat. But once they "doubled it," instead of being able to beat the game with 98/99 gear, you must do it with 99.99 or 99.999 gear, the top 0.01% or 0.001% gear. This doesn't take a few days of farming. This takes months of farming and/or blind luck. This exacerbates the RNG and turns the game into a lottery rather than something that resembles Diablo 2. Although, even with "Doubled It," the Auction House is still the worse issue.

Solution: Release Ladder, with no GAH (possibly no RMAH as well, but as argued in point #4 below, the GAH is the much bigger problem). The bartering system, while primitive, seems to avoid all the problems of the AH system.

If you are TL;DR'ing this, at least read the bulleted comparison between non-AH and AH player experiences just above. I hope they adequately explain why the AH is so problematic.

Edit: Additions, Refinements, Objections:

1. "The stat roll distribution is not uniform."

That is correct. There are 2 tiers of All Resist that can appear on ilevel 63 armors: Chromatic and Spectral. Chromatic on a chest, for example, maxes at 80, while Spectral maxes at 70. Thus, items with 71+ All Resist are actually much rarer than argued in the OP. Instead of a 1/61 chance to roll 80 All Resist, which I assumed, it is actually perhaps ~1/100 or lower. Same for the other stats. This only strengthens the argument.

2. "If i have gear that is better than 99% of all other gear, why the heck is act 3 still so hard. (or even act 2)."

Because they "doubled it." See the paragraph right before the solution.

3. "The choice of 95 for Chest A seems arbitrary."

Yes it is a bit arbitrary. But considering how hard it is to roll Chest A, how hard it is to roll ilvl 63, how hard it is to roll the correct stat tiers (e.g., Chromatic instead of Spectral), Chest A is likely much higher than a 95. Possibly, it is a 98 or 99. This would only strengthen the argument too. I understated the rarity by putting it at 95, so it would be easier to explain the rest of the argument. The understating does not subtract from the argument, and it is not a hole in logic.

4. "You talked about the "AH" but which one?"

As far as the math in this argument goes, it does not concern the difference between the AH's. It is concerned with people accessing the 99+ percentile items easily. Since the GAH is much more easily accessible than the RMAH, and because people are much more willing to use the GAH than the RMAH, I would say based on this logic that the GAH is far worse of a problem than the RMAH. According to the math in this argument, if the game was RMAH-only, many of these issues would actually be avoided, quite surprisingly.

========================================================

TL;DR

The low quality of drops is an illusion created by constantly comparing one's gear to that on the AH, which is usually in the top 1% or 0.1% of all gear.

So when you find an item in the 95th percentile (better than 95% of all items), it is still relatively bad compared to anything on the AH, so you perceive it as trash. Even 98th percentile gear gets vendored these days. So while you think that an "average" piece of gear from the AH doesn't seem that great, it's often in the 99th or 99.9th percentile of drops. The items that are worth millions, are in the 99.999th or higher percentile.

Whereas, if the AH did not exist, people would gear at the natural pace and not encounter this issue. They would perceive gear even in the 85th or 90th percentile as amazing, instead of perceiving gear in the 99th percentile as "meh."
No. AH is actually a cool feature of D3. People are just going to learn to play and adapt. You can choose not to use AH, decisions are solely based on you, yourself.
Pretty much dead accurate. The INSTANT you use the AH, you are now bypassing the entire loot progression of the game. Instead of being content with your own loot pool (ala D2) you now are utilizing the ENTIRE region loot pool. Law of large numbers then makes the D3 RNG work, but for the individual, it sucks bad.
Are you trying to say that my Ilvl 63 chest is a 100?
07/24/2012 03:56 AMPosted by Himeros
No. AH is actually a cool feature of D3. People are just going to learn to play and adapt. You can choose not to use AH, decisions are solely based on you, yourself.


I agree that the AH has its pros as well as cons. I can choose to not use the AH, and in fact, I am trying the Hardcore Absolutely-No-AH Challenge right now as I am pretty much done with softcore.

My argument is that the AH is not being used as it was intended, and is thus hurting most of the people who use it rather than helping them.

The problem is that you need that class C/D stuff to finish the game. If the class B stuff let you finish the game reasonably well and class C and D stuff meant facerolling it, sure, you'd be right.


Exactly! I think before they "doubled it," it would be indeed true that class B gear was progression gear while class C and D was faceroll gear. But after the doubling, class B is useless, C is iffy, and D can only guarantee progression at all, not a smooth or easy progression.


The chance is (31/81)*(31/81)*(131/181)*(11/61) = 1.91%

Is the chance to roll strictly better than or equal to Chest B only 10% lower than that? No, it is actually a lot lower!

The chance is (24/81)*(24/81)*(124/181)*(4/61) = 0.39%

Chest B is almost 5 times rarer than Chest A.


wtf where did these numbers come from? seems legit just wondering


The chance is (31/81)*(31/81)*(131/181)*(11/61) = 1.91%

Is the chance to roll strictly better than or equal to Chest B only 10% lower than that? No, it is actually a lot lower!

The chance is (24/81)*(24/81)*(124/181)*(4/61) = 0.39%

Chest B is almost 5 times rarer than Chest A.


wtf where did these numbers come from? seems legit just wondering


I'll use the resist all as an example. :D

You can roll Resist All between 20 and 80, integer only. So that's 20, 21, 22, 23... 79, 80. Counting carefully, we see that's actually 61 numbers, not 60. To roll a 77 or better, you only have the options 77, 78, 79, and 80. So the chance is 4/61 for the Resist All stat. Similar calculations work for the other rolls.

Hope that made sense.
I read the whole thing and agree. I didn't think about it from that point of view and I reckon you just hit the nail on the head. I trust your math and even if it's not correct you're spot on about the upgrades via the AH.

So the real question is, would you still play knowing this now?
Good post.

What do you think if they periodically improve the quality of loots to create an impression that players can find useful loot more frequently ? This can improve casual players' gaming experience and alloe newer players to catch up. They can then tune up the difficulty by a little to make it not too easy for older players
Great post, my conclusions written better than i could have. :)
nice post. also, i want to read more 'chests'


wtf where did these numbers come from? seems legit just wondering


I'll use the resist all as an example. :D

You can roll Resist All between 20 and 80, integer only. So that's 20, 21, 22, 23... 79, 80. Counting carefully, we see that's actually 61 numbers, not 60. To roll a 77 or better, you only have the options 77, 78, 79, and 80. So the chance is 4/61 for the Resist All stat. Similar calculations work for the other rolls.

Hope that made sense.


yes ty :)
great post, the problem for many of us is that havn't 0.1% of the gear does not nessasarily mean you are beyond godlike, a stage most D2 players would hope to achieve, we are still fraustrated over the fact that having seemingly GREAT items does not prevent us dying from invulnerable/deflect dmg/ fire chain or other stupid combination of mobs. Hence a feeling of endless grind and a bad taste
07/24/2012 03:56 AMPosted by Himeros
No.

Yes. Please read again and understand what is being said. The psychology of using the AH to get access to the best is that people think the game drops only crap. But that's because they already got high level gear from the AH.
The AH offers better items than players usually find for an affordable price, so it's not a free choice, the AH is the better choice.


wtf where did these numbers come from? seems legit just wondering


I'll use the resist all as an example. :D

You can roll Resist All between 20 and 80, integer only. So that's 20, 21, 22, 23... 79, 80. Counting carefully, we see that's actually 61 numbers, not 60. To roll a 77 or better, you only have the options 77, 78, 79, and 80. So the chance is 4/61 for the Resist All stat. Similar calculations work for the other rolls.

Hope that made sense.


you do realise that you are assuming that all numbers have an equal chance to roll. I can only make guesses but wouldnt it be possible that the chance of getting higher rolls decreases as they go up?
07/24/2012 03:56 AMPosted by Himeros
No. AH is actually a cool feature of D3. People are just going to learn to play and adapt. You can choose not to use AH, decisions are solely based on you, yourself.


You have no hope of playing in inferno act 2 and beyond without using the auction hall unless you play ALL the time. No one with a life has that kind of time.

The OP is completely right. The difficulty of inferno was obviously based on the assumption that players would use the AH -- and that's the problem. It takes WAY too long to save up the kind of money needed to get gear good enough to get through inferno. Again -- if you have 6 hours a day to play you may not feel that way but those of us that have families and jobs can't do that.

As an example, it takes me about two weeks of farming act 1 inferno to save up 1 million gold. Fine: but it'll take 50 million gold (at least) to get gear good enough to get through inferno. That's the problem: it's *BORING* to spend all of my available gaming time for six months repetitively playing through the same act just so that I can save up enough money to buy the gear I need. The fun of games like this comes from the sense of progression -- it's fun to feel like you're becoming more powerful. Endless farming doesn't give that feeling because you're not going up against new challenges. You're just endlessly repeating the SAME challenges.

You can't hold people's attention with a game like this UNLESS you make them feel as though they are constantly progressing, becoming more powerful.
07/24/2012 04:15 AMPosted by Rudski
So the real question is, would you still play knowing this now?


nope..

I've always said the Risk vs. Reward and the Appreciable Gains vs. Time Spent are poorly calibrated in this game. Or perhaps theyre perfectly calibrated if the intent is to entice people to use the AH.


I'll use the resist all as an example. :D

You can roll Resist All between 20 and 80, integer only. So that's 20, 21, 22, 23... 79, 80. Counting carefully, we see that's actually 61 numbers, not 60. To roll a 77 or better, you only have the options 77, 78, 79, and 80. So the chance is 4/61 for the Resist All stat. Similar calculations work for the other rolls.

Hope that made sense.


you do realise that you are assuming that all numbers have an equal chance to roll. I can only make guesses but wouldnt it be possible that the chance of getting higher rolls decreases as they go up?


Ah yes, you're completely right. The chance of actually rolling exactly 100 Str is likely lower than the chance of rolling exactly 70 Str, just because of the way the stat range tiers work. I suppose if I factored this in, it would only amplify my argument, because it would mean that the 99.999 items are actually the 99.99999 items.

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