Mathematics: Why The AH Is The Main Problem

I originally posted this article on the US battle.net forums, where it has met surprisingly positive responses. I expected to be trolled, but it seems the trolls were on vacation that day.

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 beat 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 RMAH and no GAH. The bartering system, while primitive, seems to avoid all the problems of the AH system.

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