Diablo III: Auction House Algebra

Recently Blizzard revealed more details on the Diablo III auction house.

http://us.battle.net/d3/en/services/auction-house/how-to
http://us.battle.net/support/en/article/diablo-iii-auction-house-functionality

 The flat 1$ fee on gear at the real money auction house (RMAH) versus the percentage fee on the gold auction house(GAH) suggests one of them will be more profitable than the other depending on the value of the item. For higher value, 1$ on the RMAH will be much less than 15% on the gold auction house. Of course the best way to avoid these fees is to avoid the auction house altogether and trade in-game.

The best auction house to sell your item also varies depending on what your end-goal is because there is an extra 15% fee if you sold your item on the GAH and need to sell your gold on the RMAH.

Here I make an attempt to find these limits depending on the case. Forgive me if I make mistakes, its been years since I had to calculate anything.

Real Money Auction House (RMAH):

Maximum price: 250$
Minimum price: 1.25$
Blizzard cut on gear: 1$
Blizzard cut on commodities: 15%

Gold auction house (GAH):

Maximum price: 100 000 000 000g
Minimum price: 100g
Blizzard cut: 15%

Variables:

v: The current value of the item in gold
g: The current value of 1 gold in $ (gold/$)
vg: The current value of the item in $

1) When is it more profitable to sell gear on one auction house instead of the other:

a) When your goal is Gold.

P1: Profit from selling the item on the gold auction house.
P2: Profit from selling on the $ auction and then buying gold.

P1 = v * 0.85
P2 = (vg - 1)/g

The breaking point:
P1=P2 v*.85 = (vg-1)/g
vg*.85 = vg-1
.85 = 1-1/vg
.15 =1/vg
vg = 6,67

So if you want gold, you should sell gear on the gold auction house when its value is under 6,67$

And you should always sell on the $ auction house when the value is above 6,67$ And then buy gold.

But what happens when you reach the limit? you can ever make only 249$ profit on the $ auction so you might think to sell on the gold auction house anyways, but there will be range where its still more profitable to sell at a lower price on the $ auction house because of the huge 15% blizzard cut. So at what gear value does selling on the gold auction house allow you to make more profit than 249$ in gold?

249$/g = v *0.85
vg = 293

So in conclusion, if you want gold:

• If you think that the item will sell under 6.67$, sell on the gold auction house.
• If you think that the value of the item is between 6.67$ and 292.94$, sell it on the real money auction house and then buy gold with the balance.
• Assuming than the maximum price on the GAH is worth more than the maximum price on RMAH, if you think the item will sell for more than 292.94$ in gold, sell it on the gold auction house.

b) When your goal is real money: 

P1: Profit from selling the gear on the gold auction(15%) house and then selling the gold (15% for commodities!) on the RMAH.
P2: Profit from selling the gear on the RMAH (-1$).

P1 = (v*.85)*g*.85 = 0.7225 *vg
P2 = vg -1

0.7225*vg = vg -1

0.7225 = 1-1/vg
0.2775 = 1/vg
vg = 3,60


Same thing, when nearing the limit there is a range where its still more profitable to sell on the $ auction house despite the 250$ limit.

249 = 0.7225*vg
vg = 344.63

So in conclusion, if your goal is money:

• If you think that the item will sell under 3.60$, sell it on the gold auction house, and then sell the gold.
• If you think that the item would sell between 3.60$ and 344.63$, sell it on the real money auction house.
• Assuming that maximum value on the GAH is much higher than the maximum value on the RMAH, if you think that the item would sell above 344.63$ sell on the GAH and then sell the gold in batches on the RMAH.

Other questions which could be answered with basic algebra:

2) When is it profitable to buy an item on the RMAH and sell it on the GAH:
 a) When you goal is gold.
 b) When your goal is $

3) When is it profitable to buy an item on the GAM and sell it on the RMAH:
 a) When your goal is gold.
 b) When your goal is  $.