WoW Mathematics: Your guild bank donation

Aim: To calculate the value of a deposit of gold to a guild bank (gbank) that discounts sales at a given rate.

Summary:

• A donation of G gold to a gbank with a decimal discount rate of r will be worth G/r gold. This means that G is multiplied by 1/r, the reciprocal of the discount rate. Keep in mind that Full price = Discounted price + Discount.
• As long as the discount rate is less than 100%, the donation is worth more than the amount donated (even in pure money terms)!
• A gbank donation's impact will grow according to the sum of a geometric series and approach a limiting value.

The given:

• The discount rate is fixed.
• There is no cost in performing transactions and the full proceeds of a gbank sale are deposited back in.
• All gold is used in buying other resources for resale. No gold is dispensed as a repair allowance (though it can be factored into the discount rate).

The detail:
Consider a gbank deposit of G gold. It is used in buying (for example) enchanting materials worth G gold. They are sold to a guild member at a discount rate of r, so the undiscounted proportion of the price is:

1 - r

The proceeds from that sale are:

G × (1 - r) = G(1 - r) gold

The proceeds are then used for buying potions worth G(1 - r) gold. There potions are sold at a discounted price of:

G(1 - r) × (1 - r) = G(1 - r)² gold

Continuing this process for n terms, we get the following progression of values:

G, G(1 - r), G(1 - r)², G(1 - r)³, ... , G(1 - r)^{n - 1}

The geometric series 1 + 1/2 + 1/4 + ... , with its value represented by area.

This is in fact a geometric progression with common ratio (1 - r) and first term B. Note that, while the amount deposited diminishes, it keeps making an impact with each gbank sale. The total impact for n sales is the following sum of the corresponding geometric series:

S(n) = G(1 - (1 - r)ⁿ) ÷ (1 - (1 - r)) = G(1 - (1 - r)ⁿ)/r gold

As n » ∞ , (1 - r)ⁿ » 0 , and (1 - (1 - r)ⁿ) » 1 . So, the total impact approaches the following limiting value as resales are conducted repeatedly with that amount:

S(∞) = G/r gold

Here is an example. Say 100 gold is donated to a gbank with a discount rate of 25% or 1/4. That donation will be worth:

100 ÷ 1/4 = 100 × 4 = 400 gold