+pickpocket vs. +item bonus

[ketchup monger]

With the rollout of the new limited time Crimbo zone (10% drop from unscaling candy monsters), I gave some thought on how to efficiently farm the drops. Pickpocketing is an easy way to increase drops, but is it efficient to trade +item gear for +pickpocket gear? Would sticky gloves (+20% pp) be better than a cheese eye (+20% item) I gave it some thought, and found out that it was not. Note, this is only true for monsters with single drops.

Here's why:
The way pickpockets (pp) work is that you get a second chance at getting a loot. You don't get two loots. The bonus to pickpocketing is calculated the same way as item bonus modifiers work. It is not additive. You have:

(base_rate) x ( 1 + (bonus/100)) = chance

This means a 10% base drop with a +20% bonus modifier would have a 12.5% chance of dropping, not a 30% chance. And this is true for both +item and +pp bonuses.

When the combat starts, a roll occurs to see if a pp succeed. If it fails, another roll occurs at the end of combat to see if you win the drop. Therefore, the formula for this series of activities is:

Effective_rate = (pp_chance) + (1-pp_chance)x(item_chance)
= pp_chance + item_chance - (pp_chance x item_chance)

Where the chance formula (above) is used for pp and item.

If we trade in +item gear for +pp gear, we have a bound equation. For simplicity, I'll assume that it's a 1:1 trade, and work out later the ratio needed to make the tradeoff even. As it is a bound equation, lets use Y to represent the total value of our pp_chance and item_chance.

Y= pp_chance + item_chance

Alternatively, solving for pp_chance:

pp_chance = Y - item_chance

plugging this back into the expanded formula for Effective_rate:

Effective_rate = Y - ((Y - item_chance)x item_chance) = Y + Y x item_chance - item_chance^2

This is a polynomial of order 2, so we can take a derivative and solve for 0, to find the minima. Y is a constant value, since we've assumed that pp_chance + item_chance is always going to equal the same number. However, item_chance is not, and will vary.

0= Y -2 item_chance

Plug Y back in, solve for zero

0= pp_chance + item_chance - 2 item_chance
0=pp_chance - item_chance
item_chance = pp_chance

Our lowest value will occur when our pickpocketing chance is equal to our item chance! Giving up an equal amount of +item to gain +pp results in the lowest possible value for that level of bonuses.

4 analytic examples (base rate = 30%)
+item = 200, +pp = 0
Effective rate = 93.0%

+item = 150, pp = 50
Effective rate =86.3%

+item = 100, pp=100
Effective rate = 84.0%

+item = 0, pp= 200
Effective rate = 93%

The takeaway message should be that you shouldn't substitute +pp for +item

[geronimoyang]

Thanks for writing this up. I would ask how a second chance at pickpocketing would change things with the bling of the new wave, but then I realized you would have rave steal at that point anyway, so it would be a silly question(and the answer obvious given the shape of the function). And now I wonder why they gave the DB a second shot at pickpocketing in the first place.

[ketchup monger]

A double pick pocket is kind of interesting. You'd still get a single loot, but you would have the pp_bonus applied to each attempt.

Effective_rate = (pp_chance) + (1-pp_chance)(2nd_pp_chance)+ (1-pp_chance)(1-2nd_pp_chance)(item_chance)
Effective_rate = (pp_chance) + (1-pp_chance)(pp_chance) + (1-pp_chance)^2(item_chance)

Effective_rate = (pp_chance)+(pp_chance)-(pp_chance)^2 + (1 - 2 pp_chance + pp_chance^2)(item_chance)
Effective_rate = (pp_chance)+(pp_chance)-(pp_chance)^2 + (item_chance) - 2(pp_chance)(item_chance)+(pp_chance)^2
Some terms cancel out
Effective_rate = 2 pp_chance + item_chance - 2(pp_chance)(item_chance)

comparing this equation to the single pp rate formula:

= 1 pp_chance + item_chance - 1(pp_chance)(item_chance)

It's no longer possible to say the value of pp_chance + item_chance is a constant number (and makes the math a little tricky).

plotting the function in google docs:


And may as well compare the function from post 1

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