In a recent
discussion about new affixes (i.e. randomized names for looted items), some problems were mentioned:
- stacking of attack or defense modifiers can give large values
- such values get difficult to balance (one hit kill, or mostly, no damage at all)
- such items are ridiculously rare (especially attack)
Here I'd like to start a new discussion, so the other one does not get off-topic. I don't see a better place than the wish-list, however, this does not mean that I wish the proposed ideas will be used. Rather, I just want leave some notes here that may be useful for future discussions. Please note that I don't want to discuss any imbalances between attack and defense boosters here.
So, what is it about? Multipliers for affixes and how they stack. How are item stats added up or multiplied? And what can we do about super rare items? For this, let us first look at additive stacking. We know this well from affixes that raise stats or skills, for example:
- Spidersilk Fish: Agility +10
- Eagle's Fish: Agility +20
- Spidersilk Eagle's Fish: Agility +30
Things are different for modifiers to damage (or protection), defense and attack. Here the factors are multiplied, for example:
- Ancient Fish: Defense *1.5 or +50%
- Fish of the Defender: Defense *2 or +100%
- Ancient Fish of the Defender: Defense *3 or +200%
If we add up the percentages instead, we get 50% +100% = +150% or a factor of 2.5. Not a huge difference. But we soon get there.
If you consider attack and defense modifiers, the bonus is not only applied to that particular item, but to all equipment. Consider 3 ancient items of the defender, two weapons and a helm.
- Multiplicative: 33 = 27 or +2'600%
- Additive: +150% +150% +150% = +450% or a factor of 5.5
Now that's a significant change. And it gets even more pronounced if you take into account more defense modifiers from spells. This concept applies to attack and defense equally.
Side note: For damage or protection stats (same affixes on weapons or armor), the bonus is only applied to the given item. For the combat rules, though, the overall attack or defense modifier is multiplied, respectively. So, if damage and attack stack additively each, and you want to maximize the product, you will want to ramp up both factors.
Okay, so we could make all item buffs stack additively each. But what about other random loot properties? Another term, the value in tria, is straight forward. Currently, each affix raises the price by a factor, e.g.
- Ancient Fish: Value *3 or +200%
- Fish of the Defender: Value *5 or +400%
- Ancient Fish of the Defender: Value *15 or +1'400%
It is trivial to see how the percentages could be summed up, which would greatly lower the value of such rare items. However, that does not have to be a reasonable thing to do. Exponentially large tria values for the rarer items are okay. I mention this, because it may get relevant if you address a more interesting property: Rarity.
So, we said that items with very good attack properties are incredibly rare to find. You can look up the rarity percentages in the Hydlaa Museum if you inspect the short sword collection. Single affixes typically have a rarity of 0.1% to 1.1%. There's a few more common excpetions, such as "of the Defender" with 2.7%. If you look at rare weapons in the museum, you observe rarities in the order of 0.0000005%. Whoah... that's how many zeroe? Ouch, that's a chance of 5E-9 or one in 200 millions. The fact that such items are found in the museum is just the result of a bug, which once allowed much higher loot rates. So, why are these so extremely rare?
My guess is that the rolls for random affixes are independent: For each part of the name, adjective, prefix and suffix, there is a seperate chance to get a bonus. So, if you are looking for a particular combination of affixes, the rarity values need to be multiplied. Let us look at the inverse probability N = 1/p, the expected number of attempts it takes to find an item. How does it look for affixes with a moderate rarity of 0.5% or N=200 each?
- Spidersilk Fish: N = 200 or 0.5%
- Spidersilk Clacker's Fish: N = 40'000 or 0.00 25%
- Spidersilk Clacker's Fish of Seduction: N = 8'000'000 or 0.00 001 25%
That's lottery. But it can get even worse. There's also a few affixes with N=1'000 or 0.1% rarity, and one particularly useful prefix with 0.07%. Let's consider 0.1% each, although there's no suffix "of Imbalance" with this:
- Masterwork Fish: N = 1'000 or 0.1%
- Masterwork Quake-Steel Fish: N = one million
- Masterwork Quake-Steel Fish of Imbalance: N = one billion
One in a billion? You're kidding! Okay, it's just an estimate for a worst case, but remember the best items in the museum, their rarity is not all that far from this!
So, what could we do to make rarity additive in some sense? First thought: Nothing, it's natural that independent probabilites are multiplied, and things need to be normalized. Still, we can use the following steps to get a compromise:
- Let the loot table define how many affixes the item will have, if any.
- Assign the needed affixes without any bias (uniform trial probability).
- Then decide if the set of affixes is kept or removed (acceptance probability).
- The inverse acceptance probabilities (N) of the affixes are stacked.
The first point gives us the option that the most powerful enemies can drop long item names much more frequently, but you don't get a chance to find a super rare item when molesting gobblins. Maybe that will be tricky to implement, but it could be very useful to tune those loot rates.
The second point means that each affix gets the same chance to be considered. For simplicity, assume there are 30 adjectives, prefixes and suffixes each. Then the trial probability for each affix is one to thirty. However, if you are looking for a certain combination of two or three affixes, the inverse trial probability ramps up to N = 900 or 27'000, respectively.
The third point introduces an acceptance probability. For our example and a single affix, this is simply the rarity times 30. So, the worst acceptance on a single affix is N = 1000/30 or about 33 tries.
The fourth point implies that combinations of precious affixes are not penalized too heavily in terms of rarity. We simply add up the inverse acceptances N. Let us come back to the example of 0.5% rarity on each affix:
- Spidersilk Fish: N = 30*200/30 = 200 or 0.5%
- Spidersilk Clacker's Fish: N = 302*2*200/30 = 30*400 = 12'000
- Spidersilk Clacker's Fish of Seduction: N = 303*3*200/30 = 900*600 = 540'000
So, the double affix is more than 3 times easier to find, and the triple affix goes down from N = 8 million to a bit more than half a million.
For the second example, we considered affixes with N = 1'000 each. Again, there's no such suffix, but it seems a simple estimate for a worst case:
- Masterwork Fish: N = 30*1'000/30 = 1'000 or 0.1%
- Masterwork Quake-Steel Fish: N = 302*2*1'000/30 = 30*2'000 = 60'000
- Masterwork Quake-Steel Fish of Imbalance: N = 303*3*1'000/30 = 900*3'000 = 2.7 millions
All right, the rarity of a powerful double affix goes down from N = one million to 60'000. That's a huge difference. The worst case number of 2.7 millions is still terrible, but it is almost 3 orders of magnitude better than before (one billion).
One more side note: We simplified things, as we considered 30 prefixes, adjectives and suffixes each. If you plug in the actual numbers (it's around 20 for the suffixes), the situation does not get worse, but even a little better.
Okay, that was a lot of reading and lots of numbers. Maybe some of you will make sense of it. If it helps to get new rules done one day, that would be even better.
