New Formula for Object Count PP Scaling

[Oscujic]

Currently, PP is scaled based on the number of hit objects using the following formula (used here and here):
double lengthBonus = 0.95 + 0.4 * Math.Min(1.0, totalHits / 2000.0) + (totalHits > 2000 ? Math.Log10(totalHits / 2000.0) * 0.5 : 0.0);
Which looks like this (x axis is totalHits, y axis is lengthBonus):

The problem is that extremely short maps can give nearly the same amount of PP as much longer maps, as long as the average difficulty over the course of the map is similar, meaning there is little reward for consistency.

The proposed solution is to change the formula to this:
double lengthBonus = Math.Pow(totalHits / 480.0, 0.2);

In the new formula, as the number of objects approaches zero, the multiplier also approaches zero, which makes intuitive sense - a map with zero objects should not be able to award any PP.

The values 480 and 0.2 were carefully chosen to change the overall PP landscape as little as possible.

As an added bonus, extremely long maps also get more PP than before.

Here are the new and old formulas compared:

[stanriders]

Length bonus is already absurdly arbitrary and incorrect, and adding a short map destroying curve would be a disaster. You'd be making ringtone-sized maps worth like 400pp while being 11*.

There's no reason for global short map nerfs to exist, the fact that a map is short doesn't mean it's necessarily easy.

Think about it in this way: is 10 seconds of 10* jumps much easier than 10 seconds of 10* jumps and then a 60 seconds of 4* filler? Assuming you're playing both once with no retries.

Length bonus needs to be difficulty aware first, and short maps that people dislike need to be fixed overall, not through a crutch like that. Both changes are hopefully coming soon though

[cihe13375]

I think what op said kinda makes sense, i.e. the pp should goes to 0 when the map length goes to zero. The reason is that the time required for a retry goes to zero with the map length (let's ignore the loading time lol), so the number of retries can be performed in a fixed time becomes infinity. Since the chance of FC a map is a non-zero number (it would only increase as the map becomes shorter), the probability of getting at least one FC in a fixed period goes to 1, regardless of the player's skill.

Of course what stanriders said about the filler example makes sense, so this really should base on the distribution of difficulty in a map instead of a global length bonus. What I want to point out is that the pp award by a map should really be based on the time required to get an FC, instead of the probability of getting an FC. Imagine the following three maps:

1. a consistently hard 1 min map
2. a 10 min map that is hard in the beginning 1 min and easy in the later 9 min
3. a 10 min map that is hard in the ending 1 min and easy in the beginning 9 min
   Assume that the chance of someone FC any of the three maps in one try is 2%. Then they will need ~50 min on average to get an FC on 1). This is basically same for 2), because they will almost 100% FC the later 9 min so all retries happen in the first 1 min. But for 3) they will need ~500 min to perform 50 tries, which is prohibitively long. To FC 3) in 50 min one need to have a 20% chance of FC in a single run, which need higher skill thus should reward more pp. (even if you really have 500 min the logic is still same, because you can spam retry a harder 1 min map that only has 0.2% chance of FC)

In other words, short maps should be nerfed twice: once for higher probability of FC in a single run, once for more chances of retry in a fixed time. I'm not sure the current length fix proposals consider this. Also the same can be said for misscount: once for short maps having larger standard deviation on [# of misses / # of difficult notes], and once for more retry chances to seek for more extreme outliers