1. 1) About the changes you are proposing for the object difficulty calculation, how do you plan to calibrate the parameters in order to make them work? What criteria would you use to determine if the values are correct or not?
   
2. 2) The difficulty rating of a beatmap uses the difficulty of each object and calculates a weighted sum of their values. This method is somewhat suitable for calculating the difficulty of a FC in the beatmap, but not the difficulty of non-FCs.
   
   I don't think having a single value for the difficulty of the beatmap is good enough, the difficulty of a beatmap should be a function of the goal to achieve (for example, you could have the difficulty of pressing all notes correctly, pressing 90% of them correctly, etc...).
   
   A way I think could work, is calculating a "probability of pressing each note correctly", something like:
   
   Difficulty: The difficulty value of the object, calculated according to the strain, angles, etc...
   Skill: The "level of skill" a player requires to have a 50% chance of pressing the note correctly.
   s: A parameter that changes the sensitivity the probability changes with changes in difficulty or player skill.
   
   Then, the overall difficulty of achieving certain percentage of correctly hit notes would be defined as: "The player skill that makes the probability of achieving at least the amount of correctly hit objects in the beatmap 50% (or some other fixed percentage)", in other words, the value of "player skill" that makes the cumulative density function of the associated Poisson Binomial distribution (probably approximated to a Poisson Distribution, according to Le Cam's theorem, to make calculations faster), at the amount of correctly hit notes, be equal to 0.5.
   
3. 3) You suggest weighting harder objects higher than easier objects in the score calculation of a play. I don't think that it is always the case that the results on the harder notes have the most importance when determining the worth of a play. The importance of hitting a note depends on the "s" parameter of the probability function I described (the lower "s" is, the higher the weight of the object), rather than on the absolute difficulty.

Full Tablet wrote:

1) About the changes you are proposing for the object difficulty calculation, how do you plan to calibrate the parameters in order to make them work? What criteria would you use to determine if the values are correct or not?

Full Tablet wrote:

2) The difficulty rating of a beatmap uses the difficulty of each object and calculates a weighted sum of their values. This method is somewhat suitable for calculating the difficulty of a FC in the beatmap, but not the difficulty of non-FCs.

I don't think having a single value for the difficulty of the beatmap is good enough, the difficulty of a beatmap should be a function of the goal to achieve (for example, you could have the difficulty of pressing all notes correctly, pressing 90% of them correctly, etc...).

A way I think could work, is calculating a "probability of pressing each note correctly", something like:

Difficulty: The difficulty value of the object, calculated according to the strain, angles, etc...
Skill: The "level of skill" a player requires to have a 50% chance of pressing the note correctly.
s: A parameter that changes the sensitivity the probability changes with changes in difficulty or player skill.

Then, the overall difficulty of achieving certain percentage of correctly hit notes would be defined as: "The player skill that makes the probability of achieving at least the amount of correctly hit objects in the beatmap 50% (or some other fixed percentage)", in other words, the value of "player skill" that makes the cumulative density function of the associated Poisson Binomial distribution (probably approximated to a Poisson Distribution, according to Le Cam's theorem, to make calculations faster), at the amount of correctly hit notes, be equal to 0.5.

Full Tablet wrote:

3) You suggest weighting harder objects higher than easier objects in the score calculation of a play. I don't think that it is always the case that the results on the harder notes have the most importance when determining the worth of a play. The importance of hitting a note depends on the "s" parameter of the probability function I described (the lower "s" is, the higher the weight of the object), rather than on the absolute difficulty.

Syrin wrote:

I think I see what you're saying, although I'm not sure I understand entirely. I kept difficulty totaling this way for now because it's much closer to the ppv2 method.

Later in the essay (in the performance and rankings section iirc) I mentioned that the single value approach had its issues and discussed the workaround of having the "difficulty value" as simply a value for show and not relevant to any actual calculations, and instead using the difficulty of individual objects to calculate performance/skill points.

This method has the issue however of storage, since the database storing difficulty values used for performance calculations would need access to hundreds or thousands of values per beatmap rather than 3.

This problem can be avoided however at the cost of accuracy (and becomes potentially exploitable) by using a regression of some kind to create a curve which values could be recreated from.

I will have to look into your idea further so understand it better, but it sounds very interesting regardless.

Syrin wrote:

The scoring system proposed is focused at tournament play. Rewarding harder actions more points seems like the only way this makes any sense. Also isn't "s" a constant? If not, how is "s" defined for each object/difficulty point? I may be misunderstanding what you mean by that probability model though.

Full Tablet wrote:

The difficulty calculator can calculate a difficulty value for several goals for map (for example, 100%, 99%, 95%, etc...), and then interpolate for percentages that are in-between.

The difficulty of hitting a stream is mostly physical, a player that doesn't have enough finger speed would find hitting the note correctly near impossible, while if the player can easily achieve speeds faster than that, hitting the note would be rather easy, making misses rather rare. In this case, the value for "s" would be rather small.

Meanwhile, in the slow rhythmically complex note, it is expected that most players have at least some significant chance of hitting the note correctly, but that even experienced players have some significant chance of not hitting it correctly. The value for "s" is very big in this case.