How quickly should your players be training?

A common question for managers is whether a player is training fast or slow. Provided below is a lookup table to determine the expected training days (how many days worth of training) it should take a player to reach the next level. The player's age is listed on top and the level they are training to reach next is presented on the left. https://docs.google.com/spreadsheets/d/1WvtEShpFJHOZRhShxHZJhQe-0fqjKDZJ8B4IGS3adR4/edit?usp=sharing Some possible questions about this data are proposed and answered below. --- What assumptions does this make? --- - Training speeds can be modeled by a single non stepwise function - there is not a sudden change in training speeds for 21YOs or for players training a skill past 10. - Potential and ceiling are ignored/excluded from the data - this is discussed farther down, but a guess was developed for potential to exclude those data points and ceiling was not observed. - Players rate of training is governed solely by level and age. - There have been no changes to training since the world started. - Amateurs train at the same rate as professionals. --- How was this data generated? --- This was generated from the training logs of my players and then messed with until there was a model that fit the data. The formula for calculating training times can be pulled from the spreadsheet. --- What does it mean for the training days to be fractional? --- Think of the players skill as a number somewhere between the listed value and one more. Each day, a fractional amount is added to that value based on the inverse of the listed training days. A player that requires 2.5 training days receives 0.4 to the value of that skill each day. Therefore, they are expected to do it in either two or three days, depending on how much training has bled over from the previous level. --- Why is my player training faster than the model says? --- The first reason would be that this model is wrong. Systematic failures to predict training times would suggest that there is a part of the picture that this is missing. However, training times may sometimes be faster for a few explainable reasons: - The skill was already partially trained (primarily relevant for new trades) - The player aged up while training the skill - Although an edge case, carryover from a previous level may confuse results, since it can effectively add one extra day at a faster training level. To return to the previous example about fractional values, suppose the next level takes 4 days, so each training day produces 0.25 of value. If the previous happens slightly "late" and spills over 0.3 worth of training into the next level, it would only take 3 days to train the skill, instead of the predicted 4. This can be further confounded by the aforementioned issue of players aging up, either while training the skill or in-between stretches of training that skill. Training much more than a full day faster than predicted is a sign of a problem with the model. --- Why is my player training slower than the model says? --- Again, the model could be wrong. Alternatively, the player has begun to reach their potential and training is slowing down. The training days listed in the table do not take into account potential or ceiling. --- How much impact does potential have? --- Off the cuff, the first level that bumps up against potential seems to increase training times ~17.5% and the second increases it by ~35% (applied as a multiplier to the predicted training time). That would suggest a linear growth for the potential penalty, which does not match some of the crazier training times people have seen. Hence, ~20% per level is a good rule of thumb, but likely inaccurate. Keep in my mind that for some of the early levels it can be hard to notice a 20% increase before training time seems to suddenly jump on the next level. --- How is this affected by ceiling? --- I have no idea. This data this is derivative of quite likely contains only high ceiling players. --- Is this accurate? --- I certainly hope so, but it seems unlikely that this is more than partially correct. This has proven accurate so far for my players, but that is a limited sample size and I may have overfit the model to my data.