Revista PowerPlay

After reading the guide I once again found myself facing the following statement:
"Player with attributes 180 - 25 - 25 or 70 - 90 - 90 (where the first one is the primary main attribute and the last two are secondary main attributes) is not as good for the given position as a player with attributes 120 - 30 - 50. Similarly a player with attributes 130 - 80 - 30 or 80 - 80 - 80 is not as good as a player with attributes 100 - 80 - 50."

It's not the first time I thought about it, and I could never figure out the math behind the statement, however I think if it were not true it would not be in the guide. I decided to once again review the matter. I combined facts and some logical assumptions in order to determine the optimal way to train players:

Fact:
There is an optimum ratio between the Primary Attribute (PA) and Secondary Attributes (SA) that allows the player to achieve his full performance potential. This relationship has a value that I will call RT.

Analyzing the statement of the guide, I am led to believe that it's useless to train only the PA, because it reaches a point where it's increase, if not accompanied by an increase in the SAs, will not bring any additional benefit to the player. Similarly, training only the SAs brings forth a point where it ceases to be beneficial, unless the PA increases.

Assumption:
Based on the previous analysis, I assumed that the value of the PA determines the limits of usefulness of the SAs, and likewise the value of the SAs determine the limit of the usefulness of the PA.

This means that for a player to use his PA to 100%, he should have the SAs within a certain range (the logic being that it doesn't matter how high the passing is if a goalkeeper is mediocre, or how high the goalkeepers goalkeeping ability is if his passing is too low). And the same applies to the use of 100% of SA.

Assumption:
Based on the qualification of the guide, we have one primary attribute and two secondary, rather than a secondary and a tertiary. I assumed that the relationship between the values of the PA and each of the SA should be equal. For the same reason, the PA should be higher than the SA. Thus, PA > SA1 = SA2.

I believe that this assumption may not generate consensus, but to me it seems logical that it is so, given that this is a simulation and not real life. Anyway, if you do not agree please stay with me a while longer, until I have the opportunity to finish my reasoning.

Based on the facts and assumptions presented so far, the maximum useful value of the PA is limited by the smallest of the SA, just as the maximum useful value of each SA is limited by the value of the PA. Thus we have the following:

- Maximum useful value of PA is equal to the minimum value between PA, RT * SA1 and  RT * SA2
- Maximum useful value of SA1 is equal to the minimum value between SA1 and PA / RT
- Maximum useful value of SA2 is equal to the minimum value between SA2 and PA / RT

Consider a player with the attributes 180-20-70 (PA-SA1-SA2). If we apply this theory using RT = 2, the maximum useful value of PA will be the lowest value between 180 (PA), 40 (SA1*2) and 140 (SA2*2), the maximum useful value of SA1 will be the lowest between 20 and 90 (PA/2) and the maximum useful value of SA2 will be the lowest between 70 and 90 (PA/2). Below is a table with the players useful values for various RT.

- If RT = 2, the maximum useful value for Player A would be 40-20-70 (PA-SA1-SA2)
- If RT = 3, the maximum useful value for Player A would be 60-20-60 (PA-SA1-SA2)
- If RT = 4, the maximum useful value for Player A would be 80-20-45 (PA-SA1-SA2)

This means that if the value of RT is two, Player A will only use 22% of his PA due to the limit imposed by the low SA1, but would use his SA to their full potential. If the value of RT is four, he will use 44% of his PA, however the use of his SA2 falls to 64%.
To make a parallel to the existing types of training, so as to simplify my reasoning, coaches who use the training method 100-50-50 (or 2-1-1) believe that the RT value is equal to 2, and therefore make sure that the PA is never more than twice the SAs nor the SAs lower than half of the PA.


Determination of Optimal RT:
Having explained my theory, I now move on to the purpose of all this. I am convinced that setting up a player's training, so that their attributes are always within the limits set by my theory, will allow us to take full advantage of their quality as they age as well as reach their full potential. That being said, let's go back to the claim that started my intellectual exercise..

Fact:
Player A1 (120-30-50) is better than players A2 (180-25-25) and A3 (70-90-90). Player B1 (100-80-50) is better than players B2 (130-80-30) and B3 (80-80-80).

According to the above, I am forced to conclude that the optimum RT value is such that respect all the comparisons expressed in the fact. Since most believe in training by the 100-50-50 system, I'll first test my theory for a RT = 2. Thus the useful values of the players will be:

A1(60-30-50)
A2(50-25-25)
A3(70-35-35)

B1(100-50-50)
B2(60-65-30)
B3(80-40-40)

To be able to compare them I will have to calculate the value of their quality, and to do this I need a formula. I will start by using the simplest way to compare the players and assume that their quality PQ is the sum of their attributes AP + SA1 + SA2.

A1 PQ = 140
A2 PQ = 100
A3PQ  = 140

B1 PQ = 200
B2 PQ = 155
B3 PQ = 160

It appears that A1 is not better than A3, so I have to conclude that the value of RT and/or formula of comparison are not correct. I will change the value of RT and keep the formula of comparison for now, and later question it's validity. Assuming RT = 3 we have:

A1 (90-30-40), PQ = 160
A2 (70-25-25), PQ = 120
A3 (70-23-23), PQ = 116

B1 (100-33-33), PQ = 166
B2 (90-43-30), PQ = 163
B3 (80-27-27), PQ = 134

This time all the comparisons are true, A1 is better than A2 and A3, and B1 is better than B2 and B3. Let's change the formula to see if a different one will also prove correct. 
If some attributes are called PA and others are SA, the logical thing would be for PA to have a greater weight than the others, so I will now test the formula of comparison PA * 2 + SA1 + SA2 (this is nothing more than considering that the PA contributes 50% to the quality of the player and the SA with 25% each)

A1 (90-30-40), PQ = 250
A2 (70-25-25), PQ = 190
A3 (70-23-23), PQ = 186

B1 (100-33-33), PQ = 266
B2 (90-43-30), PQ = 253
B3 (80-27-27), PQ = 214

Again, all comparisons are true. In an attempt to eliminate the value of RT = 3, I tried another formula. I used the formula PA * 3 + SA1 * 2 + SA2 * 2 (ie the PA contributes 44% to the quality of the player and SA contribute 28% each)

A1 (90-30-40), PQ = 410
A2 (70-25-25), PQ = 310
A3 (70-23-23), PQ = 302

B1 (100-33-33), PQ = 432
B2 (90-43-30), PQ = 416
B3 (80-27-27), PQ = 348

I was able to prove that three is a possible value for the RT, considering the three formulas used.
Motivated by my success, I decided to automate the process in order to verify the truth of my theory when my two variables changed, namely the value of RT and the formula for comparing players.
I organized my spread sheet according to the formula and obtained the following comparison:

- The equation PA + SA1 + SA2 is valid for RT values between 2.001 and 3.088
- The formula 2 * PA + SA1 + SA2 is valid for RT values between 2.108 and 3.198
- The formula 3 * PA + SA1 + SA2 is valid for RT values between 2.164 and 3.240
- The formula 4 * PA + SA1 + SA2 is valid for RT values between 2.198 and 3.262
- The formula 3 * PA * + 2 * SA1 + 2 * SA2 is valid for RT values between 2.064 and 3.159
- The formula 4 * PA * + 3 * SA1 + 3 * SA2 is valid for RT values between 2.045 and 3.140
- The formula 5 * PA * + 3 * SA1 + 3 * SA2 is valid for RT values between 2.080 and 3.174

Fact:
A player is better than another, whatever the form of comparison, the greater the difference is between their performance quality.

On that basis I assumed that the optimum value of the RT would be one that, not only would make the four comparisons true (A1> A2, A1> A3 and B1> B2, B1> B3) as well as the sum of their minimum differences ( min((A1-A2);(A1-A3)) + min((B1-B2);(B1-B3)) ), would reach a maximum.

Assumption:
When the sum of the minimum value between (A1-A2) and (A1-A3) with the minimum value between (B1-B2) and (B1-B3) reaches it's maximum, we will have found the optimum value of RT.

Applying this assumption to my calculations I noticed something that surprised me.

Any comparison formula is comprised of factors and attributes (in the simplest formula all factors are equal to 1; 1 * PA + 1 * SA1 + 1 * SA2). I found that in all formulas where the PA contributes with 50% or less to the performance quality of a player, or in other terms, in all formulas in which the sum of the factors applied to the SA are greater than or equal to the factor applied to PA, the optimal value of RT is 2.4 (tested up to 7 decimal places).

(...)
- 1*AP+6*AS1+6*AS2, optimum RT = 2.4
- 1*AP+5*AS1+5*AS2, optimum RT = 2.4
- 1*AP+4*AS1+4*AS2, optimum RT = 2.4
- 1*AP+3*AS1+3*AS2, optimum RT = 2.4
- 1*AP+2*AS1+2*AS2, optimum RT = 2.4
- 1*AP+1*AS1+1*AS2, optimum RT = 2.4
- 2*AP+1*AS1+1*AS2, optimum RT = 2.4
- 3*AP+2*AS1+2*AS2, optimum RT = 2.4
- 4*AP+3*AS1+3*AS2, optimum RT = 2.4
- 5*AP+3*AS1+3*AS2, optimum RT = 2.4
- 5*AP+4*AS1+4*AS2, optimum RT = 2.4
- 6*AP+4*AS1+4*AS2, optimum RT = 2.4
- 6*AP+5*AS1+5*AS2, optimum RT = 2.4
(...)

For all formulas in which the sum of the factors applied to the SA are less than the factor applied to the PA (ie. when one considers that the PA contributes more than 50% for the quality of the player), the optimal RT value tends to 2.7:

(...)
 5*AP+2*AS1+2*AS2, optimum RT = 2.44
 6*AP+2*AS1+2*AS2, optimum RT = 2.47
 3*AP+1*AS1+1*AS2, optimum RT = 2.47
 4*AP+1*AS1+1*AS2, optimum RT = 2.52
 5*AP+1*AS1+1*AS2, optimum RT = 2.55
 6*AP+1*AS1+1*AS2, optimum RT = 2.57
 9*AP+1*AS1+1*AS2, optimum RT = 2.60
20*AP+1*AS1+1*AS2, optimum RT = 2.64
(...)


Conclusion:
- Whatever the formula of comparison, the optimal RT is between 2.4 and 2.7, being always equal to 2.4 in the formulas that consider the influence of PA on the quality of the player does not exceed 50%.
- The value of 2.7 means the best training would be with a ratio of 27/10/10, while the 2.4 would mean a ratio of 24/10/10
- Most managers in PPM believe in a training with a ratio of 100-50-50 (or 24/12/12)


Final notes:
I believe that the optimum ratio is closer to 2.4 than 2.7, so according to my theory I use the training ratio 24/10/10.
This does not mean that we should only train players in those three attributes, it just means that I believe that the optimum ratio between those three attributes is 24/10/10.

We all know that for defenders, technique is also important to avoid penalties, and most prefer to have it at least equal to the value of aggressiveness, so applying my theory to defenders would mean a ratio of 24/10/10/10 (Def/Pas/Tec/Agg)
Other believe that wingers need very high shooting, for those my theory would mean something like 24/22/10/10 (Off/Sho/Tec/Agg)

I do not mean to convince anybody that this theory is valid, each one should make their own conclusions.