Training vs Earning 2

Day 2,615, 15:17 • Published in Japan Japan • by Eikyuu

First of all, I would like to thank everybody who supported the previous article on the subject - by comments, votes and endorsements. I was not expecting such interest and I hope that you will find my further thoughts and analysis of the matter just as informative.

Among other things, I wish to address one inaccuracy in what I wrote, which I realized thanks to Kagami Hiiragi's comment. It should be clear that when analyzing the income of training contracts, one should not include independent sources of strength and/or money. So, for example, the Hard Worker medal brings a profit which should be counted separately, because you earn it whether you train or not. The same is true with the daily reward (DR) of 1 or 5 strength - this increase is independent of the contract or type of the training grounds used.

Let us start with training without contracts, which will help to expand on another comment by Wigga - what are the costs at various stages of advancement? A partial answer can be found in the previous article, namely when one has all the training grounds at level 4. The daily costs are

daily cost = 0.19 + 0.89 + 1.79 = 2.87g per day.
daily strength gain = 20 + 10 + 20 + 40 = 90 strength per day.

After 30 days, we end up with 2700 = 10*250 + 200 strength, so 10 medals and 200/250 progress for the 11th medal. If we start with 50 strength or more, this will produce 11 medals. The cost is always 30*2.87 = 86.1g, so the possible losses, like Wigga wrote, are

maximal monthly loss = 86.1 - 50 = 36.1g, or
minial monthly loss = 86.1 - 55 = 31.1g.

The problem with this best/worst case approach is that it gives us two answers instead of one. The reason is that the period of 30 days produces a number which is not divisible by 250. But we are not really forced to use the monthly period. It was only chosen because of the training contract. The practical answer is to realize that we are interested not in just one month but in longer periods of time over which the numbers get averaged.

Imagine doing the above again and again, so that sometimes one earns 10 and sometimes 11 medals. What are the chances of each? To earn 10 medals the starting progress needs to be between 0 and 49 (50 different values), while to earn 11 medal it has to be between 50 and 249 (200 different values). The ratio of possibilities, and thus chances, is simply 4.

In other words, on average, for every 5 trials, 4 of them give 11 medals, while 1 of them gives 10 medals. Not surprisingly, the average turns out to be exactly 2700/250:

mean number of medals per month = 11*4/5 + 10*1/5 = 10.8 medals,
mean loss per month = 86.1 - 10.8*5 = 32.1g

Of course one does not earn 10.8 medals in one month, but 11 or 10, and after 5 months they will most probably earn 10.8*5 = 54 medals. It will not be exactly 54 because of additional, random sources of strength like e.g. missions. Also this is where the DR come in. As mentioned, they are an independent source, so they can, and should, be counted separately since it is not a problem to add them at the end if need be.

Daily rewards regularly give 1 + 1 + 1 + 1 + 5 = 9 strength per 5 days, so 54 per 30 days. Let us switch to means again. How much gold does that give on average?

mean number of medals per month = 54/250 = 0.216 medals,
DR mean monthly income = 5 * 0.216 = 1.08g.
L4 training and DR loss per month = 32.1 - 1.08 = 31.02g.

This is, again, the same as calculated by taking the monthly strength gain of 2754 and counting fractions of medals: (2754/250)*5g. Note that with high enough starting strength a whole medal can be earned in one month just with DR. This will, of course, reset the progress, and the next month no medal will be earned.

Having separated the DR, one can compare the average losses when training with and without a contract. The previous article should thus be extended with the values excluding DR

minimal L4 contract training loss = 11 * 5 - 61.7 = 6.7g
maximal L4 contract training loss = 10 * 5 - 61.7 = 11.7g
mean L4 contract training loss per month = 5 * 2700/250 - 61.7 = 7.7g.

Keep in mind that in addition to including the DR, the previous article was the best case scenario. In other words, it assumed that the starting strength was between 246 and 249 (9/250 = 3.6% chance) so that both the maximal number of 11 medals was earned from training contract and the maximal number of 1 medal was earned from daily rewards giving 6.7 - 5 = 1.7g of loss - considerably less than on average.

Finally, this approach also allows us to see how much each training facility contributes. For your convenience, I list the monthly profits of all types, so they can be added in any combination that is required with or without the DR profit.

Name
- - - - - - - - - - - - - - - - - - - -
Weights Room
Climbing Center
Shooting Range
Special Forces Center
Q1
- - - - -
+3.0
-4.2
-23.7
-47.7
Q2
- - - - - -
+6.0
-2.7
-20.7
-41.7
Q3
- - - - - -
+9.0
-1.2
-17.7
-35.7
Q4
- - - - - - -
+12.0
+0.3
-14.7
-29.7

So, e.g., the only facility that increases the basic income is Q4 Climbing Center - by the humongous amount of exactly 0.01 gold per day. On the other hand, the first three facilities at Q4 together with DR generate a monthly loss of 1.32g.

As is known, training cannot really be thought of as a source of profit in itself, but is the basis for understanding how to efficiently build one's path towards faster advancement, which requires both strength and gold. To tackle those more general problems, one needs to understand some basic concepts like the average or calculation of chances. I hope that, apart from the numbers, the readers will find the introduction of these concepts useful in the future.