Tiskové zprávy - Researching Sponsors: Part 4: The Formula

Researching Sponsors: Part 4: The Formula

Author’s note: This is the final part of a 4-part series on deriving the sponsor formula. Part 1 on Tuesday discussed the genesis of the project and the early stages of research. Part 2 on Wednesday discussed the remainder of the controlled research. Part 3 on Friday discussed the data analysis.

Without further ado, it’s time to present the formula. Screenshots will be used to preserve proper mathematical notation.

First, let’s establish some variables:
C = Charisma, rounded down to the nearest integer
E = number of employees with PR manager skill greater than 0
P1, P2, …, Pn = PR manager skill of the 1st, 2nd, …, n-th employee when ranked by that skill from highest to lowest (note that the 1, 2, n, etc. after the P should be subscript)
PT = total impact of all of your employees PR Manager skills (again, T should be subscript)
R = overall driver rating
W = weekly payment of the sponsor offer

Before we look at the full formula, the PT variable requires some explanation and a definition. Here is the formula for PT:



For those who don’t understand summation notation, here is a plain English explanation of the above. After sorting your employees based on their PR manager skill from highest to lowest, the first (highest) employee contributes their full skill rating to the sponsor equation. The second employee contributes one-half of their skill points. Each subsequent employee in downward order of PR manager skill contributes half of the fraction contributed by the previous employee (so the third employee contributes 1/4, the fourth contributes 1/8, the fifth contributes 1/16, and so on). There is no limit to the number of employees that contribute, though at some point the fraction that they contribute is so tiny that their individual effect is too small to affect the sponsor offer.

Now here’s an interesting side fact that we can derive from the above formula. The explanation relies on advanced math usually taught in calculus courses, so if you don’t understand the next paragraph, skip to the "*****END ADVANCED MATH*****" marker below, after which I state the finding in plain English.

If you have an infinite number of employees with PR manager skills equal to 100, then the summation above becomes



which is an infinite series of the form



This form is known as a geometric series. Because the absolute value of r is less than 1, we can use the properties of a geometric series to conclude that the series converges. The sum of a convergent geometric series can be expressed as



Plugging into that formula gives the sum as



*****END ADVANCED MATH*****

Therefore, no matter how many PR managers you have or how high their skills are, their total contribution to the sponsor offer can never exceed that of a hypothetical lone employee with a PR manager skill of 200. (And technically, they can never equal that either, only approach it really, really closely.)

Now that we have established the variables and the meaning of PT, we can look at the full formula. The formula as I originally derived it is as follows (refer back to the variables at the beginning of this post):



With this form of the formula, it’s hard to properly understand just what effect PR managers and rating have. This is where Numpty’s assistance that I mentioned yesterday came in. Let’s utilize algebra to manipulate the formula a bit, starting with the distributive property:



Factor out an R/500 from the second term in the numerator:



Perfect. Now we can factor the first term in the numerator completely out:



This is the formula as Numpty initially presented it to me. However, some further simplification can be done. Let’s first eliminate the complex fraction by bringing the denominator out in front as a fraction multiplier:



Finally, factor out a 640 from the first set of parentheses:



This is what I consider to be the final and best form of the sponsor formula.

So, what does this mean?

1. It means that Charisma is the base of the sponsor offer. Since it can never be zero, it always contributes something. PR managers and rating, on the other hand, each provide a multiplier on that value.

2. If your PT value is equal to 125, then your employees will double your sponsor offers over not having any. The theoretical maximum multiplier that can be provided by PR managers is 2.6 (but see discussion above about why you'll never reach that).

3. The rating multiplier increases by 1 for every 500 rating points on a continuous scale. If your rating is 1,000, your sponsor offer will be triple what it would be if your rating were zero. A rating of 4,500 is a multiplier of 10. The current #1 driver in the world is just over 6,000, which would be a multiplier of 13.

4. Where does the 1/3 come in? I didn’t understand that at first, but Numpty offered a suggestion that makes sense. Since we have three sponsors, the game designers probably developed a formula for total weekly income, and then just divided that by the number of sponsors. I have no proof of that, but it makes sense and I’m going with it.

Don't look for strategy discussion here. I'm giving you the formula, but figuring out what it means in terms of strategy is your job.

Finally, some caveats to this whole project:

1. The highest rating that I have data for is 3,982.070. While I have no reason to believe that any variable is nonlinear when other variables are held constant, I cannot guarantee that there are not some oddities above 4,000 rating.

2. Likewise, the highest individual PR manager I have data for is a 96. As above, I have no reason to believe a 97-100 PR manager would be any different, but no guarantees.

3. And the biggest caveat of all: the game designers hold the power to change the formula at any time, including in response to this research publication. Therefore, every bit of this could become completely useless without notice. You have been warned.

Thank you for reading through this, and I hope you can put this information to good use. Now I’m off to retire as a researcher and start a new career driver.