## Explanation of Prestige

### TL;DR

*Prestige is a converging process. The best time to use a prestige token is when the prestige multiplier is low (e.g., <300%), as the potential income will be realized more quickly by using the token at lower multipliers.*

### Prestige Calculation

The process of prestige is actually a converging process. The effect of using prestige tokens is essentially aiding convergence rather than obtaining additional gains. Therefore, when comparing the two strategies, one should compare their convergence speeds rather than their numerical values.

When your efficiency of gaining firestone is x times the original, the firestone gaining rate before the first prestige is F(0) = x. Let F(n) represent firestone gaining rate after the nth prestige, then approximately log(F(n)) = log(F(n-1)) + 0.6^n * log(F(0)), that is:

`log(F(n)) = log(F(0)) * (1 + 0.6 + 0.6^2 + … + 0.6^n)`

As n -> ∞, log(F(n)) -> 2.5 * log(F(0)), or F(n) = F(0) ^ 2.5.

For example, with a 1.3x firestone gaining efficiency, the actual effect is approximately 1.93x because 1.3^2.5 = 1.93. The “realizing potential income” (i.e., going from 1.3x to 1.93x) is the mathematical essence of prestige.

When considering these two strategies:

- A. Use a prestige token at +900% prestige multiplier.
- B. Use a prestige token at +100% prestige multiplier.

**A.** Let’s assume in strategy A, F(0) = 10, then F(1) = 40, F(2) = 91, … , F(n) -> 316. Using the prestige token is equivalent to accelerating the convergence; in the first prestige, the firestones becomes 1.9x (19/10 = 1.9).

The starting point for convergence changes from 10 to 19 (the endpoint remains unchanged because it’s not a permanent bonus).

**B. **Assume in strategy B, F(0) = 2, then F(1) = 3.03, F(2) = 3.89, … , F(n) -> 5.66. Using the prestige token is equivalent to accelerating the convergence; in the first prestige, the firestones becomes 1.5x (3/2 = 1.5).

The starting point for convergence changes from 2 to 3 (the endpoint remains unchanged).

It can be observed that, after using the prestige token, in strategy B, it is closer to the convergence endpoint. The effect of the first prestige is close to the original second prestige (3 vs 3.03). In strategy A, using the prestige token, the effect of the first prestige is not as good as the original second prestige (19 vs 40).

Therefore, in practice, using prestige tokens at lower rates makes the acceleration of convergence more pronounced. However, it doesn’t affect the convergence endpoint because it’s not a permanent effect. In fact, a prestigious research with +100% effect is equivalent to using a prestige token at every prestige.

Furthermore, you may notice that in some cases, using a prestige token may surpass the original endpoint. This is possible because you temporarily borrowed the “prestigious research with +100% effect” that originally did not belong to you. When you return it, the percentage increase in firestones after prestige will remain low until you truly obtain the permanent “prestigious research with +100% effect”.

Another misconception is that firestones are permanent; therefore, increasing the firestone gain rate temporarily will also have a permanent effect. However, over time, higher-order effects can cause lower-order effects to be ignored. Let’s illustrate this with an example: if you spend 10 hours on a plane, the distance you walked for 10 hours before can be neglected. If you want to travel a greater distance, you should seek a faster mode of transportation rather than walking for more time or increasing your running speed. In this game, prestige tokens are like transitioning from walking to running, and after the firestones increase, soon you can take a plane or even a faster mode of transportation.

As for prestige, it only makes those who are running (using prestige tokens) run faster, and planes (permanent effects) also fly faster, but it doesn’t provide faster planes for those who are running. On the contrary, when someone who was running gets on a plane, the increase in speed is somewhat less.

In reality, the speed of light is only eight orders of magnitude higher than walking, while the magnitude of firestone can reach several thousand. Doubling the firestone quantity at 1e1000 has no impact on the firestone quantity at 1e2000.

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