Thread by @n1punp

1/ Just read @paradigm's blog on GOO: @artgobblers & its inflationary token Goo.

Some interesting game theory & neat #MATH tricks just fit together in the design.

Anyways, here's how you can earn the highest Goo emission rate. 🧵👇

2/ If you don't alr know what Goo is, just read this thread by @_Dave__White_.

3/ To get the highest Goo emission rate, first we need to understand that Goo emission is based on:
- Each Gobbler's multiplier
- Goo amount allocated to each gobbler (can be different)

The Goo emission rate will then be `sqrt(m*a)` , where m=multiplier and a=allocation of Goo.

4/ So, here comes the 2 questions:
1. Optimal Goo Distribution -- If you hold certain Goo amount and certain number of Gobblers, how should you distribute them?
2. Goo & Gobbler Pricing -- What should you buy? Gobblers or Goos?

5/ Optimal Goo Distribution

If you hold multiple gobblers (each with different emission multiple), the optimal way to distribute your Goo is to proportionally allocate to each gobbler based on the multiplier.

But why?

6/ Optimal Goo Distribution (cont.)

This directly comes from Cauchy–Schwarz inequality, where the equality occurs at the proportional allocation (a_i) to the multipliers (m_i).

So, given the Gobbler multipliers, you can efficiently allocate your Goos proportionally.

7/ Goo & Gobbler Pricing

The problem is, given $M, how should you spend this to earn the highest Goo emission rate?

To simplify things, we assume things can be bought fractionally:
- $1 can buy `X` multiplier for Gobbler
- $1 can buy `Y` amount of Goo

8/ Goo & Gobbler Pricing (cont.)

If you spend $A on multiplier, you can then spend $(M-A) on Goo.

You'll end up with AX multiplier, and (M-A)Y Goo amount.

9/ Goo & Gobbler Pricing (cont.)

So, your max Goo emission rate (from optimal distribution) = `sqrt(AX(M-A)Y)`, which is maximized at A=M/2 from AM-GM inequality.

This means the optimal strategy is to spend "half" the budget on Goo and "half" on Gobbler multipliers.

10/ Goo & Gobbler Pricing (cont.)

NOTE: This is just a simple model of how you would get the best emission rate given a budget. In practice, Goo & Gobbler price may vary, so use the strat at your own discretion.