Re: [ANN] Bancor | Protocol for Smart-tokens, solving the liquidity problem
One of the ways to think about reserves is as "liquidity pools". Reserves are essentially common pools that provide liquidity to their smart token holders. The current solution for liquidity is based on market orders on the different exchanges (and in some cases, in different currencies pairs). If you measure the size of these current "liquidity pools" (aka as "market depth"), relative to the market-cap of the currency, and compare the event where similar amounts are unloaded to the market through the exchanges vs. through a smart token with 10% reserve, you would probably find out that the market behaves in a much more stable way using the smart token, since there is a single common liquidity pool, which is growing relative to the market cap of the smart token.
It can be tricky to wrap the mind around it since this is a model for linking between currencies which is quite different than the one being used today and for a long time now, and I really hope my attempt to explain it was clear enough.
Anyway, I'll be happy to answer any other questions you may have!
And thank you for your interest
I went through all the math and tried out some basic modeling because I am excited and curious how it all works. One question though, the first step that allowed you to make a continuous formula out of the recursive: dR = PdS and then use that equation later, is this some known approach that helps to make continuous formula out of recursive, that I don't know about because of my limited math education, or is it a clever trick that you can credit to yourself?
This is basic calculus.
Alright, given A1, A2, A3 and recursive formula An+An+1/An+2 = An+3, eg A1=1 A2=2 A3=4 therefore A4 = 1 + 2/3 =5/3
Can you show me how to make continuous formula out of it f(x) eg f(4) = 5/3
They solve a simple differential equation. Check any calculus textbook to learn how to do this.
Sure, but first you need to get to the point of having simple differential equation, hence my question, can you get the simple differential equation out of recursive formula I provided?
You "recursive formula" makes a very little sense to me. Expression for bancor price is derived by integrating a differential equation which you pointed out in you first post, dR = PdS. This is not a rocket science, this is a level of homework for undergraduate students of technical schools. It is not possible to explain how calculus works in a forum post, just check any textbook, they explain all theory very well.
You don't seem to understand the question.
The original formula Price = Reserve / (Supply * CRR) is recursive.
It's calculated in sequential steps , add unit of Reserve, increase Supply according to price, recalculate price, rinse and repeat.
According to you making a continuous formula f(x) out of it is "basic calculus".
If that's that's true then it should be easy for you to make continuous formula out of any recursive formula.
Therefore please make continuous formula out of An+An+1/An+2 = An+3 .
You don't need no "recursive formula" to solve this problem. Derivatives and integrals of many analytic functions were derived using the limit theory long before we were born, you just use them when you need. Sorry for being boring, but I'd like to direct you to calculus textbook for undergrads again, where all such derivations are given in details.
I am going to try to clarify it one last time, if it doesn't work so be it.
I don't have a problem with derivatives and understanding their proof .
My question was not about the technique or how in this particular case they got their formulas.
Initially they had recursive formula Price = Reserve / (Supply * CRR) and using derivatives they were able to transform it to continuous formula Price(x) = ((1+ x / InitialSupply)^(1/CRR -1)) * InitialPrice
My question is there a general method to transform any recursive formula into continuous, or was it the clever trick that allowed to do that in just this particular case.
Oh yes, you do have a problem understanding derivatives. No, they don't have no "recursive function" not initially, not finally, not in the middle. Price(x) = ((x / InitialSupply)^(1/CRR -1)) * InitialPrice is a fucking integral of a fucking derivative, which you can find in every fucking math textbook. Derivative of log(x) over x equals 1/x; integral of dx/x = log(x) * constant, voua la, problem solved. Just RTFM dude.