The text explains the simple math behind an influence of BCH to the BTC fee.

Before an explanation, it is important to know following facts:

BTC and BCH are sharing the same mining algorithm (SHA256), so they share miners as well.

Both blockchains suppose to mine a block every 10 minutes. BTC blocksize is 1MB, while BCH blocksize is 8MB.

Difficulty adjustment predicts hashrate necessary to find the block every 10 minutes, in the text this hashrate is called NominalHashrate. BTC changes difficulty adjustment every 2016 blocks, while BCH switches to a faster changing difficulty algorithm on November 13th 2017, since then BCH dynamically changes difficulty.

Mining reward for both blockchains is 12.5 coins + transition fee within block.

CurrentHashrate is hashrate in given moment.





Let's start.





From the figure above first and second equations define money in dollars that miners can earn in given moment. The first equation calculates dollars for BTC, and the second one for BCH. Due to an adaptive difficulty adjustment of BCH, a ratio between CurrentBCHhashrate and NominalBCHhashrate is always 1. Large blocksize(8MB) of BCH, in the future it could be unlimited, makes BCHfee approximately equal to 0.





Assume miners are soulless and they just care about money :). They would mine a coin that makes more money in given moment. In the stable regime, the first and second equations would always give the same money, once we make the equation-1 equal to the equation-2, we get the equation-3.





In the equation-4, y is ratio between BCH and BTC prices in dollars and x is proportional to a current BTC hashrate. NominalBTChashrate changes every 2016 blocks(14days), thus NominalBTChashrate is constant so C1=1/NominalBTChashrate. Finally we got the equation-5 that could be analyzed.





If y(BCHprice/BTCprice) increases, miners switch to BCH thus the BTC hashrate decreases. Once this scenario appears, BTC fee would face a quadratic growth!





Can BTC become immune to BCH?

Increase blocksize. COPY BCH! Change difficulty adjustment. COPY BCH! Change mining algorithm. VERY DANGEROUS! COPY BTG! Cobra tried it, it didn't go well. Convince businesses and users to stop using BTC thus transactions in MB would be lower than ~144MB/day (less than 300k transactions per day). THIS WORKS FINE. e.g yours.org.





Finally, BTC will have to choose between ZERO FEE and ZERO USERS!



