A miner controlling a fraction p of the mining power has a probability p to mine each block. This follows a Bernoulli distribution and the variance is given by p(1-p). Since all blocks are independent, and since in a year there are 365 days of 24 hours of 6 blocks each, the annual variance is 365.24.6.p(1-p) while the expected return is 365.24.6.p
The reason of course is that mining costs are fixed. A miner must thus compare the relative standard deviation of this revenue and his expected revenue. A miner projecting to obtain an expected return on investment of 10% and controlling 0.19% of the hashing power would have about a 16% chance of losing money by the end of the year.
The relative standard deviation is the square root of the variance divided by the expected return, that is:
But why is variance so important? After all, statistically it all boils down to the same expection. Are people so risk averse?
Does it mean that the mining landscape is doomed to be dominated by a monopoly? Not necessarily. The solution is for miners to outsource the risk taking, not the mining. This is what P2P mining achieves. P2P mining acts as a distributed insurer which offer miners a swap between between their expected returns and their actual returns.
If the Sharpe ratio falls, the risk-reward ratio becomes less attractive. At one point, it becomes better for the miner to just invest in the stock market rather than to attempt mining.
If you control 10% of the hashing power in the network, your annual relative standard deviation is 1.3%, control only 1% and it climbs to 4.3%, control a measly 0.1% and it’s 13.8%… for a tiny miner representing 0.01% of the pool (mind you, that is still a miner averaging 11 bitcoin of monthly revenues) the variance is a whooping 43.6%
One can form the ratio of these two numbers, this is known as a Sharpe ratio (interest rates should figure in there, but they’re near 0 anyway).
In bitcoin, miners join pools for two reasons. One is to avoid the hassle of running a full node — this could be mitigated by requiring miners to include proofs of blockchain retrievability. The other is to reduce the variance of the return in their mining.
The largest miners benefit from higher Sharpe ratio than their competitors. Thus, they are encouraged to invest more in mining, pushing the margins down and squeezing competitors out of the market.
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