Remember that we as miners want to maximize their payout.
Well, is there a way that we can take advantage of proportional payout schemes? This graph is a general representation of the difference between pay-per-share payouts and proportional payouts.
We know there must be some intersection between these two because the proportional payout starts off high but decreases while the pay-per-share scheme stays constant.
This graph might reveal something to you right away.
If not, don’t worry — you’re about to see how we can increase our profit margins though a little bit of cleverness.
Here, you see the area under the curve, which represents the total reward per share times the number of shares.
If we mine under just the proportional scheme, the reward gained per share decreases as more and more shares get submitted towards this block, meaning that our marginal rewards start to approach zero.
In this example, we see the same type of graph but for pay-per-share.
It’s a giant rectangle, extended a small amount by every share that we mine.
Unlike the previous scheme, this one doesn’t approach zero.
Instead, the value of each share stays constant.
A clever miner will look for ways to take advantage of both payout schemes to increase that total area under the curve, to increase their total profit from shares.
And here’s how they can do that.
Let’s say a miner starts off mining under the proportional payout scheme.
This means that each share is very valuable in the beginning.
If we start mining here and find a block quickly, then we make a large amount of profit per share.
However, if we don’t find a block for quite a while in this pool, our shares start to become less profitable.
To get maximum profit, we can reconsider where we put our mining power.
Instead of continuing to mine in the proportional scheme, we switch to the pay-per-share scheme.
Right at the intersection point.
Why? Because the reward per additional share there is higher, and will yield more profit, than the proportional payout.
To fully understand this attack, you need to see the point at which the two curves intersect.
And this is when the value per share of the proportional scheme decreases beneath the value per share of the pay-per-share scheme.
This is why the attack is known as pool hopping: the miner hops to the most valuable pool as they see fit.
A generalization is to say that a miner will rationally mine in the pool providing the most reward per share.
Because of this, honest and loyal miners will be cheated out of profit by the rational miners, who choose personal profit over the group.
Because of this, proportional pools are not feasible in practice.
Thus far, designing a mining pool reward scheme that both aligns incentives fully and is not vulnerable to pool hopping remains an open problem.
A pool with fully aligned incentives would both have miners aim to submit nonces that produce valid block headers to the pool and stay loyal to the pool.
The proportional scheme incentivizes miners to submit such nonces, but it doesn’t incentivize miner loyalty.
Pay-per-share pools are the inverse, as previously mentioned.
We’ll see how this misalignment of incentives in pay-per-share models can lead to even further attacks.