The most well-known form of pool-hopping is with pools using the proportional method, which is among the oldest, simplest, most widely used and most prone to hopping. By all accounts hopping in this context was first discussed in a paper from January 2011 by Nakamoto Ryo; a more accurate analysis was given shortly after in Optimal pool abuse strategy by Raulo; these results were extended in Analysis of Bitcoin Pooled Mining Reward Systems by myself.



In the proportional method, a block's reward is distributed between miners in proportion to the number of shares each of them submitted since the previous block; the reward per share is the block reward divided by the number of shares in the round. Because of this, the reward of a share submitted at any given time is affected by the number of shares already submitted since the last block; a share submitted early in the round will have a higher reward on average than a share submitted later.



It can be shown that until the number of shares in the round is 43.5% of the difficulty, a submitted share will have higher than normal reward on average; the optimal way to exploit a single proportional pool is to mine in it until this point is reached, hop to a different pool, and return when a block is found. The gain that can be achieved by following this strategy is up to 28.1%, depending on the ratio between the hashrates of hoppers and continuous miners in this pool (the more hoppers, the less they will gain). The gain can be higher if more than one proportional pool is taken advantage of (for example, 51.6% can be achieved with 2 pools).



The extra profits of hoppers come at the expense of the continuous miners. The exact loss depends on the ratio between hoppers and continuous miners; when they are equal the loss is about 17.1%, and the theoretical limit when there are only hoppers is 43.5%.