"When the public branch is longer than the private branch, the sel fish mining pool is behind the public branch. Because of the power di fferential between the selfi sh miners and the others, the chances of the sel fish miners mining on their own private branch and overtaking the main branch are small. Consequently, the selfi sh miner pool simply adopts the main branch whenever its private branch falls behind. As others find new blocks and publish them, the pool updates and mines at the current public head."
"When the sel fish miner pool finds a block, it is in an advantageous position with a single block lead on the public branch on which the honest miners operate. Instead of naively publishing this private block and notifying the rest of the miners of the newly discovered block, sel fish miners keep this block private to the pool. There are two outcomes possible at this point: either the honest miners discover a new block on the public branch, nullifying the pool's lead, or else the pool mines a second block and extends its lead on the honest miners."
"In the first scenario where the honest nodes succeed in finding a block on the public branch, nullifying the sel fish pool's lead, the pool immediately publishes its private branch (of length 1). This yields a toss-up where either branch may win. The selfi sh miners unanimously adopt and extend the previously private branch, while the honest miners will choose to mine on either branch, depending on the propagation of the noti fications. If the selfi sh pool manages to mine a subsequent block ahead of the honest miners that did not adopt the pool's recently revealed block, it publishes immediately to enjoy the revenue of both the first and the second blocks of its branch. If the honest miners mine a block after the pool's revealed block, the pool enjoys the revenue of its block, while the others get the revenue from their block. Finally, if the honest miners mine a block after their own block, they enjoy the revenue of their two blocks while the pool gets nothing."
"In the second scenario, where the sel fish pool succeeds in finding a second block, it develops a comfortable lead of two blocks that allow it with some cushion against discoveries by the honest miners. Once the pool reaches this point, it continues to mine at the head of its private branch. It publishes one block from its private branch for every block the others find. Since the sel fish pool is a minority, its lead will eventually reduce to a single block with high probability. At this point, the honest miners are too close, so the pool publishes its private branch. Since the private branch is longer than the public branch by one block, it is adopted by all miners as the main branch, and the pool enjoys the revenue of all its blocks. This brings the system back to a state where there is just a single branch until the pool bifurcates it again."