How to Override a Transaction

What you're really doing is creating a new transaction, spending the same coins to the same addresses, but with a higher fee this time. So it sort of overrides your previous transaction. And now the question is, why would other nodes and miners care about that? Now other nodes might not care about it at all. Initially, I think the default was that they would just ignore it because for them, there's nothing in it to relay your transaction. But eventually, a mechanism was introduced to formally asked for a fee raise,. essentially, to ask nicely, hey, this transaction has a higher fee, please replace it.