They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism.
This is Bitcoin's consensus mechanism and it's called Nakamoto consensus.
In turn, this means that 'Bitcoin' is whatever chain has the most cumulative work done to secure it.
Nothing is 'contentious' about this. It either has the most cumulative PoW, or it doesn't.
If the minority chain has some value it might survive. If it does, that's fine. You'll have coins on both chains and you can choose to care or not.