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Scenario A: A chain split occurs due to a bug or contentious rule change (i.e. the miner incentives aren't changed in any way).
Both coins initially start out with the same difficulty (and won't change for another 2016 blocks, in the worst case), but their market price will vary.
A rational economic miner will only mine the chain with the highest reward, which is the chain with the higher market price. Users of the lower value chain can offer high fees to compensate, but this is costly.
Scenario B: A hard fork chain split occurs where one chain has decreased difficulty and/or higher block rewards (not touching the 21 million limit, but pushing the release schedule forward).
Unless the new chain has a really low market value, miners are highly incentivized to mine it. It will be hard for the old chain to compete.
The question for both scenarios: How will the scenario play out? Can both chains coexist or will the less popular chain die from a negative reinforcement loop (lower value -> less blocks mined -> even lower value) and be forced to do a PoW hard fork?
And if both chains can't coexist, does that mean that any chain that tries to out-compete bitcoin for the same PoW is a threat to the network?
Why would "less blocks mined" lead to "lower value"? If anything, you would expect a currency with a slower supply increase to have a higher price. Anyway, because of difficulty adjustment, "fewer miners" -> "fewer blocks" only in the short run. In the long run the number of blocks stays the same. – Nate Eldredge – 2017-07-04T15:21:18.093
Scenario A also only exists in the short run. In the longer run, the lower-priced chain will see its difficulty fall below that of the other chain, so the returns on mining could converge again. – Nate Eldredge – 2017-07-04T15:24:13.533
@NateEldredge If less blocks are being mined, the network becomes less useful, fees will go up, etc. I am unsure whether less newly mined coins being available makes up for that, so I suspect it will negatively affect price. And to your second point: I believe there's a limit to how much difficulty can drop in any given period, but even if it drops to a point where the other chain becomes cheaper to mine on, now the other chain stops moving forward. Will an equilibrium eventually be reached or will the hash rate keep moving to the more profitable chain every time the difficulty adjusts? – Ruben Somsen – 2017-07-05T19:41:34.620