Selfish/stubborn mining isn't applicable here, since the double spender has to fork and withhold all blocks from the point where their honest transaction is included in a block until the victim sends them the goods/exchange withdrawal/whatever, at which point they need to eventually mine a longer chain and reveal.
At ~50% hash power and with a victim who waits <10 confirmations you still only have a ~1/2+o(1) probability of mining a longer private chain. The point is if you can maintain the 50% long enough, you can expect to eventually have the longer chain.
You can also have <50% hash power and double spend, it is just progressively less likely to succeed. The calculations in the Bitcoin whitepaper (p.7) for choosing a transaction confirmations threshold assume the attacker can maintain their attack indefinitely [1].
To clarify, the whole point of selfish/stubborn mining is to increase your mining profit by wasting the honest miners' time when you withhold new blocks (so they unknowingly mine on a shorter chain) and then trying to propagate your withheld block faster when you see an honest miner release a new block. Therefore you get a greater effective proportion of the network's hash power. Of course if you're more than one block ahead of the honest chain, you can just propagate the next block in your withheld chain. But, when you're attempting a double spend, your withheld chain has to remain private the entire time otherwise the victim would see the honest transaction invalidated and cancel the e.g. exchange withdrawal.
There's a cool paper on some extended selfish mining strategies combining eclipse attacks [2].
Also interesting to note is that selfish mining is only profitable if only a few miners are using it, like a kind of prisoner's dilemma [3].
At ~50% hash power and with a victim who waits <10 confirmations you still only have a ~1/2+o(1) probability of mining a longer private chain. The point is if you can maintain the 50% long enough, you can expect to eventually have the longer chain.
You can also have <50% hash power and double spend, it is just progressively less likely to succeed. The calculations in the Bitcoin whitepaper (p.7) for choosing a transaction confirmations threshold assume the attacker can maintain their attack indefinitely [1].
To clarify, the whole point of selfish/stubborn mining is to increase your mining profit by wasting the honest miners' time when you withhold new blocks (so they unknowingly mine on a shorter chain) and then trying to propagate your withheld block faster when you see an honest miner release a new block. Therefore you get a greater effective proportion of the network's hash power. Of course if you're more than one block ahead of the honest chain, you can just propagate the next block in your withheld chain. But, when you're attempting a double spend, your withheld chain has to remain private the entire time otherwise the victim would see the honest transaction invalidated and cancel the e.g. exchange withdrawal.
There's a cool paper on some extended selfish mining strategies combining eclipse attacks [2].
Also interesting to note is that selfish mining is only profitable if only a few miners are using it, like a kind of prisoner's dilemma [3].
[1] https://bitcoin.org/bitcoin.pdf [2] https://eprint.iacr.org/2015/796.pdf [3] https://arxiv.org/pdf/1411.7099.pdf