However, if the customer can broadcast the double-spend transaction privately to a miner that controls, say, 10% of global hash power, over a non-public back channel, and if the miner agrees to replace the original transaction (the payment to the merchant in our example) with this new transaction, then the double spend will be successful 10% of the time.
If a miner offers this service for profit and if customers use it to deceive merchants, would you consider the miner to be complicit in fraud?
(This is not a discussion about enforcement, just whether or not this qualifies as fraud in your opinion)
Whoever would scam like this would be the dumbest criminal of all time. The double-spending miner would need 3 petahashes to have 10% of the network. That's at least $6-7,000,000 in asics to achieve that. But they would constantly need to add as the difficulty continues to rise. All this so that 10% of the time you are successful at a small purchase scam?
If you had that much hashing power wouldn't you just use it to mine bitcoins honestly, raking in around 900 a day?
The miner can do both. The miner can add the double spend transactions to the blocks they create and these blocks are compatible with the honest network, raking in all those block rewards.
The miner chooses which transactions to add to his blocks. In this example, the double spenders are paying the miner to add their double spend transaction instead of their initial transaction. If this particular miner then finds the next block, the double spend becomes the transaction written in the block chain. If another miner finds the next block, then the initial transaction is most likely the one which will be included in the block chain. This kind of double spending would only work in blocks found by the miner being paid to include the double spend transactions.
But why would anyone do this? If you had that type of money invested in bitcoin mining why would you strive to devalue bitcoin? If small transactions failed 10% of the time merchants would be less likely to adopt it. Then at the end of the day your 900 honestly mined bitcoins a day are all valued less and you have left yourself open to legal issues. This move just does not make any sense.
And keep in mind, you couldn't scam like this with high dollar purchases. Those would undoubtedly be waited on for full confirmation by the merchant into their wallet.