> They are the only ones mining on this particular chain, but that's not an advantage either. How mining on a hidden chain is an advantage?

It's easier to see the argument if you have a head start. Imagine you've somehow created a private chain that's 10 blocks ahead of the public chain. You could publish that now and earn 10 blocks of reward, or you could continue mining until the lead diminishes to 0 blocks, earning the same 10 blocks of reward plus however many blocks you've mined in the meantime.

If you have 50%+ε of the hash rate on the network, this argument would have you bully other miners out by almost always stranding their blocks, since in expectation you'll mine blocks faster than your competitors.

The insight is that this same situation can happen probabilistically with a finite but non-majority fraction of the hash rate on the network. With 49% of the hash rate you'll still be able to build a private chain some fraction of the time, so waiting a little bit to see if this occurs might have positive expected value.

But to get 10 blocks ahead you have to withhold blocks before knowing you'll be ahead. If you don't get ahead, you'll likely lose the reward of the blocks you withheld.

So, you have to risk a lot of rewards, and for what potential gain? If you win you get to discard some blocks of others. You don't get more rewards, you just make others earn less (and you push the difficulty down a bit).

I can see how you get a chance to double spend, though. If you want to double spend a transaction with N confirmations, you've to be N+1 blocks ahead in your hidden chain, publish your first transaction, wait for N confirmations on the public chain, and you publish your chain that's still 1 block ahead (and includes your double spend transaction).

Indeed, it's not "51% expensive", but it's still very expensive because of the rewards lost during the failed attempts before you get ahead enough. Actually, it might even be more expensive, because with 51% you're guaranteed to get ahead enough at some point, so you don't really risk your rewards (if you can maintain 51%).