I think that is a clever idea to ask what guarantees Bitcoin delivers with only a single miner. So we retain double-spend protection provided inflation is non-zero but we lose censorship resistance?
For the record, I do not believe the arguments that free competitive in an open market leads to monopolies. Every single time I've seen supposed examples of this happening put forward they always fall apart under scrutiny. So until once is conclusively demonstrated, my position is that mining obeys the same economic principles as any other service in the economy.
However, for the sake of argument from now on, I'm going to accept that it is inevitable and see what happens anyway. If it turns out that Bitcoin is still useful even with a 100% monopoly on mining, then any degree of decentralization at all is just icing on the cake.
In Meni Rosenfeld's paper from 2012, he describes the situation of a majority hashpower attacker as "all bets are off". Maybe it's actually worth looking at it in more detail to see if it's a monster under the bed or just a kitten.
So we retain double-spend protection provided inflation is non-zero but we lose censorship resistance?
We retain double spend protection provided there exists revenue which the miner can only claim by continually extending the blockchain.
I think blacklist-based censorship will always be something that improved privacy techniques can render ineffective.
Whitelisting, however, is the nuclear option from a censor's perspective that probably can't be bypassed without the users exercising their own nuclear option of abandoning the currency entirely.
Do you believe we also lose double-spend protection (hash rate falls to zero) if the inflation rate is zero and there are several miners?
I'm not sure, but I can explain in more detail why I think it's the case in a single miner.
Assume an initial condition of a non-zero difficulty, one miner, no block reward, no block size limit, and enough revenue from transaction fees to make mining profitable.
In the course of an hour, the miner will receive enough revenue to pay for 6 blocks worth of proof of work.
However, with transaction fees (as they are currently implemented), there's absolutely no requirement that those six blocks actually extend the chain.
In order to collect that revenue he only needs to extend the chain by one block. He could choose to orphan the other five if he wanted to without immediately impacting his revenue, which means he could profitably engage in double spend attacks.
Of course, he can't do this for very long because his costs are measured in purchasing power terms and this kind of behaviour would reduce the purchasing power of his Bitcoin-denominated revenue (as users abandon his chain for a better-behaving currency), but let's just consider the short term impact.
If any fraction of his revenue can only be obtained by continually extending the block chain, then this represents the minimum profitable double spend. Any transactions below that value (times the number of confirmations) can be considered safe because it will not be profitable for the miner to double spend.
Right now, the block reward is the only form of revenue that can requires the chain to be continually extended to obtain, but there could easily exist other forms. New opcodes that allow users to create transaction fees that can only be claimed at a particular block height would achieve the same thing.
