>The transaction fees and the constant $a$ would have to bee chosen such that you couldn't turn a profit by sending lots of money to yourself. But I think that would allow for a fair system with a good incentive structure.
What if the goal isn't to profit from block rewards but to attack the network, though? What if you pass stuff around over and over between tons of addresses you own? (Any person can create an unlimited number of addresses, pretty much instantly.)
Even if by the end the total transaction fees result in a heavy loss being incurred, the attacker would only do it if they think they can gain more from the network attack than they'd lose in fees.
I think a secure blockchain system can never make any kind of assumption that 1 address = 1 person/entity or 1 transaction >= 2 people/entities. I think depending on addresses or transactions in any way is probably a non-starter.
One of the key anti-Sybil features of Proof of Work, Proof of Space, and Proof of Stake is that it solely relies on a single scarce, unfakeable, totally fungible resource. The network cares about only one fungible variable: the total amount of computational power, or hard drive space, or funds in the network. That way, it makes no difference if the attacker has 1 address or 1 million addresses.