Proof of Stake based blockchain systems use what design to permit valid write transactions?

A genesis “Stake” node approves or disapproves transactions announced from other nodes

Validator nodes each give or pay a stake in order to write transactions, and if malicious, will lose their stake

Stakeholders or early product investors host nodes that approve or disapprove transactions from other nodes

Every node on the network holds an equal stake, and if malicious, gets flagged for removal