In Proof of Stake (created to substitute PoW on blockchains), a miner is selected at random (almost) and then they need to pay some money as a “security deposit” - which is higher than the transaction fees you get paid if you solve the puzzle. You get your security deposit back after a certain time - if your transaction is not fraudulent.
PoS vs. PoW is really only a matter of how you ensure your chain can’t be attacked. For PoW it’s equipment cost, burnt electricity, and opportunity cost for losing rewards due to attacking the chain. For PoS, the security is ensured through a high-enough buy-in cost (32 Eth), and penalizing attackers harshly through slashing.
https://news.ycombinator.com/item?id=26325859
https://www.youtube.com/watch?v=M3EFi_POhps