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So as far as I understand, Proof-of-stake works by all nodes voting on what they think the correct next block should be and then recieving a small reward out of thin air if they guess correctly and a small penalty if they guess incorrectly. Thereby all nodes are incentivized to believe the same thing.

This then means that there is no useless incrementing-a-nonce-and-trying-to-hash-and-come-up-with-a-value-below-some-limit puzzle, which means it should greatly reduce the work of processing blocks and save miners a lot of cost, which in turn means they can offer a lower gas price.

So what is "mining" in this context? Is it just running a load of transactions, putting a block together, and then convincing everyone else your candidate is the one they should vote for? Is there still a system where the miner of a new block gets a big payout as in bitcoin, or does everyone who votes for the right thing split the payout?

Buterin wrote about all this nearly two years ago. Has it made it in to the protocol yet?

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Casper testnet is expected soon:

https://www.coindesk.com/ethereum-soon-launch-first-casper-testnet/

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