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It's literally not this. Proof-of-stake is basically saying "I'm allowed to author the next block because some math formula says a coin I hold is eligible to do this, and here's a signature proving I own the coin". It's not proof-of-work, it's proof-of-ownership.
It’s literally this (I’m relying on the dictionary defintion of literally which also includes figuratively.
You’re right about proof of stake, in that it gives the machine the ability to write the next block. But that machine still has to do work. That work still must be proved. The only difference is because the machine was picked it doesn’t have to compete with millions of other machines doing the same work.
I’m sure you realize the work involved in Proof of Work isn’t really all that much work for your machine to do, it’s more about getting that work done faster than any other machine you are competing with. So, Proof of Work is more aptly “race to the right answer” with other machines validating that correct answer. Redundency and verification in computer systems is a good thing, but we don’t need millions of machines running the same verification.
As for which machine gets picked, that’s based on the amount of coin staked in that validator / miner / processor / machine / whatever you want to call it. So in a simplified example if 50% of the coins were staked at validator A, 25% at validator B and 25 more validators each at 1%, those percentages are a sort of weighting of the roll of a dice to favor those validators to that amount.
As a validator, you still have to do the work, possibly of other validators (redundency, etc..). If the system picks validator A to mine the next block and validators B and C to confirm validator A did it correctly, validator A earns the reward for that block. The only reason validator A got a chance to do this was because of the people who staked with validator A, so validator A rewards everyone who staked a percent of the reward based on how much each address has staked with validator A. This gives person who stakes with that validator an incentive to do so.
If on the other hand validator A submits the work and it is considered invalid by the other validators, the network (not sure of specifics, contracts I imagone) can pull from the funds from the validator which would be missing from the invalid block, hurting the validator and providing the validator an incentive to correctly validate the work.
If the systems were the same in this hypthetical example, with validator A, B and 25 more validators (a total of 27), with a hypothetical 2 validators validating the chosen validator, we have the same mechanism of action going on cryptographly, (doing the work of validating transactions and trying to write to the block chain, being stopped by other machines to cross-check that it is correct), you have 3 machines running in proof of stake vs proof of work’s 27 machines. There’s still 1 winner of that 27vs1, but the proof-of-stake system has weighed the chance of the winner at who has the most at stake, or who people have entrusted the most with their coins; whereas the proof-of-work system is largely left up to the 3 fastest machines, usually determined not by processing power but by network speed and geographic location.