Does Proof of Stake Lead to a Positive Feedback Loop?

Question

(I may have horribly misunderstood PoS, so be gentle!)

Reading about PoS, the general idea seems to be that each miner has a "hit rate" proportional to their coins / total coins in the system.

Now, if there is a reward (generation, fees, etc.) for successfully mining a block, we seem to have a positive feedback loop. A richer miner has more chance of solving a block and so their wealth would increase at a greater rate than a poorer miner.

Does this means that the richest miner in the system only needs to keep mining till they are rich enough to mount a 51% attack?

Answer 1

Even if the dynamics are interpreted in the most basic way, it does not lead to this positive feedback, rather it maintains the status quo. If there are miners with stake of 40%, 30% and 30%, they will get 40%, 30% and 30% of the rewards respectively, so the ratios will remain at 40%, 30%, 30%.

For example, if they start with 40, 30, 30 BTC, and 10 new coins are generated, they will be distributed 4, 3, 3, so the new wealth will be 44, 33, 33, which is still a ratio of 40%, 30%, 30%.