Proof of Work for Proving Stake
When miners create two Bitcoin blocks at the same height, what determines which sub-chain will come to dominate the other? Which sub-chain will be more economically important, used by more people, and have a higher price?
Usually the answer is Hashrate
Usually, the sub-chain that accumulates work fastest will become dominant, hashrate on the other chain will soon collapse, and no more blocks will be found. Nodes will quickly switch to the longer chain. The sub-chain will die.
The reason hashrate usually wins, is because usually the only important difference between the sub-chains is the hashrate, and generally speaking, people want more rather than less hashrate.
Sometimes the answer is Exchange rate
If a subset of the community modifies their nodes to reject one of the sub-chains, it becomes much less certain that hashrate will determine which chain will dominate. In this situation, both sub-chains could continue to grow, and an exchange rate would develop between tokens on each chain. The hash rate ratio would be a function of this exchange rate, plus whatever short term losses miners would be willing to suffer with the goal of helping one of the sub-chains.
To say that one sub-chain has a higher exchange rate is to say that people are willing to pay more for tokens on that sub-chain. Presumably people would pay more because the token is more useful, holding it has less risk, and will have a higher percent gain over time from holding.
The answer is never Hashrate
The only time the answer appears to be hashrate is when nodes representing the majority of stakeholders and prospective stakeholders agree that the blocks being mined form a valid chain. Thus hashrate only appears to be in control while the nodes are not attempting to exert control.
In other words, this is a subset of the situation where hashpower is allocated to different subchains based on exchange rate — but since there is no economically significant faction wishing to reject the chain being mined as invalid, the exchange rates of the hypothetical alternatives are zero, and 100% of the hashrate continues to be applied to a single chain.
The value of Proof of Work
Control of Bitcoin’s blockchain is ultimately wielded by the aggregated preferences of the economic majority of potential Bitcoin buyers and sellers — proportionally distributed according to how much Bitcoin they hold and can sell, or how much fiat they are willing to trade to buy. This is a form of Proof of Stake, played out not within the virtual economy defined by the protocol, but in the real economy at the interface of tokens and the external assets and services they can buy.
The reason we care about Proof of Work is that it proves the size of the economy that is being served by the coin.