What Is the Precise Economic Cost Incurred by a PoW Miner in the “Nothing at Stake” Scenario for PoA?

The precise economic cost incurred by a PoW miner is the expenditure of electricity and hardware wear-and-tear required to find the block header. If the miner attempts to mine on multiple competing chains (a "nothing at stake" behavior), they must expend this PoW effort for each chain.

Since only one chain will ultimately be validated by the PoS stakers, the effort on the rejected chains represents a sunk, unrewarded cost.