It’s very confusing that hashrate * hashing_time doesn’t equal work. I wonder if hashrate is more important than work. I prefer the partition that has a higher rate of lottery ticket purchases over the one with the most winning lottery tickets. The security model depends on non-repurpose-able equipment (hashrate).
Let’s say we have a strange PoW protocol where difficulty is selected by miners in the event of a partition and a partition occurs that splits them into 2 groups. One group selects a difficulty 50% harder (D=1.5) than the other (D=1). That group finds their 2nd block at the same time as the other finds their 3rd block. They have equal work (W=3) but statistically we know the 2nd group with lower difficulties had 33% more hashrate (Hashrate = 3/T * (3-1) /3 verses 3/T * (2-1)/ 2) and therefore should have performed 33% more hashes.