I think the divergence game cannot be easily applied for LMD GHOST since it only considers the latest message of each validator.
For LMD GHOST, the initial state of the game matters.
(Note that divergence games in a blockchain do not start with 0 vs 0 or even tie because of small forks by an adversary or temporal network failure.)
Specifically, we must consider these things:
- A vote does not increase a score of the target block if the voter has already voted for that block: in non-LMD voting rules, if the protocol converges to one block and honest validators keep voting for it, its score will monotonically grow. However, in LMD voting rules, the score grows only if the previous latest vote of the voter is not for that block.
- If a validator votes for a block different from the block which the validator previously voted for, the score of the previous option decreases.