I use downvotes - always when I clearly recognise a scammer (clear plagiarism, preferably from someone else's Steemit articles). Or clear farming spammers. Now this is not my hobby and I'm not hunting. But if I see somebody ... ;-)
I don't think we need a game or a specific hashtag. We just need to realise that downvotes are as much a part of the ecosystem as upvotes. And the Steemit team and its loyal followers should also realise this. After all, they have an impressive stake and don't need to be afraid of "revenge".
Okay, not my stake. I use my stake for both. That's good. Do it too, dear community!
Even if Steemit decided to unleash their downvoting resources, I think it might still be more effective to start by fencing off a small area of the blockchain and expanding from there. But yeah, if more stakeholders started using their downvote capability more regularly things might be different, and something like this might not be useful. Something would need to spur the change in behavior, though.
Retaliation can take different forms. They don't have to worry about being the victims of downvote wars, but an aggrieved party can still cause a lot of chaos that might harm the value of their holdings.
And of course, I'm not suggesting that this would be the only place for downvoting. Plagiarism, spam, etc, would still be downvoted anywhere. This would just be a place where authors would be highly motivated because they know that subpar content would be curated more aggressively - and it would only work if the flip is also true that highly appealing content is more likely to be valued correctly.