Do liberate city-states or just puppet/annex them?

Yes

1. Automatic vote in the UN (not sure how that works in BNW) no matter the ally status

2. Reduces warmonger penalty (or counters)

3. Auto-ally status (I think it's like 150 or more influence).
 
Liberating cities to other civilizations can cause lack of open borders sometimes. Ais can abuse you like this, too..
 
When going for domination victories, I like to target AIs that have captured city-states. I liberate the CS to reduce the warmonger hate from capturing the AI capital; that way I can keep reasonable relations with my future targets until I'm ready to go to war with them.
 
Ditto. Liberating city-states gives you UN votes, their inherent bonus, and a rare opportunity for a major liberation bonus to offset that capital you just got greedy and took. I go for the guys who take out city states as well. Most of the world will probably hate them already, and you can often offset the penalty for conquering 90% of the territory by liberating all those CS.
 
The harder situation for me is when I take out a CS because he was allied with a major civ I was at war with.
 
The harder situation for me is when I take out a CS because he was allied with a major civ I was at war with.

If you don't want the city, one option is to sell the captured CS to another AI. The AI generally places a very high value on cities, so you can often get a very healthy amount of gold/gpt/luxuries/etc. in exchange.

The player you sell it to is very likely to choose to liberate it (I don't think I've ever seen them not do so). Now granted, they'll then be its ally, but at least it's come back to life as a city state, and you've successfully swapped a chunk of unhappiness for a chunk of nice stuff.
 
/edit: Misunderstood the post, my bad.
 
3. Auto-ally status (I think it's like 150 or more influence).

You get them as an ally for a bit, but I think it is only like 60 points influence, which is terribly low, but I almost always liberate anyway.

CS allies are better than your own city because they give you some perk that is otherwise unavailable.

When you liberate, the city pops up to higher strength than when you conquer. It is also not in resistance, so it can fire on a mutual enemy immediately. Plus no turns of unhappiness. Plus I like the role play of liberator.

I would probably make an exception for a CS on a NW, but I don’t remember having to make that call. As an ally, you get the luxes and strategic resources anyway, so no reason just to puppet for those.
 
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