Building research is helpful when it's important to discover a tech one turn earlier than otherwise. For example, in a recent game I had heavy war and knew that the best time to end the war would be after 3 turns (after capturing another city), so I rushed for constitution to be able to have the anarchy during the last turns of the war, when the citizens aren't working anyway.
Sometimes I've built a little bit of culture in a captured city to reach 10/100 culture asap, to block some other AI from moving a settler e.g. to a place where a city has been razed (and where I'm planning to settle).
And maybe, if you have a production city that has all the necessary buildings and no useful units to be built, you could build wealth/research in the meantime, but I've never really done that - there always seems to be some partially useful building missing or some old but partially useful military unit that could be built.
In general; when the city doesn't have anything better to build. For instance in the later stages of a game where you're going for a space race win building research is a good way to shave off a couple of turns.
Building culture is occasionally useful in other situations, such as when you need to rapidly expand the borders of a newly captured city to stop other civs stealing the land. Obviously it's also useful if you're going for a cultural victory.
If you need more research, and you are not at 100% research, it is usually better to build wealth. In a normal game you have better multipliers for research, so it is better to build wealth and nudge the research slider up.
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