Am I better off letting the AI steal techs from me for a SV?

When I am going for a SV, I get to renaissance first, and then put my first spy into my capital. The question is that, would I be benefited more, if I just put my first spy into a city state, probably culture, and then let the AI steal my techs and catch up? This way, I would end up getting more science from the trade routes and from the RAs.

I think you leave it in the capital so that your spy levels up.

What does leveling up a spy do actually?

The spy gets better at stealing and doing coups.

I often put spy at home, get some quick level 2/3 and then try to coup a CS with it.

Letting the AI steal you won't have any significant impact toward RAs and trade routes (some but entirely negligible).

I believe how RA works is base on the science output not number of tech, so letting them steal wouldn't benefit you.
I'd put spy on others capital to steal military tech that I missed, as I usually just rush to Renaissance and completely ignore the bottom tech tree.

fallout3dc said:

When I am going for a SV, I get to renaissance first, and then put my first spy into my capital. The question is that, would I be benefited more, if I just put my first spy into a city state, probably culture, and then let the AI steal my techs and catch up? This way, I would end up getting more science from the trade routes and from the RAs.

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You only get science from Trade Routes if the Civ you're trading with is ahead of you in techs (or maybe has individual techs that you don't have). Either way, letting the AI steal your techs won't help you out in this regard, as you'd have to be ahead of them in techs for them to steal any (or have individual techs that they don't have).

For a while I only used my spies for rigging elections and I rarely had one fail without leveling up my spy but lately I've been sticking at least one in my capital because he will shoot down enemy spies.

Which difficult level ? Below Emperor, let AI steal all techs they want if you want have some efficient RA. You have nothing to win by upgrading spies in low level. AI don't ally CSeasily and despite stealing tech will always be behind you in tech and army.
Exception is wonder race, if you want a culture victory without sacred sites.
Beyond, upgrade your spies by stealing tech to AI and after to make coup. Friendly or ally CS are quite important on Immortal, must have on Deity.

fallout3dc said:

When I am going for a SV, I get to renaissance first, and then put my first spy into my capital. The question is that, would I be benefited more, if I just put my first spy into a city state, probably culture, and then let the AI steal my techs and catch up? This way, I would end up getting more science from the trade routes and from the RAs.

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Nope, in fact very slightly counter productive if you have an active RA with him due to expire in a few turns.
If he steals a tech from you, then his science per turn from trade routes may drop by 1 beaker per trade route per turn, therefore causing his overall rate to be slightly less. With it almost always being the AI with the lower science output, that translates into slightly fewer beakers when the RA completes.

He can only steal a tech from you that you already have, so there's no possibility of you gaining beakers via trade routes via him stealing a tech. What you would need is for him to steal a tech from a different AI that had a tech that you didn't.