What about Great Person tile improvements in Civ5?

The Great Artist, Scientist, Engineer, Merchant can build tile improvements that generate additional culture/science/production/gold. The Great General can build the citadel.

These always feel like a bad tradeoff, as:

• I tend to save my Engineers for wonders (I like to think worth a lot more that a few added production).
• Putting culture and science on a tile makes most sense in cities that already are larger (more buildings means more bonuses to the base values), but for larger cities I am already hard-pressed to squeeze food and production out of them and adding a Gerat Person tile improvement will diminish the return in food and/or production of the tile or I have to add it to a tile I wouldn't normally work with this city.
• Using a great general for a citadel will also ruin any tile improvement, so I have never used it up to now.

Are the bonuses worth it?

Solution 1:

Let me rephrase the question:

When are Great Person tile improvement bonuses worth it?

Basically the rule of the thumb for all of them is the earlier you build the improvement the more profit it would bring during the course of the game. This is especially noticeable when you play for babylonians and get a Great Scientist from Writing. Such an early academy would increase your science output about two times and would really help to zoom out those early technologies.

Generally I like to build Academies. Let's do some calculations. Before University it gives you 5 science. Technologies till Education cost from 35 to 440 beakers, so roughly 237.5 in average. So your Academy would pay off in about 47.5 turns and later it would be just plain profit. So I think that's just great. Technologies from Education to Scientific Theory cost from 440 to 1300 beakers, so 870 in average. Academy with University gives you 7.5 beakers, so it would pay off in 116 turns, so that is still quite good. And if a city is dedicated to science and has a National College, then Academy would give you 10 beakers, so would pay off in 87 turns, which is even better. Later technologies seem very expensive and definitely worth using Great Scientist to discover. But the problem with them is that in modern era they almost stop increasing in value. And you obviously want to use your Great Scientist to discover later and more expensive technologies. So it kind of stops me from doing so when I see that 3rd tier modern technology would take me just 1 or 2 more turns to discover then 1st tier modern one. So IMHO you can get the best deal from your GC discovering late renaissance / industrial era technologies.

Another thing is then you are beelining tech tree to a specific unit, wonder or even era (for opening a SP branch). In that case its definitily worth using GC for instant discoveries.

I also almost always use Great Engineers for rushing wonders, but let's do some maths anyway. Lets say your best city before Steam Power could grow from 5 to 15 in size, so (very) roughly 10 in average. So rushing a wonder gives you about 300+10*30=600 hammers. A Manufactory without Factory would give you 5 hammers, so would pay off in about 120 turns. If you build buildings a Manufactory with workshop would give you 7.5 hammers, so would pay off in 80 turns. Actually its not such a bad deal when you think about it. So if you don't have any important wonders in sight it might just be worth building a Manufactory. The problem is that you always do :).

Using a Great Merchant for a trade mission would give you from 350 to 650 gold depending on era, so 500 in average. Add to that influence worth about 250 gold (I think you get about 30 points of influence, but I might be wrong) and you have 750 gold from a trade mission. A Customs House gives you 5 gold with no buildings and about 12 gold with stock exchange, so its 8.5 gold in average, which would make it to pay off in about 88 turns. Not such a bad deal. I'd say use it for trade missions when you desperately need some cash (lets say for unit upgrades) or CS influence. The economy of using Great Merchant for Golden Age inflicted gold output is explained in this answer.

The last two Great People are mostly situational. I only used Great Artist once ever for culture bombing when I needed access to coal and couldn't settle another city as I was going for culture victory. It might well be used to access some valuable resource tile, which is on that useless one tile piece of land off the coast more than 3 tiles away from a city. It might also be used in some stalemate war situation, when pushing a cultural front line just one tile is such a big deal, I've seen AI doing that once. In other cases I find it more effective to settle that land, take it by force, or if it is CS land - just ally with it.

I really like using Great Generals for building Citadels. Just put it in front of the city (for friendly lands and easy unit exchange), build a road to it, make sure there is enough room for enemy units to surround you citadel on other 3 sides (for -3 damage) and you can pretty much fend off any attack. In one game I was controlling about 1/4 of the pangaea map, and Mr Askia was controlling the rest, we were at never ending war and he was sending swarms of units at me. I successfully held the mountain range position using just several units, several forts and a citadel.

I usually use Great People for Golden Ages when I "need them". That is if I need a cash injection for unit upgrades or to bribe some CS's. Or if I'm falling behind in producing buildings. I also like to zoom through the phase of building workshops and factories using golden age, as the faster you get those buildings the more production you will have overall.

Solution 2:

The great person improvements replace any other improvement on that tile, that weakens them already.

Great Scientist

You can instantly get any researchable technology with one Great Scientist, you can also save some Great Scientist and perform a slingshot to some advanced technology. The Academy provides 5 Science/turn (affected by city modifiers).

Discovering technologies is IMHO the most powerful ability of any Great Person, I never use a Great Scientist for anything else. You'll want to use them on the most expensive technologies you can research (e.g Civil Service).

Great Engineer

The Great Engineer provides 300 + 30 * city population (See this answer by me) hammers to any production. The Manufactory provides 3 Production/turn.

The engineer can finish all early game wonders, and many mid-to-late game wonders in high-population cities. That is much more useful to me than the +3 Production.

Great Artist

You can use the culture bomb to obtain luxuries/strategic resources outside your city limits. The Landmark provides 4 Culture/turn.

The culture bomb is very situational, the landmark is pretty useless in my opinion.

Great Merchant

The Trade Mission provides 350 + 50 * Era Gold (See this answer from Oak) and 30 influence with that city state. The Customs House provides 4 Gold/turn.

I always use them for Golden ages. They might be worth it in the early- to mid-game. Later the increased income from Golden ages is most likely higher.

Great General

The combat bonus (+25%) is very significant, even if you're not China (for China it is +45%). I keep one general for each army I have and use the rest of them for Golden Ages.

The Citadel is very situational, if you have a choke point and a superior enemy on the other side it may be useful.

Golden Ages

Golden Ages are very powerful in Civilization 5, I use most artists and merchants for them. Golden Ages from Great Persons last 8 turns the first time, and decrease each time you use a Great Person for a Golden Age by 1, with a minimum of 3 turns. Golden Ages can vastly increase your production, which is hard to come by any other way, and you income.

Summary

I never use the tile improvements, the other functions of the Great Persons are just much more powerful.

The earlier parts of the game are much more important than the later game. Any advantage that I gain right now is worth more than if it was split up over 100 turns.