Historically I have always taken my great leaders in Civ V and planted them in the ground. Give me good tile improvements! Why would I want to get a one shot boost when I can set up an ongoing boost that keeps on giving turn after turn? I've never actually cared to look at the numbers, I've just always been glad to get a good square to work. Well, the video I watched had the guy save up his great scientists. I've had people tell me recently that capping the liberty policy tree was good because it would build you a free wonder when you took a great engineer. Build a wonder? But I want more production! And in the comment thread on Facebook for a post last week a small discussion broke out on if popping a great scientist early was cost effective or not... This calls for some research and math!
Let's start with the great scientist. Consuming him for the one shot effect gives you an amount of beakers which are immediately applied to your current research task. If this amount takes you over the amount needed to research it you learn the tech right away and the excess gets applied to the next tech. The amount of beakers is dependent on the speed of the game and your current research per turn. In a standard speed game this amount is equal to 8 turns worth of research. Planted in the ground the scientist will build an academy which gives 8 research per turn. If you're only making 34 beakers per turn like I am in a current game then the academy will surpass the scientist in 34 turns. If you're making 800 beakers per turn then it'll take 800 turns to break even... A game is going to last a couple hundred turns so the first option the long term plan seems reasonable. The second one is abysmally bad.
Of course that's not the whole picture. The academy planted in the ground can be augmented by buildings, tech, and policies. You're apt to plant it in your best city, so it's entirely possible to be looking at having a university, a national college, a research lab and an observatory. You may have the rationalism opener, and you may have researched things that give it +2, +2, and +4, and you'll have a multiplier of 311%. So you could actually be looking at getting 50 science per turn from the academy. It also replaces a different improvement, so it costs you something like 2 food per turn. And you get to make immediate use of the technology right away when you use the one shot. It may take 34 turns to break even but that means your Civ was weaker for those 34 turns because you were behind on tech that whole time (not to mention being behind on the food too).
How about holding on to the scientist for later? Well, if you've decided not to plant him and you're in the process of drastically increasing your science output you definitely want to hold off on it. But if you're going to be waiting a long time you probably want to plant it because you lose all of the benefits by waiting close to the hypothetical 34 turns before popping it.
Looking at it, I think planting an early scientist is going to make sense most of the time. Unless there's some tech you really want to get right away (maybe you're racing for a wonder or you have a timing attack set up with artillery or something), anyway. Later on you might as well hold on to them until you really want a specific tech right now. If you're trying for a science victory then you only really care about getting the very last tech and you get the most from your scientist by waiting for all the pop growth before using them.
How about the great merchant? His one shot gives 350 gold plus an extra 50 per era. You also get 30 influence with a city state! The downside being you need to walk the merchant over to the city state before you can use the ability. Planting him builds a customs house worth 4 gold, 5 with a relatively early tech. Just like with the scientist there are building to amplify the return from the improvement so you're probably looking at getting 8 gold per turn (with market and bank) at the cost of 2 food per turn versus scoring up 550 gold. So it would pay off in ~70 turns. 70 turns is a long time to be down money. You're also down whatever goodies the city state would have given you as a result of being friends/allies with the extra influence. The great merchant one shot seems to give a bigger return early on than the scientist and the improvement is worse. So it feels like planting them is almost always wrong. Only if there are no city states to run to? Or maybe if you have a game plan that revolves around your cash on hand solely at the end of the game? So if you're going for a big surprise diplomatic victory play and want to just horde cash all game? Ok, I can see that as being a real thing. Holding onto them seems just wrong, since the gain from waiting (50 gold per era) feels like it's probably not worth delaying the city state influence. Unless you're about to tech up into a new era, anyway. (Merchants of Venice actually give double the one shot bonus but the same planting bonus, so it _really_ seems like cashing them in with the one shot has to be right.)
The great engineer apparently doesn't automatically rush a given item. Instead he grants 300 + 30 per citizen in the city. In practice that might as well be an instant buy if the city is of reasonable size. You don't get the building/wonder/nuke until the next turn though. The planting ability is an improvement worth 4 or 5 hammers. There aren't nearly as many things that multiply hammers as there are light bulbs which makes this compare unfavourably to the scientist already. And then there's the downside of building a wonder slowly... You don't get it at all! Wonders are unique and if you're not first you're out. That said, hammers over time are also very powerful and if you're planning on pumping out a lot of units then maybe you really want 4 hammers over 2 food? I donno, I now feel like rushing a wonder that's potentially under contention is just awesome.
Great writer is one that I've been looking at closer and decided in my game today to use the one shot over the 'plant' for the first time. Like the great scientist he gives 8 times your culture per turn as his one shot ability. If you plant him you don't get an improvement, instead you get a great work that needs to go in an appropriate slot in your city and is worth 2 culture and 2 tourism per turn. It turns out 8 times your culture per turn is a pretty big number and is just going to dwarf 2 per turn for anything resembling a reasonably amount of time. If you're trying for a cultural victory then you probably need the tourism (it's not like you have many other sources of it) but otherwise using the one shot here seems like it has to be right. And you should probably save it up until it will give you a new policy? Or maybe if you're planning on rushing a specific policy that's deep in a tree you should wait? Or if you haven't unlocked the tree at all? Some of the ideology stuff is pretty great, maybe there's a good plan around saving up a bunch of great writers and blowing them all after building your 3 factories.
Great musician has the same plant ability but the work goes into a different building. It wasn't clear but it sounds like the one shot is 10 times your tourism to the targeted Civ. If you're trying for a cultural victory and one Civ has a significantly higher culture threshold to break through then I'm sure using the one shots here makes sense. Otherwise I guess you might as well plant them for the 2 culture? Frankly I'm not sure getting a great musician is even worth doing unless you want the cultural victory.
Great artist has the same plant ability with the work again going in a different building. The one shot is to start a golden age. Golden ages are awesome and unless you want the tourism for a cultural victory it feels like the golden age has to be good? It's 20% culture and production with a gold bonus of some kind for 8 turns I think? So it's actually only 1.6 times your culture per turn as a one shot which is a lot worse than the 8 times of the great writer. But it also gives gold and production so that's really good. So again, if you don't want tourism... And even if you do, you can use archaeologists to fill the same slots.