Apparently there is a flawless strategy in civ5 and if you apply it correctly, winning on deity is rather easy. And you don't even have to make some special settings for the game. Just I have to warn that its much more fun if you discover it yourself rather than get it from internet, as in that way it kind of ruins your game experience as it did to me to some level.
The easiest way (as Irony Man correctly pointed out) is setting the map type to archipelago, as AI is totally hopeless on water. Just expand all the time, build some naval units to fend off any aggressive AIs, grab those resources, trade and develop. You should be able to get any kind of victory quite easily.
You can also use continents or Pangaea maps. You need to apply a flawless strategy to get a win on those. A strategy might be slightly different depending on which faction you are playing, but there are some basic rules which work for all of them. The specific faction strategy is to get the most of the special ability. That means if you are playing for English - build naval units, if you are playing for French - build a lot of small cities, if you are playing for Romans - build all that is needed in Rome, etc.
The most important bit is a good start. I would divide it into three phases:
- Settling. My winning initial build order is worker-settler-scout. Develop technologies needed to get those resources. Scout the land using your initial warrior and try to not be delayed by any spotted barbarians. Settle the land to grab the luxury resources asap, but don't go too far from your capital. Once you got a scout use it to discover map further and use your warrior to deal with barbarians near your cities. Get Liberty SP branch and Citizenship SP next. At this point you just build settlers, workers and monuments in all your cities. Forget about any early Wonders. If you met a militaristic CS you can steal a worker from them (they are useless). Declare a war, capture a worker and make peace all in the same turn. On the other hand its way too easy and some people call it an exploit, so I don't usually do that.
- Fighting. At some point you'll meet other civs. Be nice to them, but don't forget that they are your rivals and sadly due to a crazy inherent AI warmongering its almost impossible to make a good friend till the end of the game in Civ5. You might be able to settle 2 cities or maybe 5 cities (depending on the map) before you start rubbing your shoulders with other civs. Discover archery and build some 2-3 archers. Discover horseback riding and build some 2-3 horsemen. Don't bother about infantry. Once you have 5-6 units declare a war to your nearest neighbour. Try to get other civs as your allies in this war for now. Do not attack! Select a good defendable spot of land (might be a river valley or similar), preferably inside your borders for a friendly lands fighting bonus and faster healing and get them to come. Don't lose any of your units, if necessary retreat to heal. AI should send several waves of units and run out of them at some point. By that time you should have discovered mathematics and built a catapult or two. Now heal your army and go and get those AI cities. Raze them and leave a capital as a puppet (annex it then you have some spare happiness). Settle the land with your own settlers. Now do the same to the next neighbour...
- Developing. Constantly develop your cities, prioritize them. That means that some of your cities would be appointed to making money (if they have a lot of luxury resources nearby), some to making science (if they got jungle tiles), some to production (a lot of hills) to build wonders and units, some to make culture (none of the above advantages). You still need culture to adopt those all important policies even if you are not going for cultural victory. Dedicating city to some task means that you build that aspect enhancing building as soon as you have it available (markets, banks, stock exchanges for money cities, etc.) and you appoint specialists only to that same task. Still every city should have at least a monument, library, market and a coliseum. Make city-state allies. Preference is for not hostile ones with a luxury resource you don't have. After you make allies with several maritime CS's you can stop building farms and better build trade posts. Then make allies with cultural CS's. Forget about militaristic ones, unless they have a resource you so badly need.
Now if you still haven't lost your army at this point you should be doing rather well and can continue playing the way you like. Or if you want to be really effective and crush them all you can apply an ICS (infinite city sprawl) strategy. This strategy is genuine and it works perfectly, but honestly I don't like it myself at all, as it takes all the fun off the game. So if you enjoy Civ5 so far and want it to continue better close this post now, but if you want to further ruin your game experience read on..
The basics of ICS are explained elsewhere much better than I would be able to, so I'll just quote. Here's the theory behind Infinite City Sprawl, as written by alpaca in his Infinite Rome thread on Civfanatics:
The idea of an ICS strategy is to
settle cities as close as possible to
leverage the additional growth,
production and commerce that small
cities have over large ones. In Civ5,
there are a number of game mechanics
that lend themselves well to this kind
of strategy. To wit:
1) Maritime city states. The bonus
granted from maritime city states is
per city. Each adds +2 food to the
city tile at the start, increasing to
+4 over the course of the game. Since every city benefits from this free
food, a single maritime CS ally will
allow you to set up two additional
specialists in every in the later
stages of the game. Obviously, the
more cities the better, because each
gets the bonus.
2) Happiness buildings. In Civ5,
happiness is global... or is it? In
fact, only the consumption of
happiness is really global. The
production is local. Each city can
build happiness buildings, and the
low-tier buildings are more efficient
and more effective than the higher
ones. If you have lots of small
cities, each can have a Colosseum and
a Circus where available, and some
theatres. This actually rules out
happiness as a long-term limiting
factor and turns it into a
growth-limiting factor because you
need to set up these buildings in each
new city.
3) Purchasing things with gold.
Provided you have enough gold, you can
buy buildings where and when you need
them. Spamming trade posts is a
typical strategy, and it's possibly
even better in ICS. Just spam trade
posts, make money, and buy the
buildings you need. This is especially
good for the more expensive buildings
which have a better gold/hammer ratio.
4) Certain policies, like Communism
and much of the Liberty tree, scale
with the number of cities. So do some
civilization abilities or buildings,
like the one of Harun al-Rashid or the
Chinese Paper Maker. For them, you
also want as many cities as possible.
5) Research favors large empires
because a tech will cost the same, no
matter how many cities you have. Since
having more cities usually means
having more science, you will tech
faster.
6) Trade route maintenance makes you
want to put your cities as close
together as possible. City tiles are
free roads, after all.
There is a significant drawback to ICS
which I won't hide from you: The speed
of unlocking social policies. This is
the only thing that is really better
for small empires because, roughly,
your average culture per city is what
determines policy speed. Since there
are sources of culture that don't
scale with the number of cities, like
wonders and cultural CS, you will be
slower at unlocking these SPs, so make
sure you only pick those which are
most useful to you.
You can find some great comprehensive examples of how it is applied here:
http://www.garath.net/Sullla/Civ5/liberteordre.html
or here:
http://forums.civfanatics.com/showthread.php?t=390302
and this one is very short and straight to the point:
http://forums.civfanatics.com/showthread.php?t=388793
The amount of production of a great engineer on normal speed is calculated using the formula
300 + 30 * city population
So in a city with a population of 1 the engineer only results in 330 production, in a size 15 it results in 750 production.
For quick speed the formula is:
200 + 20 * city population
For epic speed it is:
450 + 45 * city population
and for marathon it is:
900 + 90 * city population
The amount of hammers does not seem to be influenced by era, map size or difficulty level.
Conclusion:
If you want to hurry late game wonders, build it in a city with a large population.
Best Answer
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.