First, a brief comment about play styles: My own style is based on timing, not on a particular focus and therefore I have experienced military, political, cultural, and technological situations and victories. As a result, I believe that my analysis is useful and accurate regardless of a player's particular interests. I play Monarch or Emperor on standard or small maps generally, so there may be certain hidden aspects of tiny maps that I am not familiar with.
In rough order of importance (imho):
Commercial
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This is a dominating strength. The bonus trade in city home squares is cute but essentially useless, but reduced corruption is supremely powerful. This can allow small cities to produce 2 instead of 1 (due to corruption), which equals the benefits of religious or military strengths (as far as building improvements quickly is concerned). It allows large empires to outproduce other large empires and therefore secure all wonders, a larger military, and more culture than other civs. It allows more net trade, leading to increased science which allows you to start the wonders before other civs, and also to dominate militarily because of better units. Later, increased production due to the ability to instantly buy necessary units and city improvements is extremely valuable. (e.g. In the late medieval age on monarch I was producing enough gold to buy a temple and a cathedral on consecutive turns in any new village that I founded).
Of course, most the advantages here apply to larger civs, not just ones that are "commercial", but I have found that this strength has nearly the biggest impact on the game of any strength (save, perhaps, industrial).
Industrial
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This is also a dominating strength. The extra production is again a joke, but the increased worker rate is huge. Most bonuses in CivIII are +50%, but Industrial workers work _twice_ as fast as normal workers. Captured workers (which are completely free of upkeep costs) work as fast for Industrial civs as normal workers do for other civs! (great for slave drivers out there...) This alone allows one's entire unit allowance (or money expense) to be used for military units, leading to a larger military for those of you that like to kick ass first, and then ask diplomatically for those 5 tech advances in exchange for peace... The quicker development allows for larger cities which then yield all of the benefits of "Commercial", except for small towns and their temple build.
Religious
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This can be very useful. I don't believe it is quite as useful as Industrial and Commercial, but definitely good. Changing governments quickly is nice, but not necessary. Losing a total of 10 turns on two changes of government is not critical over the course of a full game. I have no problem pursuing war under a Republic because of all the happy wonders and city improvements I am able to build due to the other two strengths, so there is no need to switch back and forth. The half price temples are very useful, but if you can go from 1 to 2 shields thanks to less corruption, that has the same effect (granted this doesn't happen in every new town, but pretty close). Half price Cathedrals are cute, but not important due to the amount of money and production available by the time they can be built. I have had decent results with this strength, and in all I think it is a close third.
Militaristic
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This is not very useful for waging war. Anyone that thinks wars are won with their armies can get in line with the Germanic barbarians and Iraq (to name a few from very different eras). Wars are won with technology. I don't care how many _elite_ knights you have if I'm coming at them with infantry. Building barracks at half price is not useful compared to the extra production of Commercial and Industrial. Additional leaders are nice, but with the rule change so that armies are permanent once loaded, early armies are again useless in the face of superior technology. Everyone would like a leader to establish a victorious army, but non-Militaristic civs will normally get one anyway by the time they have infantry or tanks (wonderful points at which to take advantage of superior technology and do some stomping). If there were even more leaders produced, then their ability to help build wonders that you are late to start might be extremely useful. However, I have only received two leaders in the same game once, and that was with a non-militaristic civ! The larger army allowed under Industrial (due to less support for workers), and the greater tech and production of Commercial (and Industrial, indirectly) are more useful for waging war than being Militaristic. Remember: wars with equal tech units are a bad idea, period. (now, quick ambushes to secure additional cities, that's a different matter, and it doesn't require Militaristic...)
Overall this can at least be useful in the right circumstances, but it is a very round-about way to increase production of wonders, and the first three strengths seem to be clearly superior.
Expansionist
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Almost completely useless. Scouts are nifty for about 50 turns, and then the rest of the game they can be replaced with diplomacy or superior production allowing _more_ units to explore. Better goody huts is simply comical. Remember, the game lasts for more than 6 millennia!
Scientific
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I believe this is the worst strength. Half price libraries, universities, research labs, etc... and one extra advance per age are not nearly the equal of the extra science allowed via reduced corruption in Commercial or by quicker growth in Industrial civs. At least Militaristic (more leaders) and Expansionist (scouts and huts) have advantages that can't be better produced by a different strength. This is just completely eclipsed by either Industrial or Commercial.
OK, that's it for civilization strengths, of course a particular civ also has to factor in its unique unit and the timing of its Golden Age, but those are different things that I won't go into here. I'm not arguing for or against any particular civ, I would rather discuss the impact of the various civilization strengths individually. Essentially it seems that the intended benefits of four of the strengths can be duplicated or sometimes even bettered by the indirect benefits of the two "infrastructure" strengths.
I suppose you can at least use the different quality strengths to handicap yourself without having to be subjected to the incredible cheating of the more "difficult" A.I. levels.
In rough order of importance (imho):
Commercial
==========
This is a dominating strength. The bonus trade in city home squares is cute but essentially useless, but reduced corruption is supremely powerful. This can allow small cities to produce 2 instead of 1 (due to corruption), which equals the benefits of religious or military strengths (as far as building improvements quickly is concerned). It allows large empires to outproduce other large empires and therefore secure all wonders, a larger military, and more culture than other civs. It allows more net trade, leading to increased science which allows you to start the wonders before other civs, and also to dominate militarily because of better units. Later, increased production due to the ability to instantly buy necessary units and city improvements is extremely valuable. (e.g. In the late medieval age on monarch I was producing enough gold to buy a temple and a cathedral on consecutive turns in any new village that I founded).
Of course, most the advantages here apply to larger civs, not just ones that are "commercial", but I have found that this strength has nearly the biggest impact on the game of any strength (save, perhaps, industrial).
Industrial
=========
This is also a dominating strength. The extra production is again a joke, but the increased worker rate is huge. Most bonuses in CivIII are +50%, but Industrial workers work _twice_ as fast as normal workers. Captured workers (which are completely free of upkeep costs) work as fast for Industrial civs as normal workers do for other civs! (great for slave drivers out there...) This alone allows one's entire unit allowance (or money expense) to be used for military units, leading to a larger military for those of you that like to kick ass first, and then ask diplomatically for those 5 tech advances in exchange for peace... The quicker development allows for larger cities which then yield all of the benefits of "Commercial", except for small towns and their temple build.
Religious
=========
This can be very useful. I don't believe it is quite as useful as Industrial and Commercial, but definitely good. Changing governments quickly is nice, but not necessary. Losing a total of 10 turns on two changes of government is not critical over the course of a full game. I have no problem pursuing war under a Republic because of all the happy wonders and city improvements I am able to build due to the other two strengths, so there is no need to switch back and forth. The half price temples are very useful, but if you can go from 1 to 2 shields thanks to less corruption, that has the same effect (granted this doesn't happen in every new town, but pretty close). Half price Cathedrals are cute, but not important due to the amount of money and production available by the time they can be built. I have had decent results with this strength, and in all I think it is a close third.
Militaristic
===========
This is not very useful for waging war. Anyone that thinks wars are won with their armies can get in line with the Germanic barbarians and Iraq (to name a few from very different eras). Wars are won with technology. I don't care how many _elite_ knights you have if I'm coming at them with infantry. Building barracks at half price is not useful compared to the extra production of Commercial and Industrial. Additional leaders are nice, but with the rule change so that armies are permanent once loaded, early armies are again useless in the face of superior technology. Everyone would like a leader to establish a victorious army, but non-Militaristic civs will normally get one anyway by the time they have infantry or tanks (wonderful points at which to take advantage of superior technology and do some stomping). If there were even more leaders produced, then their ability to help build wonders that you are late to start might be extremely useful. However, I have only received two leaders in the same game once, and that was with a non-militaristic civ! The larger army allowed under Industrial (due to less support for workers), and the greater tech and production of Commercial (and Industrial, indirectly) are more useful for waging war than being Militaristic. Remember: wars with equal tech units are a bad idea, period. (now, quick ambushes to secure additional cities, that's a different matter, and it doesn't require Militaristic...)
Overall this can at least be useful in the right circumstances, but it is a very round-about way to increase production of wonders, and the first three strengths seem to be clearly superior.
Expansionist
===========
Almost completely useless. Scouts are nifty for about 50 turns, and then the rest of the game they can be replaced with diplomacy or superior production allowing _more_ units to explore. Better goody huts is simply comical. Remember, the game lasts for more than 6 millennia!
Scientific
==========
I believe this is the worst strength. Half price libraries, universities, research labs, etc... and one extra advance per age are not nearly the equal of the extra science allowed via reduced corruption in Commercial or by quicker growth in Industrial civs. At least Militaristic (more leaders) and Expansionist (scouts and huts) have advantages that can't be better produced by a different strength. This is just completely eclipsed by either Industrial or Commercial.
OK, that's it for civilization strengths, of course a particular civ also has to factor in its unique unit and the timing of its Golden Age, but those are different things that I won't go into here. I'm not arguing for or against any particular civ, I would rather discuss the impact of the various civilization strengths individually. Essentially it seems that the intended benefits of four of the strengths can be duplicated or sometimes even bettered by the indirect benefits of the two "infrastructure" strengths.
I suppose you can at least use the different quality strengths to handicap yourself without having to be subjected to the incredible cheating of the more "difficult" A.I. levels.
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