Making a case FOR colosseum

So, everyone says that when playing a non-culture game a colosseum(and also cathedral in a lesser way) is a totally worthless city improvement. I searched forums a little bit and found nothing positive about colosseums(except culture generation for culture win).
So i made a thread asking for players to write anything positive about colosseums, be it a very specific usage or anything.
I did find a specific usage, though. Triggering WLTK. First of all, dealing with unhappy people. Specifically, when playing regent/monarch, citizens go unhappy starting from population 3. To make a minimum effort to get WLTK, we need 6 population, none have to be unhappy and 3 of them have to be happy(so 3 happy, 3 content minimum). Building a temple and having a colosseum eliminates unhappy people until population 5. That means we need one more.
In ancient era(to get WLTK with temple and colosseum) we get content citizens(from unhappy) only with The Oracle(gets one more content, thus can trigger WLTK in population 6; gets obsolete later), Mausoleum(supports until 8 population to get WLTK) and the Hanging Gardens(in the built city supports until 8 population and all others until 6).
And of course if we use military in the cities in despotism or monarchy - just throw in 1 military and we have WLTK at population 6. 2 military supports until pop 7 and 3(with monarchy?) mil supports until pop 8.
So in theory, with monarchy, temple, colosseum, 3 military, The Oracle and the Hanging Gardens(not only the specific city) we get none unhappy until population 10. But that's too specific.
So let's say we have a republic and none the wonders(and of course 0 lux rate). I found that maximum population to support WLTK is 8 - temple, colosseum and cathedral. If we throw in Sistine, we get that until population 11. Bach should get it to pop 13.
Now, i could go on like this with monarchy(it's easier).
To get happy people to support WLTK we only need enough luxuries and a marketplace. Or throw in some lux rate(with enough lux rate we can trigger WLTK in every city).
What does WLTK do? Lessens shield waste(but not corruption; and a very minor effect of reducing flip chance?). So specifically we use 2 gold per turn(colosseum maintenance) to get more shields. Even with one shield bonus - 1 shield for 2 gold. Buying up improvements cost 4 gold per shield. Colosseum itself needs 120 shields and if we would make wealth instead, we would net 30 gold. That means to get even with 4g/1shield colosseum needs to 'produce'(with WLTK) 1 shield bonus for 15 turns. Double the time to 30 turns with economics. Half the time to 7.5 turns if it 'produces' 2 shields.
Of course, some say that we should just increase lux rate, but the problem is that every commerce on lux gets happy faces first and only then when we don't have any happy faces 1 commerce turns one unhappy to content. Next commerce turns that content to a happy citizen, but all other unhappy citizens stay. So in effect we have to pay more than 2 commerce to get rid of 1 unhappy citizen.
Now, those are all dandy calculations, but in overall the usage applies only when colosseum activates WLTK effect. Throw in the population growth and WLTK disappears. Only when a player actively tries to maintain WLTK for prolonged time(actually, almost the entire game) nets some results. And it's much harder to get WLTK in Republic due to no military making any content citizens.
Corrections?

albatros13 said:

What does WLTK do? Lessens shield waste(but not corruption; and a very minor effect of reducing flip chance?).

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It is not so minor. It cuts the flip chance by exactly 50% prior to garrisions and consequentlly by more than 50% with garrisions counted in. In essense it halves the amount of units you need to securely prevent flips. That can be worth something. A marketplace for 20 happy faces is important for achieving such an ambitious goal.

albatros13 said:

So specifically we use 2 gold per turn(colosseum maintenance) to get more shields. Even with one shield bonus - 1 shield for 2 gold. Buying up improvements cost 4 gold per shield. Colosseum itself needs 120 shields and if we would make wealth instead, we would net 30 gold. That means to get even with 4g/1shield colosseum needs to 'produce'(with WLTK) 1 shield bonus for 15 turns. Double the time to 30 turns with economics. Half the time to 7.5 turns if it 'produces' 2 shields.

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I quite disagree with those mathematics. First of all lets clarify whether economics apply. Noone will mass produce wealth prior to economics, therefore the utilization of economics can be assumed.

Second you mix up buying of shields with producing gold via wealth. That makes no sense at all. Only either can apply. Basically we need to differ between 3 cases:

1. Shields are so scarce that gold is used to buy shields. That costs 4 gold per shield, 3 gold per shield in case of upgrading and 1.5 gold per shield if Leonardo applies.

2. Gold is so scarce that shields are used to produce gold via wealth. That gives 1 gold per 2 shields.

3. Gold and shields are consider almost equally worth: 1 gold = 1 shield.


In Case 1 you get 1 or 2 extra shields from WLTK day at the price of 480 gold + 2 gtp and you get the worth of 4 or 8 gtp in return. That is a netgain of 2 or 6 gtp for an investment of 480 Gold. It takes 240 or 80 turns to get your money back.

In Case 2 the 12 shields could have given you 60 Gold. You get 0.5 or 1 gtp in return, but have to pay 2 gtp, so your net return per turn is -1.5 or -1 Gold. That is totaly unreasonable.

In Case 3 you pay 120 shields and get 1 or 2 shields per turn as return. Your net return per turn is -1 gold or zero. This is unreasonable.

Only Case 1 can be reasonable, but when you are willing to pay 4 gold per shield shields are so scarce that you will hardly invest into something that will take at least 80 turns just to get your money back. So case 1 is unreasonable as well.

There are a few cases where WLTKD reduces waste(corruption of shields) by 20 to 30 percentage points, that can be 3 to 5 shields prior to railroads and 8 to 12 shields with railroads and hospitals. So there might be very few cases where the colloseum becomes reasonable. Case 1 will likely still not be relevant, altougth you might get your money back within 20 turns or less. But that is likely still not fast enough unless this is meant to facilitate the production of future units that cannot be upgraded to like battleships.

The Cases 2 and 3 apparently become reasonable. In Case 3 you get your money back after 12 to 20 turns. In Case 2 it is is even 6 to 10 turns. So in the very late game colloseum can become reasonable if a strong WLTKD is reached that way. Actually the numbers get even better once factories etc. come into play. The thing however is that you will likely have 20 happyfaces via market place, and 1+3+3+2 content faces via tempels, cathedrals, sixtine chapel and Bach, so at monarch level you will have 15 happy and 1 content citizen. Add the colloseum and there will be 16 happy and 1 content citizen. The cases for a strong WLTKD are very limited. If you can use 4 units for military police it becomes even more limited.

The time I build them is:

When on a Huge map Archipelago, 80% water, it's not at all uncommon to start on an island without luxuries. In order to get to size 12 in a non-garrison government without luxuries a town requires a Temple, Cathedral, Sistine Chapel and a Colosseum:

3 natural content
1 from Temple
3 from Cathedral
3 from Sistine
2 from Colosseum
=
12

You can allay this by using the luxury slider, but if you are on quite a small island this can have a huge impact on your Tech speed and it's actually better/easier to just build the buildings @5gp per city.

I have absolutely no doubt that there are a bazillion and one ways to get around this in such a scenario, but, as you can see, it's likely in the game precisely because those 4 buildings magically add up to 12 on Regent.

The Colosseum is the first to lose favour should you either:

Prefer the Luxury slider
Use Monarchy, Despotism or Feudalism (garrisoned)
Have access to multiple luxuries + Marketpalce
Also build the (Oracle/)Hanging Gardens + J S Bach's Cathedral
Use citizen faces

While it might not necessarily be optimum to just build the 4 buildings in such a scenario, it is certainly the neatest and most satisfying aesthetically.

But it does not add up on regent. Both regent and monarch have 2 free content citizens, not 3.

You are absolutely right, I guess I must always have either Hanging Gardens or J S Bach's as well by the time I set the slider to zero lux.

I just build them when the folks in a particular city keep revolting, because it's easier than micro-managing that city every time its pop. goes up.