Garrison Effects

[Simetrical]

Where o is the public order gain, g is the number of garrison troops present, and p is the population:

o ≈ –2.8 + 701g / p
o + 2.8 ≈ 701g / p
g ≈ p(o + 2.8) / 701

If we want +20% public order, which is what monthly games give, then:

g ≈ 22.8p / 701 ≈ .0325p

If we want to express this in terms of number of peasant units required:

g = 60u
60u ≈ .0325p
u ≈ .00054p ≈ p / 1845

In other words, to get +20% public order, you need one peasant unit per 1,845 people. The cost of that is:

c = 100u
u = c / 100
c / 100 ≈ p / 1845
c ≈ p / 18.5

Therefore, given that monthly games cost 200 (IIRC), using garrisons is more profitable when

c < 200
p / 18.5 < 200
p < 3689

I'm too lazy to do daily games now. Maybe I'll do them tomorrow.

-Simetrical

[sunsmountain]

Ok, this thread is not for strategies to increase public order. I merely wanted to point out that MajorFreak made probably a mistake which lead him to think the 0.114458477 ratio is not usefull.
Something that would be on topic though is the question, what is cheaper, a maximal garrison or games. It should depend on population size, but for which size are, say peasants, more expansive than games?

Oh i'm sorry i thought a 'rule of thumb' kind of approach can solve questions players have when they brows these pages for answers. If this forum is solely intented for pure scientific research then say so and i will leave the Ludus Magna immediately.

For example, i will use:

Originally Posted by Willie McBride
Y = 701.54 X - 2.6972 R-squared = 0.9928

those 2.6972 citizens which have to be substracted from 701.54 times the population is probably just:

Y = 700*X minus zero
(for Large)

where Y is the garrison bonus and X is garrison over population. The deviation in chi-squared allows us to chose integer values that seem reasonable for the programmers to have been used.
This formula is repeated for Huge on this same page:

Y = 350*X

Now 20% of public order bonus provided by Monthly Games costs 400 denarii. You can get 4 Peasants for that, resulting in a garrison size of 4*240 = 960. So we're looking for the population where

20 = 350*960/population
(huge)

This population occurs at 16800. This is the same for Large, Normal and Small unit size.
Daily Games cost double, yet also provide double the bonus (8 peasants):

40 = 350*1920/population

gives the same cut-off population of 16800. In summary:
If you need 20% (or more) public order in 16800+ pop settlements, and you have exhausted what you can get from buildings/governor in time, use Monthly Games!
If you need 40% (or more) public order in 16800+ pop settlements, and you have exhausted what you can get from buildings/governor in time, use Daily Games!

[A.Saturnus]

Simetrical, I think you have a mistake somewhere.

Our formula is:
Garrison bonus ~= -2.8 + 701*(garrison strength/population)
If we assume that games cost 200 denarii, you get 2 units of peasants for the same price. If we insert that and the population of 3689 we get:
-2.8 + 701*(240/3689)
=
42.806 increase in public order.
I see what went wrong. You assumed that peasants have 60 men, but the formula is for large, thus 120 men.
In addition, I think sunsmountain is right that games cost 400 denarii.
If we take his calculation, we get:
-2.8 + 701*(480/16800)= 17.229
which is aproximately correct. But we can be more precise:
-2.8 + 701*(480/X)=20
where X is the desired population cut-off value.
X = (480*701)/(20-2.8)
X = 16628

Above a population of 16628, games are a better means for public order than garrison. Of course, these values are also just estimated. They are probably not better than what sunsmountain found.

[Simetrical]

Oh, the formula's for large. Didn't realize that. In that case:

o ≈ –2.8 + 701g / p
o + 2.8 ≈ 701g / p
g ≈ p(o + 2.8) / 701
g ≈ 22.8p / 701 ≈ .0325p
g = 120u
120u ≈ .0325p
u ≈ .00027p ≈ p / 3690
c = 100u
u = c / 100
c / 100 ≈ p / 3690
c ≈ p / 36.9
c < 200
p / 36.9 < 200
p < 7378

If c is actually 400, then:

c < 400
p / 36.9 < 400
p < 14756

Which is still different from what you two got. Hmm. Well, same ballpark, anyway. Don't feel like figuring out the difference right now.

-Simetrical

[sunsmountain]

That's called cumulative errors due to standard deviation in the linear fit, Simetrical & A.Saturnus.

If you plot public order against garrison/population, you can try a linear fit, something like your:

o ≈ –2.8 + 701g / p, or
Y = 701.54 X - 2.6972

Or some other Excel-trial function. The 2,8% public (ghost) order at 0 garrison is typical, and too close to 0%.

These account for the differences between me and Saturnus, where the difference is small (172 population), but also between me and Simetrical (where the difference is 1244 population).
Using his (cumbersome) notation, but my assumed ideal formula:

o = defined as = 700g/p [unit size: Large]
g = po/700
let o = 20
g = .0286p
g = 120u
u = .000238p = p / 4200
u = c / 100
c = p / 42

Now if c < 400, then
p / 42 < 400
p < 16800

Unsurprisingly, exactly what i got. I'm using this notation to get you onboard (the ideal formula wagon), Simetrical, i'm not promoting it!

The reason the difference between Saturnus and me is less than with Simetrical is simply because he used less steps to calculate the final number. The more steps you do using rounded numbers, the larger the deviations become.

[A.Saturnus]

I failed to check my results back. sunsmountain is probably right that it's a rounding error, but the deviation is a bit to big to be left with that.
If inserted into the original formula, sunsmountain estimation of 16800 gives a public order bonus of 17.229 for four peasant units.
Mine gives 17.436
Simetrical's population cut off of 14756 gives 20.003
Since what we want to know is at which population 4 peasant units give less than 20 public order bonus, Simetrical's figure is the best one.

We can conclude that the formula we have found above says that above a population of approximately 14500, games become cheaper than garrison to fix public order.

[Simetrical]

D'oh. I should've thought of just checking the answer before now. Pretty basic way to tell who's right, huh?

I agree with sunsmountain that the –2.8 thing is probably illusory. I think we need more data points to plot this properly. Ideally, we should just add population slowly until we get to the precise switchover point, then we should adjust the garrison precisely to make sure the ratio is exact and won't have to be rounded. This would take a lot of time, which unfortunately I don't have, but the exact ratios required for each step will ipso facto give us a perfect model.

How many data points did you use, therother?

-Simetrical