Garrison Effects

[therother]

From this post:


This is rough and ready to be sure, but it gives you a reasonable guesstimate of the garrison bonus:

Garrison bonus ~= -2.8 + 701*(garrison strength/population)

So –2.8 + 701*(1600/30000) ~ 35%, which is about right

Health warning: this won't even be remotely accurate as the expression in the brackets approaches 1! I will try to simplify this ASAP.

These calculations only hold true for large units sizes. The game appears to scale the garrison effect according to the average size of the units in the game. So two units of Hastati have the same relatively effect from small to huge unit sizes. Ergo you need to scale appropriately. Multiple the result of the equation by the following factors to get the right result:

4 for small
2 for normal
0.5 for Huge

[Willie McBride]

I just joined the Guild, so I don't have any priviledges, which is why I am posting here. However, I just did some Excel-based statistics upon the effect of Garrison size upon Public Order. I had been led to believe I could find such information on this forum, and although my quest was not fruitful, I decided that this was the proper place for any conclusions I might reach after studying Garrisons.

I used a fairly limited sample size of 26 cities (those West of and including my Capital, plus my two largest cities, which happened to be East of my Capital) and collected data on their individual Populations, Garrisons, and Public Order bonus due to Garrison. My hypothesis was that there would be a direct relationship between the ratio of Garrison to Population and the Public Order bonus. I quickly observed that the PO bonus levels off at 80%, so I eliminated both of the cities in my sample (regretably, also the smallest Populations) that had PO bonus of 80%.

I then graphed the remaining 24 data points on a scatter plot in Excel, with the Ratio of Garrison to Population on the X-axis and the Public Order bonus on the Y-axis. The result:

Y = 701.54 X - 2.6972 R-squared = 0.9928

Once again, we're seeing a pretty simple linear relationship, regardless of buildings present.
In case you're wondering, the Golden Ratio that optimizes Garrison size for Population is 0.114458477; for example, a Garrison of 1376 soldiers assures a Public Order Bonus of 80% in a city of 12000 Population without wasting a single fighting man.

[therother]

Y = 701.54 X - 2.6972 R-squared = 0.9928That is in good agreement with the formula I concocted. There is a slight complication due to unit size scaling though. See here (https://forums.totalwar.org/vb/showpost.php?p=599476&postcount=18) for details.

This information is also contained in the A short investigation of squalor (https://forums.totalwar.org/vb/showthread.php?t=37586) Ludus Magna thread. You make a good point though - perhaps it should be a little more accessible...

PS Welcome to the Org. ~:wave: Glad to have you aboard! :medievalcheers:

[warbeetle]

I posted this on the totalwar.com forum and someone said you guys may be interested in such things! (Maybe I can add some stuff regarding the whole public order thing later):

I realized this the first time working in 1.1. BUT it seems to have not changed in 1.2.! It works without changes in different difficult settings! And maybe also with different unit sizes, but havn´t tested this right now!

How to plan garrisons
--------------------------------------------------------------------------------
Did anyone already notice this?

You can exactly count the numbers of men you need, to give a town the maximum effect of garrison(80%)

formula:

([number of soldiers] : ([Number population]:100) ) * 3,5 = %Garrisson influence

shorter:
(x:(y:100))*3.5=z

If you want the number of people a town can have with a special amount of soldiers at 80% garrison try this:

population= ((soldiers*3,5) : [80]%) *100

shorter:
y=((x*3,5):z)*100

Example:
1.Minimum:A town with 700 people Population needs exactly 160 Men for a garrison of 80%. This means you don´t need a second unit of 160 men until population gets over this number

2.Maximum:20 units 0f 240 peasants can give a town of 21900 people 80% garrisson

As a rule: have one unit of 160 men for every 700 people or one unit of 240 men for every 1050 people

[player1]

I don't know about this calc posted, by my own testing show, that in order to get 80% loyalty, you need to have number of soliders equal to 1/8 of city population size (large unit scale). Sometimes something smaller number is enough, but I gess that's because of some sort of rounding up effect.

[MajorFreak]

ahhhhh...it's good to be back (finally remembered my hotmail password to get THIS forum's password. lol)



In case you're wondering, the Golden Ratio that optimizes Garrison size for Population is 0.114458477That's wonderful, but hardly useful unless you're less than 16 squares from the capital. To make that formula useful we'll need the modifier for distance from capital, and since i'm no math wiz i'm hoping someone out there will provide this and make life simple for me. *g*

BTW, that's hilarious CA scaled garrison affect for unit size. lol.

[sunsmountain]

I don't think the public order bonus provided by garrison troops has anything to do with 'distance to capital', i'm pretty sure it is dependent ONLY on:

number of soldiers/population

I do not recommend using more than 5 garrison troops (if each unit has an upkeep of 100) at all. The only reason why you would want a large garrison is to maintain a Very High tax rate. The difference between low and very high tax rate is usually roughly equal to 500 denarii. If you use more than 5 troops you are spending more than you are gaining, which is not a good idea.

[A.Saturnus]

I don't think the public order bonus provided by garrison troops has anything to do with 'distance to capital', i'm pretty sure it is dependent ONLY on:

number of soldiers/population


I think what MajorFreak ment was that although garrison effect is not influenced by distance to capital, public order is. A garrison size of 0.114458477*population won't ensure a public order of 80%. What he may overlook though is that garrison effect is cut off at 80%. What McBride meant with "Golden Ratio" is how to get maximum possible garrison effect without wasting money for a garrison that doesn't have an effect. If that doesn't bring your public order in a yellow area, you need other means to improve it.
I'm not sure why you say 5 units is enough. If unrest is high, you surely want to maximize your garrison effect before using more expansive means like games.

[sunsmountain]

To make that formula useful we'll need the modifier for distance from capital,

And what i'm saying is that these formulaes are useful, are linear, and are not missing a modifier.


What he may overlook though is that garrison effect is cut off at 80%
Both garrison and distance to capital effects are capped or cut off at 80%. That way they can compensate each other, leading to larger garrisons further from the capital.


If that doesn't bring your public order in a yellow area, you need other means to improve it.
Whereas i would advise people to use other means before using garrisons to improve public order.

Distance to capital does influence public order, and games are more expensive than troops below a population of say, 12000. However that does not mean you should use troops to get a Very High tax rate or a high public order for large populations. That will simply end up costing you more than it's gaining you.

To maintain public order, you should (in order of being cost effective):
1. Install a governor if you have a spare: you're paying for them anyway!
2. Build temples.
3. Build arenas/execution squares/odeons.
4. Build health buildings and the level 5 market, though these only compensate the problems they generate themselves in terms of squalor, long term.
5. Lower the tax rate. Aim for 85% at high rate, or 75% at low/normal (to avoid the traits efficient taxman & poor assessor).
6. Use as much garrisons as you need if the population is below 16800.
7. Use Games/Races above 16800, where you mix and match. Obviously if you need 25% you can throw Monthly Games and use some troops, you don't need Daily Games in that case.

The main weapon you have to avoid these latter two is controlling the growth rate: Make sure it finishes on or slightly above 8% (total growth rate, see settlement details, and add up all the contributions to growth: farms, buildings, health, grain, etc.).

[A.Saturnus]

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?

[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

[ZZR Puig]

(Deleted)
-----------
Seemed to me an interesting suggestion but proved wrong.

[sunsmountain]

First of all, monthly games cost 400 denarii, not 200, at least for me. Anyway when you have a high population it's better to use them, as you can't get the same garrison bonus with only 4 peasant units, that would cost the same 400 denarii.

If you actually read all the posts, you would have read that A.Saturnus already corrected Simetrical for this. I will quote A.Saturnus for you here:


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.




I figured out that the troops in a town may be also counted as population for this purpose. I haven't confirmed it, but I heard that somebody said that the garrison can contribute to squalor, so lets consider the possibility it's also taken into account as population.

- If garrison contributes to squalor, then why do i get 0% squalor in my 400 pop town, garrisoned by 20 units of peasants?
- Governors can raise squalor due to certain character traits.

Because of these two reasons, lets not take the possibility into account.

[therother]

Not very many. About a dozen spread throughout the levels. I agree that of someone wants to do this properly, they'll need far more data points and more accurate data. As for the -2.8, it may indeed be illusory. However, especially at low values of X, the -2.8 helps to get the garrison right, at the slight expense at high values. Which suggest to me that the 700 is the central problem, not the -2.8.

I think it highly unlikely that garrison affects squalor.

[sunsmountain]

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.


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

@A.Saturnus, @Simetrical
First of all, my number is NOT an estimation, like yours, it's an EXACT number. So use the EXACT formula:
Y = 700*X
and you get
20 = 700 * 480/16800
exactly.

If you use your approximation formula which is wrong (-2,8% public order with no garrison present), then Simetrical's value of 14756 is correct:

Y = 701*X - 2.8
and you get
20,003 = 701 * 480/14756 - 2.8

which is what you got:


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.

So Simetrical's figure is the best one for the approximate formula, not the real or exact one.

Meanwhile, A.Saturnus number of 16628 is wrong:

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

X = (480*701)/(20+2.8)
X = 14757.895

which is what Simetrical got. You made a math mistake in bringing the -2.8 to the other side of the = sign.



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.

No, we cannot. We can conclude that above a population of exactly 16800, games become cheaper than garrison to fix public order.

@therother


As for the -2.8, it may indeed be illusory. However, especially at low values of X, the -2.8 helps to get the garrison right, at the slight expense at high values.

I contest this, the -2.8% never helps.
1) The 2.8% does cause the big deviations in numbers between myself and A.Saturnus, Simetrical.
2) The 2.8% does not deviate significantly from 0%.
Because of these reasons, i urge you to discard the -2.8%, and change that 701 to a number with a significance level you have the data for: 7,0*10^2, or 700.

I challenge anyone to find any public order bonus due to garrison which is not equal to (after rounding to remain within +/- 2.5%):
Y = 700X
[Unit size: Large, Huge: 350X, Normal: 1400X, Small: 2800X]

Where X is the number of soldiers divided by the population, Unit size: Large. Y is the public order bonus due to garrison in %. (so Y = 20 means 20%).
At least I haven't witnessed a single city/garrison to disobey my formula yet.

[therother]

@therother
I contest this, the -2.8% never helps.)
=Soldiers|=Population |=Actual PO%|=Raw|=Round to 5|=Difference|-2.8
=80|=10000|=0|=5.60|=5|=5|2.81
=80|=6000|=5|=9.33|=10|=5|6.55
=80|=4000|=10|=14.00|=15|=5|11.22
=80|=1260|=40|=44.44|=45|=5|41.71
=80|=1140|=45|=49.12|=50|=5|46.39

[sunsmountain]

At what unit size (i assume Large)? I don't Rome:TW with me here, but when i get home i will double check those numbers.

Even if you substract 2.8 from the first value 5.6, you're still left with 2.8, which should be rounded to 5. If you then argue that all numbers get rounded down anyway, then -0.7% is already enough to account for your data. Why 2.8%???

[therother]

At what unit size (i assume Large)? I don't Rome:TW with me here, but when i get home i will double check those numbers.

Even if you substract 2.8 from the first value 5.6, you're still left with 2.8, which should be rounded to 5. If you then argue that all numbers get rounded down anyway, then -0.7% is already enough to account for your data. Why 2.8%???
The table uses Large unit sizes.

The formula is based on a linear regression of my data. You'll notice that 701X - 2.8 is of the form y = mx+c. Which is why the -2.8 is there. It was only ever meant to be a guesstimate, so I'm well aware it's not 100% accurate. Until recently, I didn't see a need for it to be.

But it doesn't change the fact that it is sometimes more accurate with the -2.8 than without. If we have a case where the garrison % is 0%, and the formula says that the garrison % should be 2.500001%, and so rounds it to 5%, it is not as inaccurate as if it said 7.499999% and similarly rounded it to 5%.

[sunsmountain]

But it doesn't change the fact that it is sometimes more accurate with the -2.8 than without.

I cannot believe yet that 80 soldiers do not cause 5% public order for a population of 10000. But if they do there must indeed be some correction factor for the data which remains odd.

We need more data.

[therother]

Okay, far more accurate data for 80 men. Did it quickly, so some of it may be out, but I think it's pretty accurate.

=Transition%|=Population
=80|≤700
=75-80|=700
=70-75|=736
=65-70|=800
=60-65|=848
=50-55|=1000
=45-50|=1120
=40-45|=1217
=35-40|=1400
=30-35|=1555
=25-30|=1866
=20-25|=2153
=15-20|=2800
=10-15|=3500
=5-10|=5600
=0-5|=9333
=0|=>9333


This table contains a lot more data than it first appears. In the transition column, it tells you the point at which garrison changes from one % to another. So 5-10 tells you the point at which garrison percentage jumps from 5% to 10%. Therefore every point between 5601 and 9333 is 5%.

[ZZR Puig]

There is something odd happening here if the last table of population values is right.

If we apply the hypothesis of y=700·X to that data we could see a few things.

First of all, that the garrison bonus that appear on the city details is not a rounded value to the nearest 5% but to the lowest one. So it seems to be working by thresholds, so until the exact real value doesn't get over the 5% the value displayed is 0%. It's not very surprising that it works like this, if not, the first 5% would be twice as easy to reach than the others. If we instead suppose that the values correspond to 2.5%, 7.5%, 12.5%, ... the resulting behavior of the garrison ratio is not linear, so it's unlikely to be the real one.

The most strange thing we can see from that data table is that the previous formula gives exactly the right population threshold for half the data (10%, 20%, 30%, ..., 80%), but on the other half it overestimates the effect by 1% (6%, 16%, 26%, ..., 76%) if we use the population values shown on the table. I we agree that the behavior is linear the formula must be correct, as it gives the right value in more than 2 points. The 1% mismatch must be some effect related to the accuracy or rounding that the game is doing.

We could get the same results if we consider the values to be rounded to the nearest 5% and use the formula y=700·X - 2.5, which is very close to the other proposed solutions, but these previous formulae (701·X -2.8, ...) get slightly wrong values for the population thresholds posted on the previous table.

So, IMO, the only thing to solve at this stage is that 1%, but not the formula.

[therother]

Okay, presenting the data for 80 men a slightly different way:

3^=Population|=% Garrison
=0-700|=80
=701-736|=75
=737-800|=70
=801-848|=65
=849-933|=60
=934-1000|=55
=1001-1120|=50
=1121-1217|=45
=1218-1400|=40
=1401-1555|=35
=1556-1866|=30
=1867-2153|=25
=2154-2800|=20
=2801-3500|=15
=3501-5600|=10
=5601-9333|=5
=9334-∞|=0

[sunsmountain]

Right, i found the same data, so the exact formula becomes:

Y = 700*X, - 1 if the 2nd digit of Y is 5 (so it becomes 4)
always round Y down.

This 1% mismatch is very odd. It makes no sense at all. If we ignore it then 700*X gives good results. If you want to correct i dont suggest using -2.5%, i suggest using -0.5%. -2.5 screws it up too much. It'll be a bugger to explain to newbies anyway.

[WarHawk1953]

Guys,
I have read somewhere and I just don't remember where, searched my down loaded guides and info could not find. I seem to recall using a simple
.12 X pop size would give you the troops you need to maintain PO 80% figure. 24000 pop X .12= Garrison of 2880 men or 12 units of 0f 240 peasents on huge setting. I am no math wiz but using .12 factor with pop size works.

[player1]

I can also say that BI peasants only give half of their value to the garrison effect.

RTW peasants work as usual.

[Alastair Haines]

I couldn't find any research on how garrisons affect public order, so I did some myself.
They seem to work rather like several other features of RTW.

Results
Garrisons have a maximum effect of 80%, rising by 5% "blocks", determined simply by
the percentage of garrison troops to total population, multiplied by a factor of seven (7).

Applications
1. As a rough rule of thumb, this means one peasant unit per thousand population will provide maximum garrison effect.
2. It is cheaper to use peasants than monthly games for cities with populations below about 16,666.
Daily games are more expensive than peasants until populations reach about 33,333.

Discussion
Essentially there is only one simple magic number here -- the factor of seven that modifies garrison as a percentage of population. There are small rounding issues due to the 5% PO increments. I have only tested the results with the Julii. It is just possible other factions may be more or less efficient with garrisons. The number seven is quite probably no mystery. The developers probably set it so that the "one unit of peasants per thousand pop" would work.

Simple examples
Monthly games cost 400 denarii for 20% PO mod.
Daily games cost an additional 400 denarii for an additional 10% PO mod --
twice as expensive, hence requiring twice the pop to justify.

At 12,000 pop, 3 units of peasants give 20% PO mod for 300 denarii upkeep.
Monthly games: 20% PO mod for 400 denarii.
At 24,000 pop, 6 units of peasants give 20% PO mod for 600 denarii upkeep.
At 24,000 pop, 3 units of peasants give 10% PO mod for 300 denarii upkeep.
Daily games: 10% PO mod for 400 denarii. (daily only effect)
Daily games: 30% PO mod for 800 denarii. (total effect)
At 36,000 pop, 9 units of peasants give 20% PO mod for 900 denarii upkeep.

At 1,000 pop, 120 peasants are 12% of pop, times 7 exceeding 80% for full mod.
At 1,200 pop, 120 peasants are 10% of pop, times 7 for 70% PO mod.
At 2,000 pop, 240 peasants are 12% of pop, times 7 exceeding 80% for full mod.

There are rounding issues, but queuing units deals with this.

Strategy
Of course, there are times when I find I need all the PO I can muster, regardless of cost.
Corrupt governors with high influence are a way of "buying" PO,
just like daily games and races -- I'd rather have the city and its income than not!

Remember, even when the display says a city is "negative" income,
it is still making money for you and paying for your army.
The negative is just a way of showing its income relative to other cities its size.

Unhappy cities are cities at risk, but negative income cities simply need economic buildings.
Good public order should be exploited for tax. Tax provides a way of turning PO into cash.
But the reverse is also true, high income cities can afford garrisons and games if needed.
But, even better, the larger your empire, the more you can afford to hold "difficult" cities.

Happy gaming!