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This article presents some of the mathematics of Civilization V. It was originally published by alpaca at the CivFanatics forum. Notice that this article represents the state of the game before the December 2010 patch!

Updated on 6/13/2015: Unit Maintenance updated to reflect the actual formula in the DLL - monju125

Contents show]

Calculating Culture Edit

Border growth Edit

The cost of a border growth is, with t the number of tiles already claimed

$ 20 + (10 (t - 1))^{1.1} $

Formula for Social Policies Edit

The first thing we need to know when talking about anything culture-wise is how to calculate it. From the game core XML files and in-game observations, it works like this: You calculate a base cost that depends on the number of policies, then multiply with a factor that depends on your number of cities. As follows:

$ n $: The number of cities in your empire

$ p_{base} (k) $: The base cost of policy k. Depends only on the number of policies

$ p_{cities}(n) $: The cost factor depending on the number of cities in your empire

$ p(n,k) $: The total cost of policy k. Depends on both the number of cities and policies already unlocked

The policy cost has to be modified depending on your difficulty setting, game speed and map size. I took the numbers from a post by Yamian. Define these modifiers as:

$ m_t $: The game speed modifier (3 for marathon, 1.5 for epic, 1 for normal and 0.67 for quick)

$ m_d $: The difficulty coefficient (0.5 for settler, 0.67 for chieftain, 0.85 for warlord, 1 otherwise)

$ m_z $: The map size modifier (0.3 for normal size and below, 0.2 for large, 0.15 for huge)

$ p_{base}(k) = m_t m_d (25 + (6 k)^{1.7}) = m_t m_d (25 + 21.03k^{1.7}) $

$ p_{cities}(n) = 1 + m_z (n - 1) $

The policy cost scales linearly with the number of cities and something between linear and quadratic with the number of policies. These factors are combined by multiplication, so the total cost is simply:

$ p(n,k) = p_{base}(k) * p_{cities}(n) $ rounded to the next multiple of 5

Cost for each policy on standard settings k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 n=1 25 45 90 160 245 345 465 595 745 905 1075 1260 1460 1670 1890 2120 2365 2620 2885 3160 3445 3745 4050 4365 4690 5025 5370 5725 6090 6465 6845 2 30 55 120 205 320 450 605 775 970 1175 1400 1640 1900 2170 2460 2760 3075 3405 3750 4110 4480 4865 5265 5675 6100 6535 6985 7445 7920 8405 8900 3 40 70 145 255 395 555 745 955 1190 1445 1725 2020 2335 2670 3025 3395 3785 4195 4620 5060 5515 5990 6480 6985 7510 8045 8595 9165 9745 10345 10955 4 45 85 175 305 465 660 885 1135 1415 1720 2050 2400 2775 3175 3595 4035 4500 4980 5485 6010 6550 7115 7695 8295 8915 9555 10210 10885 11575 12280 13010 5 55 100 205 350 540 765 1025 1315 1640 1990 2370 2780 3215 3675 4160 4670 5210 5770 6350 6955 7585 8240 8910 9605 10325 11060 11820 12600 13400 14220 15065 6 60 115 230 400 615 870 1165 1495 1865 2265 2695 3160 3655 4175 4730 5310 5920 6555 7215 7905 8620 9360 10125 10920 11730 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How fast do I acquire new policies? Edit

This is very simple, and I'll just list it here to introduce some variable names. It depends on the culture your empire yields. With

$ c $: The total culture yield of your empire

$ p $: The cost of your next social policy (I will omit the dependencies for readability)

$ t $: The time in turns to unlock the next policy

we get the simple formula

$ t = p/c $

if we assume we just unlocked a new policy. If we introduce

$ c_{acc} $: The culture already accumulated in the culture bucket

we get

$ t = (p - c_{acc})/c $

How do I calculate the total culture in my empire? Edit

This is shown in the UI but again to introduce some concepts. Let

$ {\bar c} = c/n $ : the average culture of each city in your empire

It makes sense to split c up into a part that depends on n and represents your "typical city culture", the culture each city you newly create will add to your empire, and a part that is constant in n and represents bonuses from wonders, landmarks and city states. This is exact if you immediately buy your typical culture buildings but normally it's an approximation (it represents an equilibrium you will not normally have reached)

$ c_t $: The typical culture per city

$ c_c $: The city state culture

$ c = n c_t + c_c = n {\bar c} $

Before I continue let's look at the policy speed for large numbers

$ \begin{align} t &= \frac{p_b(k) (1 + p_m (n - 1))}{n {\bar c}} \\ &= \frac{p_b(k)}{\bar c}\left(\frac{1 - pm}{n} + pm\right) \\ \end{align} $

$ \lim_{n \to \infty}t = \frac{p_m p_b}{\bar c} $

Also: $ \lim_{n \to \infty}{\bar c} = c_t $

So for large n, the average amount of culture per city is a good measure for the policy speed.

Will expanding increase or decrease policy speed? Edit

This is the question Paeanblack and I discussed in said thread. To analyse this, we have to calculate the number of turns to the next policy and see how it's affected by going from n->n+1.

When we do the city number increase, both p and c change. Let p and c be the policy cost and culture yield before founding the new city and p' and c' be the respective numbers after the founding.

$ p' = p_b(k) (1 + p_m n) = p + p_m p_b(k) $

$ c' = (n + 1) c_t + c_c = c + c_t $

Then calculate t' and check when it gets smaller than t (this would signify an increased policy speed)

$ t' = p'/c' = \frac{1 + p_m n}{c_t n + c_t + c_c} < \frac{1 + p_m n - p_m}{c_t n + c_c} $

$ (1 + p_m n) (c_t n + c_c) < (1 + p_m n) (c_t n + c_c) + (1 + p_m n) c_t - p_m (c_t n + c_t + c_c) $

$ p_m c_t n + p_m c_t + p_m c_c < c_t + p_m n c_t $

$ p_m (c_t + c_c) < c_t $

$ r_c = \frac{c_c}{c_t} < \frac{1}{p_m} - 1 = \frac{1 - p_m}{p_m} $

For standard-sized maps, $ p_m = 3/10 $ so

$ r_c < \frac{7}{3} = 2.\hat3 $

So if the base culture from city states is less than 7/3 times larger than the typical city culture (let's call this the culture ratio r_c), you will get an increased policy speed from expanding. If it's exactly equal to this, the speed will stay the same, and if it's more, policy speed will slow. It should be noted that, no matter the value of r_c, for large numbers of cities the increase or decrease for founding an additional city will be very small.

For larger maps, this will be a little different.

Large: r_c < 4

Huge: r_c < 17/3 =5.\hat6

Numbers Edit

Now we can plug in some example numbers into our calculations. I will discuss the results only for the standard map size

Let's assume you only build a monument in each of your cities. This is equivalent to a value of cT = 2. So if cC is five or greater, for example because you have at least one cultural city state as a friend, expansion will slow down your social policy speed.

Standard: cC > 4, Large: cC > 8, Huge: cC > 11 Now assume we build a monument and a temple, or cT = 5. Then, cC <= 11 will still yield an increase in your social policy speed. A cC of 10 is still pretty low, though. You normally still get it later on if you have at least one cultural city state ally.

Standard: cC > 11, Large: cC > 20, Huge: cC > 28 Looking at France, with a monument cT = 4 (true also for Egypt with Monument and Burial Tomb) and with both monument and temple, cT = 7. The corresponding cC values are 9 and 16. For 9 the same as above is true, but for the case with temples, you will actually gain an increase in policy speed if you don't have at least two city state allies or a city state and a few wonders.

Standard: cC > 9 or 16, Large: cC > 16 or 28, Huge: cC > 28 or 40 The Songhai have the excellent Mud Pyramid, so they share the cT = 7 case with France. The same goes for adding two artists in each city.

To sum up, expanding will in almost all cases slow down your social progress. The only cases where it will speed it up are if you either aren't interested in city states and wonders, or if you play a civ with a culture bonus. The only somewhat realistic scenario where expansion could speed up your policy gain is in my opinion if you play Songhai because you'll really want the Mud Pyramid and a Monument isn't that expensive.

I'm not sure if city state bonuses scale with the map size but if they don't, expansion increasing your policy speed is a lot more likely on larger maps, probably happening at some time if you just have a monument and a temple or are playing France. If you play Songhai, or France with temples it will even happen pretty often in fact. This is another case where the game doesn't scale well (well in the sense of preserving the same effects on gameplay as on standard size) with map size.

Food Edit

Food cost Edit

The food cost for a city to grow is calculated as follows

$ n $: The number of citizens in the city

$ f $: Amount of food to grow to size n+1

$ f(n) = 15 + 6 (n - 1) + (n - 1)^{1.8} $

Comment: It is OUTDATED! Patch on Dec 2010. Now the formula is:

$ f(n) = 15 + 8 (n - 1) + (n - 1)^{1.5} $

Need to edit everything below.

Food required to grow to level n+1

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 f 15 22 30 40 51 63 76 90 105 121 138 155 174 194 214 235 258 280 304 329 354 380 407 435 464 493 523 554 585 617 650 684 719 754 790 826 863 901 940 979

Integrated food values (total amount food it takes to grow to size n)

Code:

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 f 15 37 67 107 158 221 297 387 492 613 751 906 1080 1274 1488 1723 1981 2261 2565 2894 3248 3628 4035 4470 4934 5427 5950 6504 7089 7706 8356 9040 9759 10513 11303 12129 12992 13893 14833 15812

What this means is that growing from size 5 to 6 costs 51 food, while growing from size 10 to 11 already costs 122 food. I would like to present a few ways of looking at this problem from different angles in the following.

Constant Food Surplus Edit

This simplification assumes that each new citizen will work a tile that's worth 2 food and you therefore have a constant amount of food surplus that is put into growth. For a normal city without any bonuses, the amount of turns you need to grow is simply given by

$ t(n) = g(n)/f $

where n is the number of citizens, f the food surplus and t the number of turns until growth, assuming you start at 0 food. Let's look at the numbers for df = 8

Code:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 t 2 3 4 5 6 8 10 11 13 15 17 19 22 24 27 29 32 35 38 41 44 48 51 54 58 62 65 69 73 77 81 86 90 94 99 103 108 113 118 122

This kind of explains why growth feels so slow once you hit the teens: The number of turns cities need to grow becomes pretty large, even with a pretty decent food surplus. A city takes about the same time to grow from size 12 to 13 as from size 1 to 5.

Constant food per citizen Edit

Here, the assumption is that each citizen will yield a certain mean amount of food. To understand what happens there better, let's define another function, the average growth cost per citizen

$ ga(n) = g(n)/n = 9/n + 6 + (n - 1)^{1.8}/n $

(In blue the exact function, in red the rounded one the game uses)

The last term is approximately n^0.8 and dominates for large n. As you can see, this function has a minimum at size 4 (about 3.75 in the continuous case) and then constantly grows a little less than linearly - but linear growth is an excellent approximation.

As you can see below, the function 0.41 n + 8.18 fits the data over this range perfectly but is much simpler (the functions agree at all integer places).





So how do you interpret this function? What it tells you is that, with a constant amount of food produced per citizen, your city will grow more quickly than before until it hits size 4, and then starts slowing down again. Or, to put it more simple, this function tells you the amount of food each citizen has to produce to let the city grow.

An alternative, but equivalent, point of view is that, if you're aiming for a constant city growth speed, each citizen has to become more efficient as your city grows, in an approximately linear function.

You should note that the amount of food generated per citizen will usually be larger for small cities due to the city square itself (especially with maritime CS).

Slightly more realistic Edit

Cities start with a center square that yields some food (this catches granary, maritime city-states and water mill food, too). So the assumption that the amount of food per citizen will stay the same isn't really completely applicable. So let's introduce a new variable, f_csq, the city square food amount.

The assumption of an average amount of food per citizen now makes sense if you leave the city square food out. So let's call the amount of food each citizen generates f_c. The total food per turn, f, is then given by

$ F = f_c * n + f_{csq} $

More interesting for us is the food surplus. This is given by

$ f = F - 2*n = (f_c - 2)*n + f_{csq} $

The amount of surplus food produced per citizen is then

$ fa = f/n = f_{csq}/n + f_c - 2 $

To get the number of turns we need for growth, we need to combine this with the food growth cost and get

$ t = g/f $

Since this function doesn't read too pretty, I'll just plop in some numbers and give you a graphical representation. Let's say f_csq = 2, which is the case if you don't have city state allies. If the city is to grow in a reasonable amount of time, f_c should be greater than 2.5, as you can see below

You will notice a local minimum evolving in the f_c = 4 case, which signifies the onset of the constant food per citizen approximation, which is the limiting case for f_csq = 0

Are granaries worth it? Edit

Ultimately, that's for you to decide. I can give you some information so you can make an educated decision, though. Granaries increase f_csq from 2 to 4, so let's look at what happens to t if we make that change. Dashed are the values without granary, the full lines are with a granary

More useful to judge the granary's effects is looking at the difference between the case without a granary, and the case with a granary

As you can see, the difference quickly becomes essentially constant at a city size of 10 or more for any case of f_c > 2. From this analysis, I'd say that for f_c values up to 3, a granary is generally worth it, because it saves you a turn or two per growth step. For f_c = 2.5 it's very much worth it, and this seems to be a more realistic case than the higher values because not every citizen will work a farm (the f_c = 4 case represents each citizen working a Civil Service/Fertilizer farm) and some won't produce any food at all, like specialists.

Other f_csq effects Edit

We can continue this analysis by increasing f_csq in steps. For example, a maritime CS will yield 2 extra food, as will a water wheel. Let's see what happens if we go from f_csq = 4 to f_csq = 6. I will omit the more unrealistically high cases from now on for a better overview. Shown is again the total number of turns needed and the difference between 4 and 6. This time, f_csq = 4 is dashed

So getting the second city-state isn't as good as getting the first, and getting a granary when you already have a city-state isn't so great, either. It still shaves off a turn (or three in the f_c = 2.5 case), though.

The next increase step, from 6 to 8 (the "before" being dashed as usual)

Now things start becoming somewhat underwhelming. As I said, if things are worth it for you is up to you to decide, but I'd definitely not build that water mill if I already have a granary and a city-state ally because the difference will only be something like two turns in three growth steps or so, which isn't exactly a lot.





f_c values < 2 Edit

After reading a comment from ehrgeix, I think it makes sense to extend the analysis to f_c values that are smaller than 2. The values greater than 2 are applicable if you want to let your city continue to grow for the rest of the game. Values smaller than 2 still make sense in transitionary periods, if you want a growth cap, or if you're fine with your growth speed slowing down even more than in the constant food surplus case discussed above.

Qualitatively, we can already see from looking at $ f = (f_c - 2)*n + f_{csq} $, which occurs as the denominator in the formula for t, that these functions will have a pole at a finite n, because f_c - 2 becomes negative. The locus of this singularity is given by $ n = f_{csq}/(2 - f_c) $. This singularity signifies the number of citizens where the city stops growing.

In the following are some graphs in the same way as above. First, f_csq = 2

If you increase f_csq you shift the position of the pole to the right, so the difference between t(csq = 2) - t(csq = 4) has a singularity at the same points as t. See the f_csq = 4 case below. Dashed is the "previous", in this case f_csq = 2

f_csq = 4 -> f_csq = 6

f_csq = 6 -> f_csq = 8

Unit maintenance Edit

As of 6/2015, the formula for unit maintenance as defined in all versions CvGameCoreDLL is calculated as follows:

b = Base unit cost (INITIAL_GOLD_PER_UNIT_TIMES_100)

f = Free units from handicap (GoldFreeUnits), specific unit types, or policies

u = Total paid units

n = max(0, u-f) = Number of actual paid units (if u-f is less than 0 then n = 0)

m = Multiplier (UNIT_MAINTENANCE_GAME_MULTIPLIER)

d = Divisor (UNIT_MAINTENANCE_GAME_EXPONENT_DIVISOR)

t = Current turn

e = Estimated end turn (based on all entries for the current GameSpeed in GameSpeed_Turns)

g = t/e = Game progress factor

final cost = (n*b(1+g*m)/100)(1+(g/d))

Unit cost modifiers from traits are applied before the exponent. Unit cost modifiers from policies, handicap, and AI difficulty level are applied to the final cost.

Since there's no easy "each unit in turn t will cost this much" here's a table you can use as a rough reference. The first row is the number of turns, the first column the number of units

Total unit maintenance costs Turns Units 1 20 50 100 150 200 250 300 350 400 4 2 2 3 5 7 9 11 14 17 20 8 4 5 7 11 15 19 24 30 36 43 12 6 8 11 16 23 30 38 47 57 68 16 8 10 14 22 31 41 52 64 78 94 20 10 13 18 28 39 52 66 82 100 121 24 12 16 22 34 47 63 80 100 122 148 28 14 18 26 40 56 74 94 118 145 176 32 16 21 30 46 64 85 109 136 168 204 36 18 24 34 52 73 96 124 155 191 232 40 20 26 37 58 81 108 138 174 215 261 44 22 29 41 64 89 119 153 193 238 291 48 24 32 45 70 98 131 168 212 262 321 52 26 35 49 76 107 142 184 231 287 350 56 28 37 53 82 115 154 199 251 311 381 60 30 40 57 88 124 166 214 270 335 411 64 32 43 61 94 133 177 229 290 360 442 68 34 45 64 100 141 189 245 310 385 473 72 36 48 68 106 150 201 260 330 410 504 76 38 51 72 112 159 213 276 349 435 535 80 40 54 76 118 167 225 292 370 461 567 84 42 56 80 124 176 237 307 390 486 598 88 44 59 84 130 185 249 323 410 512 630 92 46 62 88 137 194 260 339 430 537 662 96 48 64 92 143 203 273 354 451 563 694 100 50 67 95 149 211 285 370 471 589 727

Here's an equivalent table detailing the cost per unit