Mathematics of Civilization V

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
==Calculating Culture==
===Border growth===
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===
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 exponentially 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	12570	13435	14320	15230	16160 	17115
7 	70 	125	260 	450 	690 	975 	1305	1675	2085 	2535 	3020 	3540 	4090 	4680 	5295 	5945 	6630 	7340 	8085 	8855 	9655 	10485	11345	12230	13140	14080	15045	16040	17055	18100 	19170
8 	75 	140	285 	495 	765 	1080	1445	1855	2310 	2805 	3345 	3915 	4530 	5180 	5865 	6585 	7340 	8130 	8950 	9805 	10690	11610	12560	13540	14550	15590	16660	17755	18885	20040 	21225
9 	85 	155	315 	545 	835 	1185	1585	2035	2535 	3080 	3665 	4295 	4970 	5680 	6430 	7220 	8050 	8915 	9815 	10755	11725	12735	13775	14850	15955	17100	18270	19475	20715	21980 	23280
10 	90 	170	345 	595 	910 	1290	1725	2215	2760 	3350 	3990 	4675 	5405 	6180 	7000 	7860 	8760 	9700 	10685	11705	12760	13855	14990	16160	17365	18605	19885	21195	22540	23920 	25335
11 	100	180	370 	640 	985 	1395	1865	2395	2985 	3620 	4315 	5055 	5845 	6685 	7570 	8495 	9470 	10490	11550	12650	13795	14980	16205	17470	18775	20115	21495	22915	24370	25860 	27390
12 	105	195	400 	690 	1060	1500	2005	2575	3205 	3895 	4635 	5435 	6285 	7185 	8135 	9135 	10180	11275	12415	13600	14830	16105	17420	18780	20180	21625	23110	24630	26195	27800 	29445
13 	115	210	425 	740 	1135	1605	2145	2755	3430 	4165 	4960 	5815 	6725 	7685 	8705 	9770 	10895	12065	13280	14550	15865	17230	18635	20090	21590	23135	24720	26350	28025	29740 	31495
14 	120	225	455 	785 	1210	1710	2285	2935	3655 	4440 	5285 	6195 	7160 	8190 	9270 	10410	11605	12850	14150	15500	16900	18350	19850	21400	23000	24640	26330	28070	29850	31680 	33550
15 	130	235	485 	835 	1280	1815	2425	3115	3880 	4710 	5610 	6575 	7600 	8690 	9840 	11045	12315	13635	15015	16450	17935	19475	21070	22710	24405	26150	27945	29790	31680	33620 	35605
16 	135	250	510 	885 	1355	1920	2570	3295	4100 	4980 	5930 	6950 	8040 	9190 	10405	11685	13025	14425	15880	17395	18970	20600	22285	24020	25815	27660	29555	31505	33505	35560 	37660
17 	145	265	540 	930 	1430	2025	2710	3475	4325 	5255 	6255 	7330 	8475 	9690 	10975	12320	13735	15210	16750	18345	20005	21725	23500	25330	27220	29170	31170	33225	35335	37500 	39715
18 	150	280	565 	980 	1505	2130	2850	3655	4550 	5525 	6580 	7710 	8915 	10195	11540	12960	14445	15995	17615	19295	21040	22845	24715	26645	28630	30675	32780	34945	37165	39440 	41770
19 	160	290	595 	1030	1580	2235	2990	3835	4775 	5795 	6905 	8090 	9355 	10695	12110	13595	15155	16785	18480	20245	22075	23970	25930	27955	30040	32185	34395	36665	38990	41380 	43825
20 	165	305	625 	1075	1650	2340	3130	4015	5000 	6070 	7225 	8470 	9795 	11195	12680	14235	15865	17570	19345	21195	23110	25095	27145	29265	31445	33695	36005	38380	40820	43320 	45880
21 	175	320	650 	1125	1725	2445	3270	4195	5220 	6340 	7550 	8850 	10230	11700	13245	14870	16575	18360	20215	22145	24145	26220	28360	30575	32855	35205	37620	40100	42645	45255 	47930
22 	180	335	680 	1175	1800	2550	3410	4375	5445 	6615 	7875 	9230 	10670	12200	13815	15510	17285	19145	21080	23090	25180	27340	29575	31885	34265	36710	39230	41820	44475	47195 	49985
23 	190	345	705 	1220	1875	2655	3550	4555	5670 	6885 	8200 	9605 	11110	12700	14380	16145	18000	19930	21945	24040	26215	28465	30790	33195	35670	38220	40845	43540	46300	49135 	52040
24 	195	360	735 	1270	1950	2760	3690	4735	5895 	7155 	8520 	9985 	11550	13200	14950	16785	18710	20720	22815	24990	27250	29590	32010	34505	37080	39730	42455	45255	48130	51075 	54095
25 	205	375	765 	1320	2025	2865	3830	4915	6115 	7430 	8845 	10365	11985	13705	15515	17420	19420	21505	23680	25940	28285	30715	33225	35815	38490	41240	44070	46975	49960	53015 	56150
26 	210	390	790 	1365	2095	2970	3970	5095	6340 	7700 	9170 	10745	12425	14205	16085	18060	20130	22290	24545	26890	29320	31835	34440	37125	39895	42750	45680	48695	51785	54955 	58205
27 	220	405	820 	1415	2170	3070	4110	5275	6565 	7970 	9495 	11125	12865	14705	16650	18695	20840	23080	25410	27835	30355	32960	35655	38435	41305	44255	47295	50410	53615	56895 	60260
28 	225	415	845 	1465	2245	3175	4250	5455	6790 	8245 	9815 	11505	13300	15210	17220	19335	21550	23865	26280	28785	31390	34085	36870	39745	42710	45765	48905	52130	55440	58835 	62315
29 	235	430	875 	1510	2320	3280	4390	5635	7015 	8515 	10140	11885	13740	15710	17790	19970	22260	24655	27145	29735	32425	35210	38085	41055	44120	47275	50520	53850	57270	60775 	64365
30 	240	445	905 	1560	2395	3385	4530	5815	7235 	8790 	10465	12265	14180	16210	18355	20610	22970	25440	28010	30685	33460	36330	39300	42370	45530	48785	52130	55570	59095	62715 	66420
31 	250	460	930 	1610	2470	3490	4670	5995	7460 	9060 	10790	12640	14620	16710	18925	21245	23680	26225	28875	31635	34495	37455	40515	43680	46935	50290	53740	57285	60925	64655 	68475
32 	255	470	960 	1655	2540	3595	4810	6175	7685 	9330 	11110	13020	15055	17215	19490	21885	24395	27015	29745	32585	35530	38580	41735	44990	48345	51800	55355	59005	62755	66595 	70530
33 	265	485	985 	1705	2615	3700	4950	6355	7910 	9605 	11435	13400	15495	17715	20060	22520	25105	27800	30610	33530	36565	39700	42950	46300	49755	53310	56965	60725	64580	68535 	72585
34 	270	500	1015	1755	2690	3805	5090	6535	8130 	9875 	11760	13780	15935	18215	20625	23160	25815	28585	31475	34480	37600	40825	44165	47610	51160	54820	58580	62445	66410	70475 	74640
35 	280	515	1045	1800	2765	3910	5230	6715	8355 	10145	12085	14160	16370	18720	21195	23795	26525	29375	32345	35430	38635	41950	45380	48920	52570	56325	60190	64160	68235	72415 	76695
36 	285	525	1070	1850	2840	4015	5370	6895	8580 	10420	12405	14540	16810	19220	21760	24435	27235	30160	33210	36380	39670	43075	46595	50230	53980	57835	61805	65880	70065	74355 	78745
37 	295	540	1100	1900	2910	4120	5510	7075	8805 	10690	12730	14920	17250	19720	22330	25070	27945	30950	34075	37330	40705	44195	47810	51540	55385	59345	63415	67600	71890	76295 	80800
38 	300	555	1125	1945	2985	4225	5650	7255	9030 	10960	13055	15295	17690	20225	22895	25710	28655	31735	34940	38275	41735	45320	49025	52850	56795	60855	65030	69320	73720	78235 	82855
39 	310	570	1155	1995	3060	4330	5790	7435	9250 	11235	13380	15675	18125	20725	23465	26345	29365	32520	35810	39225	42770	46445	50240	54160	58200	62365	66640	71035	75545	80170 	84910
40 	315	580	1185	2045	3135	4435	5930	7615	9475 	11505	13700	16055	18565	21225	24035	26985	30075	33310	36675	40175	43805	47570	51455	55470	59610	63870	68255	72755	77375	82110 	86965
41 	325	595	1210	2090	3210	4540	6070	7795	9700 	11780	14025	16435	19005	21725	24600	27620	30790	34095	37540	41125	44840	48690	52675	56780	61020	65380	69865	74475	79205	84050 	89020
42 	330	610	1240	2140	3285	4645	6215	7975	9925 	12050	14350	16815	19440	22230	25170	28260	31500	34880	38410	42075	45875	49815	53890	58095	62425	66890	71480	76195	81030	85990 	91075
43 	340	625	1265	2190	3355	4750	6355	8155	10145	12320	14670	17195	19880	22730	25735	28895	32210	35670	39275	43025	46910	50940	55105	59405	63835	68400	73090	77910	82860	87930 	93130
44 	345	635	1295	2235	3430	4855	6495	8335	10370	12595	14995	17575	20320	23230	26305	29535	32920	36455	40140	43970	47945	52065	56320	60715	65245	69905	74705	79630	84685	89870 	95180
45 	355	650	1325	2285	3505	4960	6635	8515	10595	12865	15320	17955	20760	23735	26870	30170	33630	37240	41005	44920	48980	53185	57535	62025	66650	71415	76315	81350	86515	91810 	97235
46 	360	665	1350	2335	3580	5065	6775	8695	10820	13135	15645	18330	21195	24235	27440	30810	34340	38030	41875	45870	50015	54310	58750	63335	68060	72925	77930	83065	88340	93750 	99290
47 	370	680	1380	2380	3655	5170	6915	8875	11045	13410	15965	18710	21635	24735	28005	31445	35050	38815	42740	46820	51050	55435	59965	64645	69465	74435	79540	84785	90170	95690 	101345
48 	375	695	1405	2430	3725	5275	7055	9055	11265	13680	16290	19090	22075	25235	28575	32085	35760	39605	43605	47770	52085	56560	61180	65955	70875	75940	81150	86505	92000	97630 	103400
49 	385	705	1435	2480	3800	5380	7195	9235	11490	13955	16615	19470	22515	25740	29145	32720	36470	40390	44475	48715	53120	57680	62395	67265	72285	77450	82765	88225	93825	99570 	105455
50 	390	720	1465	2525	3875	5485	7335	9415	11715	14225	16940	19850	22950	26240	29710	33360	37185	41175	45340	49665	54155	58805	63615	68575	73690	78960	84375	89940	95655	101510	107510
===How fast do I acquire new policies?===
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?===
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?===
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===
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==
===Food cost===
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===
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===
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===
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?===
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===
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===
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==
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	Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns	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
Maintenance cost per unit
Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns	Turns
Units	1 	20 	50 	100 	150 	200 	250 	300 	350 	400
4 	0.50 	0.50 	0.75 	1.3 	1.8 	2.3 	2.8 	3.5 	4.3 	5.0
8 	0.50 	0.63 	0.88 	1.4 	1.9 	2.4 	3.0 	3.8 	4.5 	5.4
12 	0.50 	0.67 	0.92 	1.3 	1.9 	2.5 	3.2 	3.9 	4.8 	5.7
16 	0.50 	0.63 	0.88 	1.4 	1.9 	2.6 	3.3 	4.0 	4.9 	5.9
20 	0.50 	0.65 	0.90 	1.4 	2.0 	2.6 	3.3 	4.1 	5.0 	6.1
24 	0.50 	0.67 	0.92 	1.4 	2.0 	2.6 	3.3 	4.2 	5.1 	6.2
28 	0.50 	0.64 	0.93 	1.4 	2.0 	2.6 	3.4 	4.2 	5.2 	6.3
32 	0.50 	0.66 	0.94 	1.4 	2.0 	2.7 	3.4 	4.3 	5.3 	6.4
36 	0.50 	0.67 	0.94 	1.4 	2.0 	2.7 	3.4 	4.3 	5.3 	6.4
40 	0.50 	0.65 	0.93 	1.5 	2.0 	2.7 	3.5 	4.4 	5.4 	6.5
44 	0.50 	0.66 	0.93 	1.5 	2.0 	2.7 	3.5 	4.4 	5.4 	6.6
48 	0.50 	0.67 	0.94 	1.5 	2.0 	2.7 	3.5 	4.4 	5.5 	6.7
52 	0.50 	0.67 	0.94 	1.5 	2.1 	2.7 	3.5 	4.4 	5.5 	6.7
56 	0.50 	0.66 	0.95 	1.5 	2.1 	2.8 	3.6 	4.5 	5.6 	6.8
60 	0.50 	0.67 	0.95 	1.5 	2.1 	2.8 	3.6 	4.5 	5.6 	6.9
64 	0.50 	0.67 	0.95 	1.5 	2.1 	2.8 	3.6 	4.5 	5.6 	6.9
68 	0.50 	0.66 	0.94 	1.5 	2.1 	2.8 	3.6 	4.6 	5.7 	7.0
72 	0.50 	0.67 	0.94 	1.5 	2.1 	2.8 	3.6 	4.6 	5.7 	7.0
76 	0.50 	0.67 	0.95 	1.5 	2.1 	2.8 	3.6 	4.6 	5.7 	7.0
80 	0.50 	0.68 	0.95 	1.5 	2.1 	2.8 	3.7 	4.6 	5.8 	7.1
84 	0.50 	0.67 	0.95 	1.5 	2.1 	2.8 	3.7 	4.6 	5.8 	7.1
88 	0.50 	0.67 	0.95 	1.5 	2.1 	2.8 	3.7 	4.7 	5.8 	7.2
92 	0.50 	0.67 	0.96 	1.5 	2.1 	2.8 	3.7 	4.7 	5.8 	7.2
96 	0.50 	0.67 	0.96 	1.5 	2.1 	2.8 	3.7 	4.7 	5.9 	7.2
100 	0.50 	0.67 	0.95 	1.5 	2.1 	2.9 	3.7 	4.7 	5.9 	7.3
Category:Civilization V
Category:Guides (Civ5)