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The Number Crunching Thread

Discussion in 'Civ5 - Strategy & Tips' started by alpaca, Oct 8, 2010.

  1. alpaca

    alpaca King of Ungulates

    Joined:
    Aug 3, 2006
    Messages:
    2,322
    I thought I'd open up a home for the more mathematically inclined among us to post their theories and calculations. Statistical data like average yields for cities and per citizen, average city size, population, GNP, etc. at turn n are very useful to plug into mathematical models so it is very welcome here, too.

    As for conventions, in anything I write a space is equivalent to a times sign (*). If there's no space, it is usually a compound variable name because indices aren't really available in BB code. Anything else should be self-explanatory

    To start off, I'll post a bit about culture that's a result of a discussion between Paeanblack and me in a thread about the Songhai.


    Calculating Culture

    Border Growth

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

    20 + (10 (nT - 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:

    pb(k): The base cost of policy k. Depends only on the number of policies
    n: The number of cities in your empire
    p(n,k): The total cost of policy k. Depends on your number of cities and policies already unlocked

    pb(k) = 25 + (6 k)^1.7 = 25 + 21.03*k^1.7

    p(n,k) = pb(k) (7 + 3 n)/10 rounded to the next multiple of 5

    So the policy cost scales linearly with the number of cities and something between linear and quadratic with the number of policies.

    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. Let

    pm: The map size modifier (0.3 for normal size and below, 0.2 for large, 0.15 for huge)
    pd: The map difficulty coefficient (0.5 for settler, 0.67 for chieftain, 0.85 for warlord, 1 otherwise)
    ps: The game speed modifier (3 for marathon, 1.5 for epic, 1 for normal and 0.67 for quick)

    pb(k) = ps pd (25 + (6 k)^1.7)
    p(n,k) = pb(k) (1 + pm (n - 1)) rounded to the next multiple of 5


    Below is a table detailing the cost for each policy on standard settings:

    Code:
    	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

    cB: The culture already accumulated in the culture bucket

    we get

    t = (p - cB)/c


    How do I calculate the total culture in my empire?

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

    cA = 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)

    cT: The typical culture per city
    cC: The city state culture

    c = n cT + cC = n cA

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

    t = pB (1 + pm (n - 1))/(n cA) = pB/cA ((1 - pm)/n + pm) -> pm pB/cA and cA->cT for n->inf

    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' = pb (1 + pm n) = p + pm pb
    c' = (n + 1) cT + cC = c + cT

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

    t' = p'/c' = (1 + pm n) / (cT n + cT + cC) < (1 + pm n - pm)) / (cT n + cC)
    (1 + pm n) (cT n + cC) < (1 + pm n) (cT n + cC) + (1 + pm n) cT - pm (cT n + cT + cC)
    pm cT n + pm cT + pm cC < cT + pm n cT
    pm (cT + cC) < cT
    rC := cC/cT < 1/pm - 1 = (1 - pm)/pm

    For standard-sized maps, pm = 3/10 so

    rC < 7/3

    So if the base culture from city states is less than 7/3 (=2.333) times larger than the typical city culture (let's call this the culture ratio rC), 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 rC, 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: rC < 4
    Huge: rC < 17/3 (=5.67)


    Numbers

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

    1. 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

    2. 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

    3. Looking at France, with a monument cT = 4 (true also for Egypt with Monument and 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

    4. 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 rounded down to the next integer.

    Here are some tabularized and plotted values:

    Food required to grow to level n+1


    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
    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

    See a first analysis in post 16



    Unit Maintenance

    Unit maintenance is calculated as follows:

    c(t,n) = ((0.5 + 8/1000 t) round(n,2))^(1 + 2/7000 t)

    where n = number of units, t = number of turns and the round function is meant to take the next lowest even number if n is odd.

    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

    Code:
    					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

    Code:
    					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
    
     

    Attached Files:

  2. alpaca

    alpaca King of Ungulates

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    Reserved. Get those suggestions and analyses coming :)
     
  3. Yamian

    Yamian Chieftain

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  4. alpaca

    alpaca King of Ungulates

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    2,322
  5. Paeanblack

    Paeanblack Prince

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    518
    One thing that is important to keep in mind is the diminishing effects of further expansion

    Even if your rC is very high or very low, the effect of another city on t' gets fairly small after n > 5 and negligible after n > 10 (Assuming cT > 0 ! Else rC is infinite, which is more than what I have in mind for "very high")

    Once you have a bunch of settlements/annexes, the damage is done or the fruits have been picked, so to speak. Unless you are continuing to expand out exponentially the exponential growth of pb(k) will soon marginalize the effects any linear expansion of your empire.

    I haven't played any Huge games, but on smaller maps with my playstyle, the REX phase ends relatively quickly, since the good spots are gone and I'm bumping into other players. I rarely have several cities simultaneously pumping out settlers who will found cities that will join in the settler production. I assume if you are doing that, you don't really care what it does to your policy costs.

    Since only the first few settlements really make a difference, the early timing is dominated by when you get your first few policies. It's obviously better to expand right after buying a new policy than right before. The question that raises is: where is the threshold between policies when it's better to wait for the next policy than settle now? Even if you are running with a low rC and rushing your monuments (buy/chop), there will still be times when the expansion window closes.


    Also, warmongers need to pay attention to this, since my inner prognosticator tells me that puppet cities will increase policy costs Real Soon Now. It is much harder to time the capture of a city than it is to time a settler build. If you attack too soon, you can't easily wait around a few turns to capture a city until your next policy buy. Reinforcements will usually arrive.
     
  6. alpaca

    alpaca King of Ungulates

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    Somehow I thought you'd find your way here :D

    You're mentioning an interesting point. At the moment I'm having fun with a close-packed ICS strategy (where in an ideal case every city only has its 7 inner hexes). In that regime, you don't care about building another city but you do care about cT, or cA which it approaches once you have a lot of settlements.

    Since I usually spend my money on buying colosseums or libraries and hard-build monuments - or chop them if I have enough forests - my own analysis doesn't work very well for me because I play in a way where I will settle four or five cities at any given time that won't yield culture for a while, and my culture and whole game is generally far from equilibrium.

    I decided to include the steady-state analysis here because it's a lot simpler and I don't actually want to do a time-dependent analysis. Maybe I'll do another boundary case for n->inf which should actually be even easier.


    As for larger maps, they significantly influence the analysis (see the updated OP). Does anyone know if cultural city state bonuses scale with map size?
     
  7. CharonJr

    CharonJr Warlord

    Joined:
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    Once you include the added cultural gain from additional cities it starts to get fuzzy, especially in cultural games where you will stay at a very small number of cities.

    Besides directly producing culture an additional city will enable you to gain culture via conquest, better research, splitting wonder building between cities, getting additional strategic/luxury ressources and being able to produce more units, too.

    Unless my calculations are wrong you need about 20k culture to fill up 5 social trees (with the culture cost adjustments from Christo/social policy included), one additional city will raise this by 30%. I think that most often the 2nd cities benefits will be enough to compensate.

    But since there are diminishing returns from adding more cities at some point the gains will no longer be enough to compensate.

    CharonJr
     
  8. alpaca

    alpaca King of Ungulates

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    30 policies without any modifiers cost 73179.5 - this doesn't include free SPs, etc.

    You can easily calculate if the second city will speed up things from a purely culture point of view: Just include any extra culture your first city has in cC and use the second city's culture as cT.

    It's obviously way out of scope to analyse any second-order effects like money from the second city being used to buy cultural buildings, etc. I think the only way to reliably tell whether a one-city or multiple-city approach makes sense is to test it by playing games and logging the end date, then doing statistics on a large enough sample. That's part of the reason why I asked for data in the first post :)
     
  9. CharonJr

    CharonJr Warlord

    Joined:
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    BTW, the current gauntlet should be a treasure trove for cultural data, we are playing Arabia at prince level for earliest cultural victory there, till now 1620AD is the fastest, so far I only managed to finish in 1700, maybe due to having 2 cities ;)

    And yes, the 20k should only be valid for 27 policies and with full boni.

    CharonJr
     
  10. vmxa

    vmxa Deity

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    Location:
    Oviedo, Fl
    Interesting thread, I would like to know what the cost is for each break point. I should know it, but I don't. I know the first one is 30 and I think the second is 60. After that I do not know.

    The reason I wish I had paid attention and wrote them down is I am saving for a later ages selection. I now see I have no idea of how close I am to the next one as I only have my accumulation number and current out put per turn.
     
  11. Paeanblack

    Paeanblack Prince

    Joined:
    Dec 4, 2001
    Messages:
    518
    The diminishing marginal returns are paired with diminishing marginal costs, that's why rC is important. If rC is below the threshold, then every additional city will help give you policies faster.

    The impact of each additional city is what actually diminishes. If you have 2 cities, the third one will speed up or slow down your policies by X%, based on your rC. If you have 10 cities, you need to acquire five more cities to have the same effect.

    Your city count will not continue to increase exponentially through the whole game, but with the return of viable ICS strategies, it may stay in the exponential model longer. Those strategies seem to revolve around waiting for key policies and expanding in big jumps...which doesn't fit nicely into any exponential/polynomial/linear model. Ugh.
     
  12. Paeanblack

    Paeanblack Prince

    Joined:
    Dec 4, 2001
    Messages:
    518
    Random thought...with the massive cost of the last few policies, does it really matter if you didn't have expansion-city monuments built back when you thought 30 culture per turn was a big deal? A 20-turn delay on a monument is peanuts. A 20-turn delay in 50 cities, is still

    This question is geared for a culture victory, where you want all 27-30 policies, and you are strongly considering saving up 3 for the Renaissance to get the 33% discount. If you are just going for a strong play at 15-20 policies for your empire, then sooner is better.
     
  13. alpaca

    alpaca King of Ungulates

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    In an empire with constant number of cities, I don't think early culture really matters. I think there are strategies where you save your policies for when you have Cristo Redentor and the Freedom -33% but I haven't really been trying for a culture victory yet except when I started playing, with Gandhi. I'm thinking about turning my current ICS game into one but I'm not sure if I can still pick up enough policies because I have a lot of cities (>60) and right now I'm already at turn 264 with a policy speed of 19 turns - recently started investing heavily into culture, though, so cT is steadily increasing towards museum-levels (atm cA is about 10).

    Edit: Here's a table for you vxma.

    Cost of increase:

    25
    45
    95
    160
    245
    350
    465
    600
    745
    905
    1080
    1265
    1460
    1670
    1895
    2125
    2370
    2625
    2890
    3165
    3450
    3745
    4050
    4370
    4695
    5030
    5375
    5730
    6095
    6465
    6850

    Integrated cost of all policies up to this point:
    25
    70
    165
    325
    570
    920
    1385
    1985
    2730
    3635
    4715
    5980
    7440
    9110
    11005
    13130
    15500
    18125
    21015
    24180
    27630
    31375
    35425
    39795
    44490
    49520
    54895
    60625
    66720
    73185
    80035
     
  14. vmxa

    vmxa Deity

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    Very gracious of alpaca, thanks.
     
  15. alpaca

    alpaca King of Ungulates

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    Added numbers pertaining to food cost for growth. If anyone would like to do some calculations, be my guest ;)
     
  16. alpaca

    alpaca King of Ungulates

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    I'd like to understand food growth better so here's an attempt at analysing it a bit.


    Food Cost

    Let's start by repeating the food cost function. When your city has n citizens, the amount of food required to grow to n+1 is

    g(n) = 15 + 6 (n - 1) + (n - 1)^1.8

    This function consists of a linear part and a more quickly increasing part with an exponent of 1.8 (about 2 if you want to visualize it). The function, up to 40, is plotted here



    What this means is that growing from size 5 to six 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 CS 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 CS 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



    -----------------------------------------------------

    Comments? Ideas? Suggestions?
     

    Attached Files:

  17. ehrgeix

    ehrgeix Chieftain

    Joined:
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    Posted in granary thread, but OP requested I repost here:

    This is a great thread, but: with maritime CS/granaries you do not have anything like +2f per citizen in terms of working tiles.

    Also, realistic games on deity/immortal tend to be won or lost before many cities hit size ten. Often earlier.

    To clarify: You want to work as many mines/lumbermills/trading posts as possible and the maximum you'll realistically get from a square is 2f, but often 0-1. I would be really interested to see similar graphs focusing on size 1-10 cities with each citizen generating average 1f at all times.

    -

    Followup in response to questions on typical games:

    This is a game report I posted recently: http://forums.civfanatics.com/showthread.php?t=390164

    Games can go faster if you use horsemen/companion cav, or if you rush sci revolution for t100 riflemen, but most of mine are over by t200. I do tend to play tiny or small maps though. I think the important thing is that even if they are not over by t200 (going for a non-domination win on a standard map, say), they are still *decided* by t150~, and usually much earlier. I think I have some normal-city shots in that report, but especially early working a lot of mines and 2 cities with sci specialists is important, and it crushes your average food per citizen/you rely on granary and maritime CS more.

    Edit: If I could harass you into doing even more math, 1f is important and normal, but 0.5f (not really likely, but possible with many cities/specialists/mines) would be neat too!

    Edit #2: Alternative potentially viable early deity option: build just monument + library early, skip granaries, work less production and more gold/food squares, and then focus more on production a little later. I'm not sure how you'd go about mathing this out though.
     
  18. alpaca

    alpaca King of Ungulates

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    Thanks for re-posting. I'll give it a look tomorrow because I want to get some play-time in. You're right that I didn't put enough thinking time into the f_c numbers, they are more appropriate for a scenario where you don't have maritime CS and therefore rely on farms and such to grow your cities.
     
  19. ehrgeix

    ehrgeix Chieftain

    Joined:
    Oct 9, 2010
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    12
    No problem. Thanks for doing the math at all in such detail - it's very helpful. I also want to say that after looking at it more I think that it's quite possible that I am wrong, or that you do want to run more food tiles and skip granaries or something. I basically feel like granaries are like... not a big return. You might invest 100p 1gpt and get back 100p 2gpt +1scipt, and I think that if I was playing a leader like ramesses I would consider something like monument, library, burial tomb early. There's just so little in the way of buildings you actually want to build early when looking at optimal play. I think I would not be having this discussion if you could build another early building as good as a library or something. I guess you could always rush horsemen. =(

    It feels like you want to work so many strong gold/production tiles early that you almost never get more than +1f/tile, and I think that if you do build a bunch of farms you end up with big cities that are doing nothing but getting bigger and also producing not enough production/gold while growing to be at all worth it. The granary feels like a sort've happy medium that when combined with maritime states lets you keep growing while working tiles that are heavily focused on +production/gold (and running specialists in 2-3 cities), but not growing too fast and getting maximum prod/gold from your city while it is growing.
     
  20. Dizzy75

    Dizzy75 Warlord

    Joined:
    Mar 22, 2006
    Messages:
    245
    You were assuming normal speed, normal size or smaller, prince+ difficulty for your first formula for pb, right?
     

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