Math of Freeciv

Here are formulas that are used by FreeCiv to calculate some things, presented in a form better for mathematicians.

Tile output

Tiles produce three types of output: food, shields (for production), trade. To calculate the output of a tile:

1. Start with the base output for the terrain type.
2. Add output from the special resource, if the tile has any.
3. Add the point bonuses from terrain improvements:
   • Add bonus to food, if the tile has some irrigation or is the center of a city.
   • Add bonus to shields, if the tile has a mine. (desert, hills or mountains)
   • Add bonus to trade, if the tile has a road. (grassland, plains or desert)
   • Add bonus to trade, if the tile has a river.
4. Add to shields, the railroad bonus [50% in default ruleset], if the tile has a railroad.
5. Add point bonus from effect "Output_Add_Tile" [as from Harbor or Offshore Platform].
6. Add point bonus from effect "Output_Inc_Tile_Celebrate", if the tile already has some output of the same type, and the city is celebrating [like the +1 trade bonus for Monarchy or Communism]
7. Add point bonus from effect "Output_Inc_Tile", if the tile already has some output of the same type [like the +1 trade bonus for Republic or Democracy].
8. Add percentage bonus from effect "Output_Per_Tile" [as from Superhighways or Supermarket].
9. Subtract 1 point if the output exceeds the value of effect "Output_Penalty_Tile" [like the penalty for exceeding 2 points during Anarchy or Despotism].
10. Subtract pollution penalty [50% in default ruleset], if the tile has pollution.
11. Subtract fallout penalty [50% in default ruleset], if the tile has nuclear fallout.
12. Apply minimum values [1/1/0 in default ruleset], if the tile is the center of a city.

See get_worked_tile_output() and city_tile_output() in common/city.c.

City output (food/production/trade)

Cities produce six types of output: food, shields (for production), trade, gold, luxury and science. To calculate the output of any single city:

1. Start with the base citizen output from all citizens who are working tiles, or who are specialists.
2. If the city has any trade routes, then add the bonus from these routes to trade.
3. If the city pays tithes (like from Fundamentalism in civ2 ruleset), then add the tithes to gold.
4. Calculate waste and corruption.
5. Apply tax rates to trade. Add result to gold, luxury and science.
6. Apply city bonus effects (like 50% shield bonus from Factory, and the 100% science bonus from Library.)

References:

• enum output_type_id in common/fc_types.h
• output_type_iterate() in common/city.h
• set_city_production() in common/city.c

Food box

The amount of food needed to fill the food box is a function of the game.ruleset values granary_food_ini and granary_food_inc. The server option foodbox adjusts the amount.

• Recursive formula:
  • $city\_granary\_size = f(city\_size) \times \frac{foodbox}{100}$
  • $f(city\_size) = \begin{cases} granary\_food\_ini[city\_size] & \mbox{if } city\_size \le num\_inis \\ f(city\_size - 1) + granary\_food\_inc & \mbox{if } city\_size > num\_inis \\ \end{cases}$
• Where:
  • city_size is the current size of the city.
  • num_inis is the number of values in granary_food_ini (which is a list).

When you add a settler the food in the foodbox is kept. With a granary the foodbox will be half full when the city is decreased because of starvation, but when you grow a settler the foodbox will be empty(?). When you add a settler you will not gain half a foodbox

References:

• load_ruleset_game in server/ruleset.c
• city_granary_size in common/city.c

Production

Gold needed to finish production ($T$ - shields left to produce):

• For units: $2T+\frac{T^2}{20}$
• For buildings: $2T$
• For wonders: $4T$

Multiply these values by 2 if there is no shield in the city stock ($C$ - cost in shields):

• For units: $4C+\frac{C^2}{10}$
• For buildings: $4C$
• For wonders: $8C$

References:

• impr_buy_gold_cost in common/improvement.c
• utype_buy_gold_cost in common/unittype.c

Trade routes

(Freeciv 2.1 uses the formulas in this section. Freeciv 2.2 uses different formulas, see Gna bug 15092).

Trade route revenue generated every turn in cities: ($T_1$ = trade production in the city 1, $T_2$ = trade production in the city 2. These values do not include the extra trade revenues from other trade routes

• With yourself, on the same continent: $\frac{(T_1 + T_2 + 4)}{16}$
• With another player, on the same continent: $\frac{(T_1 + T_2 + 4)}{8}$
• With yourself, on another continent: $\frac{(T_1 + T_2 + 4)}{8}$
• With another player, on another continent: $\frac{(T_1 + T_2 + 4)}{4}$
• With experimental trade route revenues (on Warserver): $\frac{(T_1 + T_2 + 4)}{4}$

Caravan bonus, the gold and science you get when establishing a new trade route: ($T_1$ = trade production (including trade routes revenues) in the city 1, $T_2$ = trade production in the city 2, $D$ = real distance between the cities):

• $\frac{(D + 10) (T_1 + T_2)}{8}$

The possession of the technologies Railroad and Flight each reduces the bonus by $\frac{1}{3}$. Then:

• You have one of this technologies: $\frac{(D + 10) (T_1 + T_2)}{12}$
• You have two of this technologies: $\frac{(D + 10) (T_1 + T_2)}{18}$
• With experimental caravan bonus style (on Warserver): $(2 \log(D + 20 + T_1 + T_2)) ^ 2$

Bonus from a caravan, when it enter in a existent trade place, you get only $\frac{1}{3}$ of the bonus: ($T_1$ - trade production in the city 1, $T_2$ - trade production in the city 2, $D$ - real distance between the cities):

• $\frac{(D + 10) (T_1 + T_2)}{24}$

The possession of the technologies Railroad and Flight each reduces the bonus by $\frac{1}{3}$. Then:

• You have one of this technologies: $\frac{(D + 10) (T_1 + T_2)}{36}$
• You have two of this technologies: $\frac{(D + 10) (T_1 + T_2)}{54}$
• With experimental caravan bonus style (on Warserver): $\frac{(2 \log(D + 20 + T_1 + T_2)) ^ 2}{3}$

These formulas are the base of the calculation. Different governments output different amounts of trade in each city. Republic and Democracy get a +1 trade bonus in tiles that are generating trade, so usually, you will make a lot more profit under these governments. Monarchy and Communism get the +1 trade bonus only if they are celebrating.

Taxation of trade

For each city, multiply the trade output (after subtracting corruption) by the tax rates to yield the gold/luxury/science output taxed from trade. (See add_tax_income() in common/city.c.) The problem is how round the result to exact integers, without increasing or decreasing the trade output (which must exactly equal the sum of the three tax outputs).

From distribute() in utility/distribute.c:

  /* 
   * Distribution of a number of items into a number of groups with a given
   * ratio.  This follows a modified Hare/Niemeyer algorithm (also known
   * as "Hamilton's Method"):
   *
   * 1) distribute the whole-numbered part of the targets
   * 2) sort the remaining fractions (called rest[])
   * 3) divide the remaining source among the targets starting with the
   *    biggest fraction. (If two targets have the same fraction the
   *    target with the smaller whole-numbered value is chosen.  If two
   *    values are still equal it is the _first_ group which will be given
   *    the item.)
   */

The order of the three groups in Freeciv is science, gold, luxury.

In Wikipedia, this method is the Largest remainder method using the Hare quota. (Wikipedia discusses elections, so pretend that the trade points are senate seats, and that the tax rates are 100 votes for the Gold, Luxury, or Science party lists.)

Example 1

• Trade surplus: +2
• Tax rates: 60% science, 40% gold

Initial multiplication:

• $science = +2 \times 60\% = 1 + \frac{20}{100}$
• $gold = +2 \times 40\% = 0 + \frac{80}{100}$

This gives 1 science and 0 gold, but uses only 1 of 2 trade points. Gold has the highest remainder (80 > 20), so Freeciv gives the other point to gold.

Final result:

• 1 science, 50% of +2
• 1 gold, 50% of +2

Example 2

• Trade surplus: +17
• Tax rates: 30% science, 30% gold, 40% luxury

Initial multiplication:

• $science = +17 \times 30\% = 5 + \frac{10}{100}$
• $gold = +17 \times 30\% = 5 + \frac{10}{100}$
• $luxury = +17 \times 40\% = 6 + \frac{80}{100}$

This uses 5 + 5 + 6 = 16 trade points. Luxury has the highest remainder (80 > 10) and receives the 17th point.

Final result:

• 5 science, ≈29.4% of +17
• 5 gold, ≈29.4% of +17
• 7 luxury, ≈41.2% of +17

Example 3

• Trade surplus: +18
• Tax rates: 30% science, 30% gold, 40% luxury

Initial multiplication:

• $science = +18 \times 30\% = 5 + \frac{40}{100}$
• $gold = +18 \times 30\% = 5 + \frac{40}{100}$
• $luxury = +18 \times 40\% = 7 + \frac{20}{100}$

This uses 5 + 5 + 7 = 17 trade points. Science and gold tie for the highest remainder. To break the tie, Freeciv would give the 18th point to the group with fewer points, but both groups have 5 points. In this case, Freeciv always prefers science to gold, and gives the 18th point to science.

Final result:

• 6 science, ≈33.3% of +18
• 5 gold, ≈27.8% of +18
• 7 luxury, ≈38.9% of +18

Example 4

• Trade surplus: +24
• Tax rates: 60% science, 10% gold, 30% luxury

Initial multiplication:

• $science = +24 \times 60\% = 14 + \frac{40}{100}$
• $gold = +24 \times 10\% = 2 + \frac{40}{100}$
• $luxury = +24 \times 30\% = 7 + \frac{20}{100}$

This uses 14 + 2 + 7 = 23 points. Science and gold again tie for the highest remainder, but now gold has fewer points. To break the tie, Freeciv gives the 24th point to gold.

Final result:

• 14 science, ≈58.3% of +24
• 3 gold, 12.5% of +24
• 7 luxury, ≈29.2% of +24

City health

Settings

The health of a city is defined by the possibility of an illness (plague). The following ruleset options are available in game.ruleset, section [civstyle]:

; Whether plagues (illness) are possible 
illness_on = 0 
; the base factor for illness (of percent) 
illness_base_factor = 25 
; minimum city size for illness 
illness_min_size = 3 
; factor for how much trading with a plagued city increases our city's 
; chance for plague (in percent) 
illness_trade_infection = 50 
; factor for how much pollution within a city increases its chance for 
; plague (in percent) 
illness_pollution_factor = 50

Equations

The possibility of an illness is given by a base value, defined in the game.ruleset file (illness_base_factor: maximum propability for illness in percent). Illness due to trade infection and illness due to pollution are added to this value. The illness values calculated below are given in tenths of percent.

The base illness value depends on the city size reduced by the minimum size for illness (illness_min_size) and is given by

$illness\_size = (1- \exp(- \frac{size - illness\_min\_size}{10})) \times 10 \times illness\_base\_factor$

This function starts at 0 for $size - illness\_min\_size = 0$ and goes asymptotic to its maximum value (illness_base_factor).

If a trade city had an illness within the last 5 turns the illness value is increased by

$illness\_trade = \frac{illness\_trade\_infection}{100} \times \sqrt{size \times trade\_city\_size}$

Pollution also increases the illness value:

$illness\_pollution = \frac{illness\_pollution\_factor}{100} \times pollution$

Remember: The illness value is the illness in tenth of percent, i.e. a value of 56 means 5.6% propability for an illness within the city.

Example

City health example (using this m-file)

Taking into account a city size of 1 to 50, a pollution level which equals six times the city size (pollution = 6 * size) and four trade cities of size 10 the illness possibility of the city is given in the following table (in percent).

#	(illness)			aqueduct			aqueduct + sewer system
	base	+trade	+pollution	base	+trade	+pollution	base	+trade	+pollution
5	9.800	11.200	12.700	4.900	5.600	6.350	1.960	2.240	2.540
10	15.800	17.800	20.800	7.900	8.900	10.400	3.160	3.560	4.160
15	19.400	21.800	26.300	9.700	10.900	13.150	3.880	4.360	5.260
20	21.600	24.400	30.400	10.800	12.200	15.200	4.320	4.880	6.080
25	22.900	26.100	33.600	11.450	13.050	16.800	4.580	5.220	6.720
30	23.800	27.300	36.300	11.900	13.650	18.150	4.760	5.460	7.260
35	24.200	27.900	38.400	12.100	13.950	19.200	4.840	5.580	7.680
40	24.500	28.500	40.500	12.250	14.250	20.250	4.900	5.700	8.100
45	24.700	28.900	42.400	12.350	14.450	21.200	4.940	5.780	8.480
50	24.800	29.300	44.300	12.400	14.650	22.150	4.960	5.860	8.860

Health effect

Furthermore for the aqueduct and the sewer system a health effect is defined.

[effect_aqueduct_health] 
name	= "Health_Pct"
value	= 50 
reqs	= 
    { "type", "name", "range" 
      "Building", "Aqueduct", "City" 
    }

[effect_sewer_system_health] 
name	= "Health_Pct"
value	= 30 
reqs	= 
    { "type", "name", "range" 
      "Building", "Aqueduct", "City" 
      "Building", "Sewer System", "City" 
    }

The possibility of an illness is reduced by <value> percent defined by the effect Health_Pct (i.e aqueduct + sewer system = Health effect of 80; an base illness possibility of 10% is reduced by 80% to 2%).

Migration

Settings

Migration between cities is controlled by the following game settings. They can be changed by using the set server command. help <setting> will give you more information.

• migration - Whether to enable citizen migration (default: 0 = off)
• mgr_turninterval - Number of turns between migrations from a city (default: 5; min:1; max: 100)
• mgr_foodneeded - Whether migration is limited by food (default: 1 = on)
• mgr_distance - Maximum distance citizens may migrate (default: 3; min: 1; max: 7)
• mgr_nationchance - Percent probability for migration within the same nation (default: 50; min: 0; max: 100)
• mgr_worldchance- Percent probability for migration between foreign cities (default: 10; min:0; max: 100)

Equations

Helper function to calculate a "score" of a city. The score is used to get an estimate of the "migration desirability" of the city. The higher the score the more likely citizens will migrate to it.

The score depends on the city size, the feeling of its citizens, the cost of all buildings in the city, and the surplus of trade, luxury and science.

• base formula: $score = (city\_size + feeling) \times \prod factors$
• feeling of the citizens: $feeling = 1.00 \times [happy~citizens] + 0.00 \times [content~citizens] - 0.25 \times [unhappy~citizens] - 0.50 \times [angry~citizens]$
• factors
  • if the city has at least one wonder a factor of 1.25 is added
  • for the capital an additional factor of 1.25 is used
  • an additional factor is given by the effect 'Migration_Pct': $f = 1 + \frac{\sum_{effect} Migration\_Pct}{100}$
  • the build costs of all buildings: $f=(1+{\frac {1-\exp(-{\frac {[build~shield~cost]}{1000}})}{5}})$
  • the trade of the city: $f=(1+{\frac {1-\exp(-{\frac {[city~surplus~trade]}{100}})}{5}})$
  • the luxury within the city: $f=(1+{\frac {1-\exp(-{\frac {[city~surplus~luxury]}{100}})}{5}})$
  • the science within the city: $f=(1+{\frac {1-\exp(-{\frac {[city~surplus~science]}{100}})}{5}})$

The last four factors f have values between 1 and 1.2; the overall factor will be between 1.0 (smaller cities) and 2.0 (bigger cities)

[build shield cost], [city surplus trade], [city surplus luxury] and [city surplus science] must be >= 0!

The score of the source city is multiplied by 3 to take into account the persistence of the citizens.

$score\_source = city\_score * 3$

The score of the target city is weighted by the distance between both cities. The weight factor is calculated as

$weight = \frac{[maximal~distance] + 1 - dist}{[maximal~distance] + 1}$

$score\_target = city\_score * weight$

Example

For the following three city the migration score is calculated:

city characteristics
	city 1	city 2	city 3
city size	17	5	3
citizens (happy/content/unhappy/angry)	7/9/1/0	1/4/0/0	0/3/0/0
city has a wonder	yes	yes	no
city is capital	yes	no	no
effect Migration_Pct	30	0	0
build shild cost	2500	1000	250
city surplus (trade/luxury/science)	62/16/15	10/2/5	6/0/6
distance
city 1	-	3	5
city 2	3	-	3
city 3	5	3	-
calculated values
	city 1	city 2	city 3
city size	17.000	5.000	3.000
feeling	6.750	1.000	0.000
factor: wonder	1.250	1.250	1.000
factor: capital	1.250	1.000	1.000
factor: effect Migration_Pct	1.300	1.000	1.000
factor: build shield cost	1.184	1.126	1.044
factor: surplus trade	1.092	1.019	1.012
factor: surplus luxery	1.030	1.004	1.000
factor: surplus science	1.028	1.010	1.012
basic city score	66.044	8.726	3.208
weight: distance city 1	-	0.625	0.375
weight: distance city 2	0.625	-	0.625
weight: distance city 3	0.375	0.625	-
migration checks
source city / target city	city 1 (source)	city 2 (source)	city 3 (source)
city 1 (target)	-	26.179 / 41.2773	9.623 / 24.7664
city 2 (target)	198.131 / 5.4539	-	9.623 / 5.4539
city 3 (target)	198.131 / 2.0048	26.179 / 3.2076	-

Migration will take place from city 2 to city 1 and from city 3 to city 1 with the possibility given in the corresponding game settings (mgr_nationchance and mgr_worldchance).

Migration Effect

TODO: exact description of the effect; increase / decrease the possibility of migration (i.e for different governments)

Science

Technology cost formula depends on several game settings: techcoststyle, researchcost, techcostdoubleyear, techleackage, techleackagerate. There might be differences between calculated values and real-game ones due to rounding to integers in several places.

• Future techs: calculated like common techs with techcoststyle=0.
• Common techs:
  • If techcoststyle=2, ruleset might set its own values, not calculated by FreeCiv. For the techs not overrided by the ruleset, cost is calculated like if techcoststyle=1.
  • If techcoststyle=1 (the default case), $\textrm{basecost}=\frac12 \times (n+2)^{1.5} \times \textrm{researchcost}$, but not less than researchcost. $n$ is number of all (including already researched) technologies required by given tech.
  • If techcoststyle=0, to discover nth technology you need $\textrm{basecost}=(n-1)\times\textrm{researchcost}$ (like in Civilization I/II games).

Additional rules.

• If techcostdoubleyear parameter is not zero, cost doubles starting from this year.
• If techleakage parameter is not zero, you get bonus when other players already have developed the tech. The formula is:$\textrm{bonus}= \textrm{basecost}\times \left( 1-\frac{\textrm{techleakagerate}}{100}\times\frac{p_2}{p_1} \right)$, where $p_1$ is total number of players and $p_2$ is:
  • number of players that know the technology and you have embassy, when techleakage=1,
  • number of players that know the technology, when techleakage=2,
  • number of players that know the technology, excluding barbarian players, when techleakage=3 (in this case total number of players is calculated without barbarians too).

Note: $\textrm{techleakagerate}$ is a Warserver setting. On standard server, $\textrm{techleakagerate} = 100$.

• For AI players there might be a penalty/bonus depending on AI skill.
• If the calculated cost is zero, the real cost is 1 bulb.

Getting tech without research

You can get techs, when you conquer a city, use a diplomat, or from Great Library. The chances and costs of this is decided by server settings conquercost, diplochance, freecost etc.

Research

You can research either by science out put from your trade income, or by using Specialists when your cities are at size 5 or above.

Specialists	Default output	With library	With university
1 scientists	3 bulbs	6 bulbs	10 bulbs
2 scientists	6 bulbs	12 bulbs	21 bulbs

Probability of winning a fight

The chances of winning a fight are sometimes counter-intuitive. E.g., what would you say are your chances of winning in the following situation (after taking into account the land advantages and other modifiers): you have an attack of 2 against a defense of 3, and the same number of hit points. Considering that the probability of winning each round is 2/5, you might think that this will still give a reasonable chance of victory, something like 40%. Lost. You have only 19% chances of winning. The fact that the battle lasts several rounds tendz to give a larger advantage to the biggest power than one could think beforehand.

Supposing that the rules of the fight are known (for each round, take a random number, decide who wins that round, etc.), let's think about a formula representing the chances of winning a fight.

Notations and basic computations

Consider a unit having a strength (all modifications, field, city, unit experience of combat, etc. being applied) $s$ and an enemy unit having a defense $r$. The number of hit points of the defending unit is written $d_\text{hp}$, the damages per victorious rounds from the attacking unit is $a_\text{fp}$. Let us note $k$ the number of victorious rounds required to win the fight. And $l$ is the number of rounds that, if lost, implies a defeat ($l$ is for "losing allowance").

When the attacking unit cause one damage per victorious rounds ($a_\text{fp}=1$), $k$ is the number of hit points of the defending unit ($d_\text{hp}$). More generally, $k = \lfloor\frac{d_\text{hp} + a_\text{fp} - 1}{a_\text{fp}}\rfloor$.

The chance of winning each round, let us note it $p$, is $p = \frac{s}{s+r}$. (Thus the chance of losing a round is $1-p$.)

Reasoning

Now let us suppose that the fight takes n rounds to achieve (note that we don't know n in advance), and let us compute the probability of victory. For a given n, the probability of victory is the probability that k rounds have been won over these n tries and less than l rounds have been lost. If n < k + l (see below why), then the probability of victory is simply the probability that k rounds have been won over these n tries (as this implies that less than l rounds have been lost). In case of victory, the last round must be the final hit (otherwise the fight would have been stopped earlier). Apart from that, there must be (k - 1) hits among the (n - 1) other rounds. That probability is given by a binomial B(n-1,k-1) and is (n-1) choose (k-1), times p raised to (k-1), times (1-p) raised to (n-1-k+1). As the last round must be won, we multiply by p and get a probability of victory with n rounds of (n-1) choose (k-1) times p raised to k times (1-p) raised to (n-k).

For the attacking unit to win the fight, n must be greater or equal to k and smaller than k + l. Thus we get the final probability of winning equal to:

sum over n from k to k+l-1 of (n-1) choose (k-1) times p raised to k times (1-p) raised to (n-k).

Alternative reasoning

The freeciv source code uses a slightly different reasoning (look at ./common/combat.c:unit_win_chance() and ./common/combat.c:win_chance()). Let us note d the number of rounds lost, which must be smaller than l for a victorious attack. Then we have the following probability of victory.

sum of d from 0 to l - 1 of (k - 1 + d) choose (d), times p raised to (k-1), times (1-p) raised to d, times p.

Using n = k + d, we get the initial formula. (Also have to use the fact that (n-1) choose (n-k) equals (n-1) choose (k-1).) In the code, k is def_N_lose, l is att_N_lose, p is def_P_lose1, (1-p) is att_P_lose1, d is lr.