Abstract



My nickname and current clan:

This document provides quantitative insight into the farming of gold resource in “Clash of Clans” multiplayer game. I collected the data while farming in Gold III league with Town Hall level 9. The data on raids were recorded during five weeks (Feb.– Mar. 2014), and include 2236 attacks (about 360 million of farmed gold). In this document I analyze i) distribution of trophies of attacker and defender, ii) distribution of town halls, iii) looted gold, iv) amount of attempts to find opponent, v) time-dependent farming, and provide corresponding recommendations on improving the farming efficiency.

Introduction

Clash of Clans (CoC) is a popular strategy game from the company supercell; over the last months the game stayed the top-grossing one on ios devices. Clash of Clans is a multiplayer strategy game with interaction between opponents separated in time. At the beginning of the game, each CoC player is provided with a village where the player builds resource, defensive, and army-training buildings. The latter allows to train troops for attacking villages of other players. All the players are ranked according to the trophies they have, and one of the ultimate goals in the game is to collect more trophies than opponents. Trophies are distributed among players, and they can be taken by a player from his opponent after wining attack on his village; on the other hand, losing such attack will give some part of the attacker trophies to the defender. This emphasizes the importance of both strong attacking troops as well as good village defense on the way of collecting trophies.

Defence of the village consists of active and passive buildings: various “shooting” structures (like canons, archer towers, mortars etc.) deal damage to the attacking troops while walls are passive obstacles slowing down the enemies. Clash of Clans is balanced such that enhancement of both active and passive parts is necessary to create well-defended village. Creation and improvement of troops and buildings, including defensive ones, requires resources: gold, elixir, and dark elixir. (All these resources can be bought using “gems”, which in turn are mainly available via in-game purchases and provide revenue stream to Supercell company.) Resources can be looted (“farmed”) while attacking villages of other players or obtained from the special buildings (mines/collectors/drills), which constantly produce resources. The improvement of resource buildings increases their production rate, reaching maximum of 21.000/21.000/300 per hour for gold/elixir/dark elixir, respectively. After necessary amount of a given resource is obtained, the construction or improvement of a building can be initiated. The active defensive buildings require gold for their improvement, and their total costthe estimates here were done for the version

of game available in March 2014, i.e. with

maximum levels of 7 and 12 for Hidden Tesla

and Archer Tower, respectively, as well as

no possibility of upgrading walls with elixir is about 550 million; at the same time, the total gold price of walls is significantly higher, and reaches 2.2 billion. The maximally improved gold mines will produce 2.75 billion (550 million + 2.2 billion) of gold in about 15 years which results in almost meaningless reliance solely on the mines on a way to maximize the village defense. As will be mentioned later, a good farming strategy may result in about 10 million of gold per day which is about 20 times faster than the mines production rate, and makes the time scale of collecting 2+ billion of gold more realistic. Improving the active defensive buildings require one additional resource — builder’s time (the total time required to improve all active defence is about 1150 days, or 8 months when using all 5 available workers simultaneously), while the walls can be upgraded instantly and therefore are limited only by the player’s productivity in collecting gold.

On the one hand the process of farming looks pretty simple: the player trains troops, finds opponent, and attacks it. But on the other hand any thoughts on improving particular farming approach raise not-so-easy-to-answer questions including the following: 1) which troops to chose?, 2) in which trophy range to stay?, 3) is there any preferred time when loots are richer?, 4) opponent with what minimal amount of gold to consider as suitable for attack? The last question arises due to the fact that each attempt to find new opponent requires gold, and therefore looking for very rich villages (they appear relatively seldom) becomes prohibitively expensive and results in investing more into the opponent search rather than getting back after successful raid. Currently, I am not aware of any attempts to provide quantitative insight into the questions 2–4, which motivated me to create this document. Here I briefly address question No. 1 (section 1) and provide quantitative analysis based on the questions No. 2, 3, and 4 (rest of the document).

1. Farming approach and collection of data

In CoC the process of resource (gold) farming can be split into two steps: 1) player trains troops, and 2) the player searches for the opponent and attacks it. In principle, the effectively organized farming will rely on maximization of the (g−c)/t ratio, where g is the looted gold, c is the cost of opponent search, and t is the time to train consumed troops. In other words, one can estimate efficiency of each raid using the ratio above and calculating GPS (Gold Per troops Second ratio). Please note that this ratio does not explicitly contain the amount of battles or gold per battle meaning that two “small” raids with (g−c) = 100k gold each are equivalent to the one larger raid with (g−c) = 200k if the larger raid consumed two times more troops training time than each smaller ones. Actually, keeping in mind maximization of GPS the following troops were chosen:

In this troops set the amount of archers and goblins could vary from time to time, but the rest stayed almost always fixed. As you see, this choice does not include long-training-time troops such as wizards, healers, balloons, dragons etc. (Please note that dark troops have twice large effective training time due to only two dark barracks available, and therefore training 4 hog riders will take the same time as training 8 giants) The troops above require 215 army camp capacity, and such a choice can be characterized as “farming-oriented” one: say, replacing goblins and part of archers with wizards will dramatically increase the overall army damage as well as its training time. Actually, I do not assume that this troop set is the most optimal one for farming, and believe that after reaching good GPS values the choice of troops may become subjective, i.e. will depend on the player's mastery with particular troops as well as ability to see efficient ways to loot gold with lower troops consumption. To my knowledge good farmers have the farming efficiency similar to mine using popular BAM troop set.

During farming heroes were used very rarely (due to their “permanent maintenance”:) while spells where used occasionally. From time to time I used gems to boost up to two barracks, and approximate gem usage was about 1200 ("bag of gems") per 1.5 month. All the time estimations below are done for non-boosted barracks. Elixir cost of the indicated troops was so low in comparison with what was farmed that I did not farm the elixir intentionally and still the storages were usually full.

Not all the troops shown above were used in each raid, and typical consumption was about 50 archers and 50 goblins. This motivated to fill two barracks with archers and two with goblins when starting the opponent search. Usually, I was back from the raid before 50 archers and 50 goblins were ready, and if the rest of troops was consumed I started to train them, otherwise just was waiting for goblins and archers and continued to farm. I did not track troops consumption for each raid, but approximate GPS can be estimated using i) average looted gold (164k), ii) average cost of opponent search (25 attempts * 0.75k per attempt with TH 9 = 19k), and iii) consumed troops of 50 archers and 50 goblins: g is 164k, c is 19k, and t = 50*(25 + 30) / 4 (where 50 is the amount of archers/goblins, 25 and 30 are their training times in seconds, and 4 is the amount barracks), resulting in GPS = 211 (0.211k of gold per troops second) or about 760k gold per hour. I did not include training time of dark barrack troops (minions) because their consumption was very low (to save dark elixir), and often was between 0 and 5 minions in the case of presence of Barbarian King or buildings not protected by air-shooting defence. This also means that the indicated troops choice can be improved replacing some troops with dark ones, to equalize the training times between normal and dark barracks (and therefore reduce training time in normal barracks).

The collection of data was done in February and March 2014, covers the time span of about five weeks, and includes statistics for 2236 attacks (360 million of farmed gold). All the farming was done with Town Hall level 9. The trophy range was chosen near 1520 trophies (Gold III league) as the lowest suitable for TH 9 from my previous experience. I tried to keep trophy range approximately fixed to reduce complexity of the analysis and fixing at least one key parameter (attacker trophy count). During farming the following data was collected:

attacker (my) trophy count before starting the opponent search;

amount of attempts to find the opponent "suitable" for attack;

opponent trophies before attack;

town hall level of the opponent;

available gold loot;

looted gold;

gold located only in storages (and therefore, only in mines), which was estimated after replaying the raid and/or checking the opponent village;

time (date, hours, and minutes) when the opponent search was started;

victory or defeat;

levels of the opponent and attacker.

Despite the choice of “suitable” for attack opponent was subjective one, according to my experience with clan mates from different clans we attacked very similar villages with gold easy accessible for the farming troops. The only difference was sometimes that I played more than they and often tried to consume less troops (to improve GPS). The choice of a given village for attack was influenced by several factors including filled bins of gold mines, available gold loot close to or more than its cap value (for the details see here) as well as how well defended in general were storages and mines. The lowest available loot could be as low as 50k, if the gold was lootable with few troops (keep in mind that farming efficiency if determined not by available loot but by GPS).

The fraction of victories was about 72.9%, which is close to 75% reflecting the dominance of one-star victories with 50+% destroyed buildings: in the case of only one-star victories each defeat (25%) must be compensated with three victories (75%) to keep the attacker trophy count fixed. This points out that farming strategy must be designed in a way to win on a regular basis (for example, destroying 50% of buildings), and “hypothetical” strategies with deploying couple of troops, taking resources, and losing the battle will not work on a regular basis.

The chosen trophy league was “Gold III” resulting in the loot bonus of 5.2k, and “effective” bonus (keep in mind not 100% but 72.9% of victories) of about 5.2k * 0.729 = 3.8k gold. This bonus comprises about 2.3% of the average gold loot (164k) and therefore was omitted in the analysis below.

The data were collected after change in the loot system in December 2013 but before clan wars (CW) update (April 2014); the latter added resource storage ability to the clan castle building (CC). However, during two-month farming experience after CW update I find that CC stores very little resources available for looting (well, very often it is just zero) and therefore has almost no impact on farming as well as on the quantitative analysis below. The analysis is based only on mines and storages, with up to 1k gold stored in town hall considered as storage.

2. Trophies and town halls

2.1 Trophies of the attacker and defender

Let’s start analysis from the distribution of attacker (i.e., my) trophies. During collection of data I tried to stay close to 1500–1550 trophy range, which is reflected by Figure 1.

Figure 1. Distribution of the attacker and defender trophies for the whole collected data.

Figure 1 shows histogram where the height of each column (bin) reflects how often I attacked starting search for the opponent from the trophy range covered on the horizontal axis by a given bin. For example, the highest blue bin is centered at 1520 trophies, includes 430 battles when I started the opponent search when my trophy count was from 1511 to 1530. As you see, the distribution of my trophies is Gaussian-like reflecting random nature of winning and losing battles to keep my total trophy count approximately constant. The corresponding defender trophies (i.e., the players I attacked) span much wider range, and demonstrate slight bimodality, appearing like a combination of two distributions. As will be shown later (Figure 2), the difference between attacker and opponents trophies approximately follows the uniform distribution meaning that during opponent search the game will offer to the attacker villages with Y trophies, where Y is taken from some range [Y1, Y2] with equal probability; this process also can be seen as the transformation of the attacker trophy distribution into the defender one using uniform distribution. Such transformation should preserve the shape of the initial attacker distribution (bell-shaped here): it can make the shape wider and/or shift the distribution center, but cannot introduce the local minimum in the middle, as observed for the defender distribution in Figure 1. I assume that the observed minimum is not due to the data scatter, and then it can arise from two reasons: i) during opponent search I have chosen fewer opponents having about 1450 trophies, or ii) there is actually fewer opponents at ~1450 trophy mark. The second reason appears to me more logical: the higher concentration of players below and above ~1450 trophies can be attributed to the players trying to enter the “Gold III” league (peak at ~1400 trophies) as well as players trying to stay in “Gold II” and therefore keeping their trophy count above 1500 (peak at ~1520). Such reasoning could be also applied to ~1320 and ~1600 trophy marks, but there are no peaks visible in Figure 1 because the actual amount of (my) attacks to these opponents was significantly lower. It would be straightforward to exploit this non-uniformity in players concentration and shift maximum of the attacker trophies to match one of the defender maxima, but as will be shown later (and particularly seen in Figure 1), the uniform distribution transforming attacker trophy value to the defender one is too wide (about 280 trophies), and covers several maxima of players concentration.

2.1 Difference between attacker and defender trophies

The next straightforward step is to analyze the difference between attacker and defender trophies, dT. For this difference Figure 2 reveals approximately uniform distribution with sharp left and right edges located at −223 and 58 trophies, respectively, resulting in the trophy difference width of 281. Such a distribution means that on average the game will offer the opponents with 82 trophies below the attacker one. I note that such a difference is valid only for the studies trophy range of 1400–1600, and for higher values of attacker trophies (such as the champion league) the absolute difference may reach 700 trophies or more.

Figure 2. Distribution of the difference between defender (T def ) and attacker (T att ) trophies dT = T def − T att for the whole collected data. Distribution appears to be approximately uniform, with sharp left and right borders located at dT = −223 and dT = 58, respectively. Circles indicate centers of bins used later in Figure 3.

Top of the dT distribution in Figure 2 is approximately horizontal resulting in close-to-uniform selection of the offered opponents over the whole dT range. However, there is slight increase on the right meaning that on average I attacked more “stronger” opponents than “weaker” ones, while the game most likely offered all opponents from the indicated range with equal probability. In other words, Figure 2 suggests that the trophy range where I stayed was not optimal and should be increased. This idea is examined in Figure 3 where I plot the difference for three sets of the attacker trophies, the whole distribution considered before (the same as in Figure 2) as well as 25 lowest and highest attacker trophy values located below 1450 and above 1610 trophies on the blue histogram in Figure 1, respectively. Due to the low amount of data points for the two last trophy sets, the histograms were created for several bin sets and then all the bins were used for the analysis. All histograms are normalized (i.e., value of each bin is divided by some constant to have unity total area under the bins of each histogram) to make meaningful comparison between each other. As Figure 3 shows, the data are quite scattered, and to roughly estimate how close they are to the uniform distribution, one can draw a straight line as close as possible to the top centres of bins of each histogram (or in other words, to perform linear fitting). In the case of “optimal” or uniform-over-the-whole-range sampling the opponents, such a line should be horizontal.

Figure 3. Distribution of the difference between defender (T def ) and attacker (T att ) trophies dT = T def − T att for the whole collected data (blue circles), for T att T att > 1610 (red dots). Due to the low amount of data points for T att T att > 1610 three histograms with different amount of bins were prepared, all three sets of bin centers are shown in the figure. The color lines denote the corresponding linear fits to each set of symbols with the same color.

Figure 3 reveals that at the low trophy values (1450 and less) the slope of the line is much higher than at higher ones, and the slope decreases with the increase of average attacker trophies, becoming approximately horizontal for the highest considered trophy range (T att > 1610). This suggests that my original trophy choice close to 1520 was not optimal in a sense that on average I attacked more stronger opponents than the weaker ones. This non-uniformity may originate in, for example, larger amount of attempts to find a suitable opponent or in the average looted gold. Unfortunately, I did not find confirmation of these assumptions in the collected data, however analysis of the town hall distribution (next section) will support the idea that the 1520 trophy range may be not optimal.

2.2 Distribution of town halls

The Town Hall (TH) is the main building in village, and, among other in-game functions, its level determines the penalty applied to the resources available by default to attacker (see explanation here). In particular, the penalty is i) zero if the difference between defender and attacker THs is zero or positive, ii) 10% if the difference is −1, and iii) 50% if the difference is −2. The data were collected for the TH level 9, and therefore corresponding raids on the villages with TH levels ranging from 8 to 10 can be assumed as (almost) not affected by penalty while the loot available in TH7 villages is actually halved. (However, I still attacked TH7 in the case of easily accessible loot and therefore high GPS.) Figure 4 shows distribution of defender TH levels (shown are only centers of bins of corresponding histograms). Three most left and right intervals bin centers correspond to T att T att > 1610 (i.e., to the green and red symbols in Figure 3). Horizontal dashed lines denote the relative fraction of each TH for the whole considered trophy range (this corresponds to blue symbols in Figure 3). Figure 4a reveals that the relative fraction of TH 7 significantly decreases with the increase of T, from about 40% at T att T att > 1610. This can explain the observed non-uniformity in the distribution of trophy difference shown before in Figure 2 and Figure 3. Figure 4 shows prevalence of TH8 (more that 50%) for the considered trophy range which means that efficiency of farming will decrease significantly after upgrading attacker TH to the level 10 (in this case the loot from TH8 will be penalized by 50%). For the sake of completeness, Figure 4b shows the same distribution of town halls as Figure 4a, but now as a function of the defender trophies T def , reflecting reduction of the TH7 fraction with the increase of T def .

Figure 4. Fraction of defender town halls as the function of attacker trophies (a) and defender trophies (b). To reduce the data scatter, values were obtained for three sets of bins with different width, and all three sets are shown in (a) and (b). Each point/circle denotes the center of a bin; all histograms are normalized.

3. Looted gold

Figure 5 shows the distributions of available G avl and looted gold G ltd gold per attack. Both distributions are relatively close to each other, and the ratio between average looted (164k) and average available gold (192k) is about 85%. Regular raids on TH8 villages with filled gold storages but empty mines result in a small peak on the looted gold distribution located near G avl =270k, which is maximum available gold for looting from storages for TH8 (300k) penalized by 10% due to raids with TH9. Figure 5a reveals that main farming source are the villages with moderate (<200k) amount of taken gold comprising about 70% of all battles, and very rich loots (>300k) appeared quite rarely, in 5% of all the attacks. Separate contributions of villages with fixed TH level demonstrate that loots coming from TH7 (penalized by 50%) give much smaller average value (99k) than from TH8, TH9, and TH10 (170k, 181k, and 206k, respectively). The collected data allow to simulate the situation when the same attacks would have done not with TH9 but with TH8 or TH10, by recalculation of the corresponding loot penalties for each attack (Figure 5b).

Figure 5. a) Distribution of the available (G avl , red dots) and looted (G ltd , blue dots) gold for the whole collected data. Vertical solid line is the average of looted gold, and vertical dashed lines denote the average G ltd for each defender TH level separately. b) Distribution of the looted gold (blue dots) as well as its estimations for the same attacks done with loot penalties of TH8 and TH10 (green and red circles, respectively). Solid vertical lines are the corresponding average values, and dashed lines are individual contributions of each level of defender TH. For estimated loot with TH8 (TH10) penalties individual contributions are 178k, 189k, 180k, and 206k (49k, 95k, 163k, and 206k) for defender TH levels 7, 8, 9, and 10, respectively.

Figure 5b shows that the estimated distributions for TH8 and TH9 are quite similar while the one for TH10 stays apart, shifting towards smaller values. Estimated outcome of TH10 attacks shows strong differentiation in the mean values of each attack set on fixed TH level (red dashed lines in Figure 5b), while gold looted with TH8 oppositely shows very similar values for all considered TH levels (green dashed lines). This points out crucial impact of the penalty on available loot, demonstrating that in the case of zero/low loot penalty even TH7 could provide high average loots (its mean value for estimated TH8 loot is 178k).

According to my current two-month experience with TH10, the average looted gold is higher than the estimated value of 110k. However, troops consumption is much higher too due to attacking mainly TH9 and TH10, which finally reduces GPS. The estimated TH10 distribution in Figure 5b (red circles) can be seen as indication of the overall difficulty in farming with TH10, which is significantly higher compared to TH8 orTH9.

4. Level of defender

In CoC, the level of a player serves as the integral measure (i.e., just a single number) of the overall village/player strength, but in some cases the level can be “artificially” increased with the intensive troops donation to clan mates. Figure 6a shows distribution of the attacker and corresponding defender levels, demonstrating that most of the raids were performed on opponents having mostly levels between 50 and 80, which implies that they typically have good resource buildings but still weaker defence.

Figure 6. a) Distribution of defender (histogram) and attacker (vertical lines) levels. b) Defender level vs. defender trophy count.

For the studied range of the opponent trophies, there is no apparent dependency of the opponent level on its trophy count, as confirmed by Figure 6b. Currently, I don't have any information on the dependency of the defender level on the attacker one, and Figure 6a suggest that there is no strict boundary from below (i.e., very weak/low level opponents can be offered to high level attacker). On the other hand, it is not clear if there is such boundary from above. Figure 6a can be useful for low-level attackers as an orienting point where (in terms of player level) the high-loot opponents can be found.

5. Attempts to find opponent

5.1 Distribution of attempts

After training troops and starting the opponent search, the game offers to attacker villages of opponents which are currently offline. While farming the attacker ends his search when the “suitable”/”passing” opponent is found. The choice is typically based on the available gold loot as well as possible troops consumption, and my approach is briefly described in Section 1. Figure 7a shows that the distribution of the attempts A to find the opponent looks similar to the to the probability density function (PDF) of the exponential distribution. The exponential distribution on itself describes the occurrence of events in a Poisson process, i.e. events occurring independently on each other.

Figure 7. a) Distribution of the attempts A to find “passing” opponent. b) Distribution from panel a) but now normalized (i.e., value of each bin was divided by some constant) to have unity area under the distribution. Red line denotes the corresponding probability density function of the exponential distribution, with L equal to the reciprocal of the average attempts (which is about 24.75 for my attacks).

Exponential distribution is characterized by a single parameter L, which is the rate of occurrence of a given event. In terms of gold looting in Clash of Clans, the rate L can be understood as the reciprocal of the average amount of attempts the attacker should make to find the opponent with “suitable” loot. (For example, L = 1/20 = 0.05 means that on average “suitable”-loot opponent will be found after 20 attempts.) The value of L is straightforward to estimate from the collected data (by just taking the average and inverting it), and using equation for the PDF of exponential distribution as well as normalized histogram from Figure 7a (i.e., making unity area under the histogram), we find excellent agreement of the collected data with the corresponding mathematical model of the independent events.

Figure 8 shows the dependency of looted gold G ltd on the number of attempts A to find opponent. Horizontal orientation of the “cloud” in Figure 8 means that during the collection of data amount of looted gold was actually independent on A: very rich loots (350k and higher) can be found with few attempts as well as more than hundred attempts may be not enough to find moderate loot with at least 100k gold.

Figure 8. Looted gold G ltd vs. amount of attempts A to find “suitable” opponent. Left top corner corresponds to fewer attempts with finding rich loots while right bottom corner reflects relatively long searches resulted in low looted gold.

5.2 Merge of events

Exponential distribution of the observed events (to find “suitable” loot) followed by assumption of their independence allows to analyze impact of looted (or available) G ltd,min (or G avl,min ) on the overall farming efficiency, introducing explicit boundary for what is considered as “suitable for looting” opponent village. In other words, one can combine events introducing different minimum gold boundary (of taken or available gold) for accepting given village for the raid, without actual recollecting the data. Figure 9 illustrates this approach. In the top row, events No. 1, 2, and 3 occurred after 14, 7, and 21 attempts, and resulted in looting 133k, 80k, and 122k gold, respectively. Now, let us assume that raids were performed only on villages with at least 100k of looted gold (G ltd,min = 100k, middle row in Figure 9). In this case i) event 1 stays unchanged, and ii) attempts for the events 2 and 3 (7 and 21, respectively) are merged while looted gold for the event 2 (80k) is omitted because it is lower than G ltd,min = 100k. Bottom part of Figure 9 illustrates the described approach but with G ltd,min = 130k.

Figure 9. Schematic illustration of the events collected during play, where each event is characterized by i) amount of attempts to find “suitable” opponent, and ii) corresponding gold looted after attacking this opponent. Top row is the collected data, while middle and bottom rows are estimates when particular events N not satisfying given criterion (for example, for the middle row that gold loot must exceed 100k) are merged with the events O obeying this criterion. Events are merged such that attempts of each N are added to the nearest O while looted gold for N is discarded, which reflects skipping village N and just continuing search up to village O.

Using the approach described above, the merged attempts to find “suitable” opponent demonstrate exponential distribution (Figure 10) similar to the one obtained from originally collected attempts (see Figure 7 and the corresponding explanation in text), but due to more rare occurrence of merged events the value of L increases making the distribution more "horizontal".

Figure 10. Distribution (normalized histogram) of the attempts to find “suitable” opponent. Blue: originally collected data (the same as in Figure 7b); green and red: estimation based on merged events (with G ltd,min = 110k and G ltd,min = 150k, respectively) as described in Figure 9. Circles denote centers of the histogram bins while lines are plots of the exponential distributions with L calculated from the corresponding set of attempts.

5.3 Effective loot

The next straightforward step is to analyze impact of the minimal looted gold (which actually determines to consider or not a given village as “suitable” for attack) on the overall farming efficiency. The obvious point here is that the more richer opponent is searched for, the longer attacker performs the search, and therefore the more gold is given for attempts to find such opponent. Figure 11a shows the effective gold loot G ltd,eff , where G ltd,eff = (gold looted after attack) − (gold given for attempts to find the corresponding “suitable” opponent). Dashed horizontal line in Figure 11a denotes G ltd,eff (of about 145k) for the whole collected data, with average G ltd = 164k and average attempts A = 1/L = 25 (with TH9 one attempt costs 0.75k gold). This horizontal line also corresponds to low values of G ltd,min , when almost all recorded attacks are accepted, and therefore no events are merged. The curves shown in Figure 11a are obtained for G ltd,eff with different thresholds for events merge: blue curve is calculated using looted gold (G ltd,min , i.e. the gold taken after attack), while green curve is based on the available gold (G avl,min , reflecting what is shown in CoC as “available loot” before attack). With increase of G ltd,min (G avl,min ) blue (green) curve starts to grow, reach its maximum at G ltd,min = 195k (G avl,min = 225k) and then starts rapid decrease when very rich loots begin to appear to rarely. Two maxima observed in Figure 11a indicate optimal opponent selection, based on the looted and available gold thresholds. The relative difference between two maxima is about 13%, which is close to the relative difference of 15% between the total average looted gold (164k) and the total average available gold (192k) mentioned in the Section 3.

Figure 11. a) Effective gold loot as a function of minimal looted gold (G ltd,min , blue dots), minimal available loot (G avl,min ), and minimal loot taken solely from gold mines (G mns,min ). Optimal values are indicated with corresponding dashed lines. b) Average amount of attempts (1/L) to find opponent with looted, available, or available only in mines gold not smaller than the indicated on horizontal axis value of G ltd,min , G avl,min , or G mns,min , respectively. Vertical dashed lines are the same as in panel a), and horizontal ones indicate average amount of attempts (about 75) for the corresponding optimal values.

The green curve (for which events merge is based on the available gold G avl,min ) in Figure 11a is located below the blue one (based on the looted gold G ltd,min ) for G ltd,min G ltd,eff ; at the same time, looted-gold-based curve lacks this negative bias and therefore demonstrates higher G ltd,eff at moderate G ltd,min . With further increase of G ltd,min , above 220k, looted-gold-based curve (blue) starts rapid decrease because average amount of attempts (to find “suitable” loot) for this curve grows significantly faster than for the available-gold curve (see Figure 11b), resulting in faster drop of G ltd,eff for the loot-based merge of events.

Efficiency of farming is determined not only by the effective loot but also by the time to train troops (see Section 1). Farming based on attacking only gold mines typically requires less troops because usually mines are less protected compared to gold storages, and full gold bins in mines in addition indicate inactive villages with often empty clan castle as well as x-bows and inferno towers. Using collected data and merge of events, it is straightforward to calculate efficient loot G ltd,eff based on the gold available only in mines (G mns,min ) assuming that 100% of this gold was looted (I did not record how much gold was taken solely from mines, but may assume that this value is close to 95–100%). The red dotted curve in Figure 11a shows the corresponding effective loot: at low G mns,min , as in the case of G ltd,min and G avl,min , almost no events are merged, and G ltd,eff is close to its value averaged over the whole collected data. Slight overestimation of the red curve above the dashed line (denoting G ltd,eff based on the whole collected data) is due to the fact that less than 100% of gold was actually taken from mines while calculation of G ltd,eff based on G mns,min assumes 100% exactly. At moderate values of G mns,min mines-based curve reaches its maximum at G mns,min = 150k, and then start to drop down. The estimation based on mines gold demonstrate fastest growth of average attempts to find opponent, as shown in Figure 11b. The maximum of mines-based curve is reached at G ltd,eff = 184k while for the curve estimated from the total loot (from mines and storages) is located at G ltd,eff = 197k meaning that relying solely on gold from mines is less efficient but still may be an efficient farming strategy.

5.3 Effective loot estimation for Town Halls 8 and 10

Combining the collected data and events merge approach, we can recalculate penalties and find dependencies G ltd,eff vs. G ltd,min (effective loot as a function of minimal looted gold to be considered as “suitable” for attacking) for the case when the same attacks would be performed with TH8 or TH10. Similar to the analysis of gold loot distribution (Section 3), recalculated dependencies illustrate crucial impact of loot penalties (due to the difference in TH levels between attacker and opponent) on the effective loot, and corresponding shift of the optimal G ltd,min value (Figure 12).



Figure 12. Effective loot (G ltd,eff ) as a function of minimal looted gold (G ltd,min ): originally collected data (blue dots) and estimates for the same attacks performed with TH8 (green dots) and TH10 (red dots) obtained from recalculation of the looted gold using the corresponding TH-difference penalties.

6. Time-dependent farming

Collecting time stamps for each raid, or, more precisely, the time when each opponent search was started, gives possibility to visualize the time dependency of the collected loot. (Here I address only looted gold, without subtraction of the opponent search cost.) Figure 13 shows looted gold as the function of time with minute accuracy (dots in each panel of Figure 13). The timestamps were recorded using Central European (Summer) Time due to my location in Germany. For the analysis I chosen a) each weekday, and b) loots combined for working days and weekend.



Figure 13. Dependency of the looted gold on the time of starting opponent search. Each dot denotes individual attack while each blue (red) circle is the average for a given hour (two hours). The error bars for each average value denote 95% confidence intervals calculated as 2*1.96*s/sqrt(n) where s is the standard deviation of the loots for a given time interval (one or two hours) and n is the corresponding number of loots for this interval. The data are combined for each week day separately (7 top panels) as well as for all working days and weekend (two bottom panels). Very "rich" loots (400k and higher) are located at >400 mark.

Figure 13 reveals large scatter of data, and to reduce this scatter I performed averaging of gold loots for each hour (0, 1, 2 ...; blue circles) and each two hours (0–1, 2–3, ...; red circles). I also calculated the corresponding error bars which help to visualize uncertainty in the averaged loot values and demonstrate how far the "true" loot for a given time interval might be from its indicated value. During the play my impression was that there must be some temporal patters (say, evenings of working days) when the loot is richer. However, the data in Figure 13 suggest that it is difficult to distinguish any preferred farming time interval: many of error bars do overlap, and some significantly higher values (like Thursday 4:00–5:00 or Friday 2:00–3:00) also have much larger error bars suggesting that these rich loots were accidental and unlikely to be found on a regular basis. This finding is also supported by independence of looting events found in previous section as well as by the fact that people play CoC all over the world and day/night separation for a given time zone is actually only nominal.

Conclusion

This document is based on the data collected with Town Hall 9 while farming in “Gold III” league near 1520 trophy mark using “farming” troop set of 10 giants, 10 wall breakers, 15 minions, 50+ goblins, and 50+ archers. Analysis of the trophy and town hall distributions revealed slightly non-optimal selection of the original trophy mark, and for the lower boundary for the farming with TH9 I suggest at least 1600 trophies (when the fraction of attacked opponents with TH7 in my case decreased to about 5%, see Figure 4). Please note that current analysis did not take into account amount of spent troops, and therefore in some cases TH7 with moderate available loot can provide very high GPS values due to low troops consumption.

Analysis of the looted gold distribution as well as its estimates for the same attacks done with loot penalties of TH8 and TH10 (Figure 5) shows that penalty due to the difference in TH level is the main farming obstacle, and without it attacking even TH7 opponents could result in high gold loots. Combining this conclusion with the previous paragraph it follows that in general for optimal gold farming the choice of trophy range can be based on the rate of occurrence of opponents with target TH level: from below these are THs with at most one level below of the attacker TH (i.e., defender TH8 for attacker TH9). The upper trophy boundary can also be determined based on the occurrence of opponents TH, which should be close to the attacker TH, otherwise while attacking strong THs like 9 and 10 the attacker will be forced to consume a lot of troops resulting in reduced GPS.

Distribution of attempts to find opponent (events) demonstrated exponential behavior (Figure 7) suggesting that these events are independent, and therefore can be combined (merged). Merge of events allowed to simulate situation when the same set of opponent searches would be performed with predefined acceptance condition of minimal looted (G ltd,min ), available (G avl,min ), or available only in mines (G mns,min ) gold, and merge the events not satisfying this condition accordingly (Figure 9). The optimal values maximizing (looted gold) − (gold given for opponent search) are found at G ltd,min = 195k, G avl,min = 225k, and G mns,min = 150k (Figure 11) with the corresponding average amount of attempts to find opponent close to 75 for all three conditions. Such average amount of attempts means that searches with 300+ attempts will occur, and if each attempt requires about 3 seconds, resulting in about 900+ seconds in total, time for opponent search may dramatically reduce GPS (all the consumed troops will be ready in village while the player will still search for opponent). This actually means that the suggested optimal values should be taken with care, and they are valid in the case of fixed troops consumption. On the other hand these values can be used to calculate some reference GPS value (say, for the case of using all troops for a given attack), and then use this value to consider given opponent for attack after estimating possible troops consumption (and therefore GPS) before attack.

Time-dependent analysis with grouping loots on per-day basis and averaging them over one and two hours did not reveal any temporal patterns with preferred farming time (Figure 13).





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