Maintenance Update - cRPG 0.223

[WaltF4]

I have completed an updated study of the rate of maintenance in cRPG 0.223. This study used the same system of data collection as my previous study on this topic that was preformed in February 2011. This amounts to recording the amount of gold and experience gained over a given set of rounds, the multiplier each of these rounds was player with, and value of the items equipped each round. To simplify the experiment, I play every round with the same set of equipment. All rounds were played on the NA battle servers. From a set of 915 rounds, I have determined the rate at which gold is lost to maintenance.

I have reused the same equations and variables as the last study since they are still correct (with a modification for the newer generational bonus) and are still the most direct method I have found. To save everyone else the trouble of switching back and forth between posts, I will reproduce the calculations here in their entirety.

The following values are need to for the complete derivation and are taken from records explained above:
starting experience: 1041390 exp
ending experience: 9853970 exp
starting gold: 1499164 gold
ending gold: 1771701 gold
generation: 12
average multiplier: 2.23
equipment cost (buying price): 9686 gold

My character gained 8812580 experience and 272537 gold over this period. This amount of gold gain has been reduced due to maintenance. The gold gain without maintenance is proportional to experience gain and character generation such that:
gold gain/exp gain = 50 gold/(1000 exp+30 exp*(generation-1))

With a values of generation greater than 16 counting as 16. For my character:
gold gain/exp gain = 50 gold/1330 exp = 0.0376 gold/exp

So without maintenance I would have gained: 
8812580 exp * 0.0376 gold/exp = 331353 gold

Therefore, over this period I lost 58816 gold to maintenance. As gold gain, experience gain, and equipment breakage are incremented at regular intervals, or clock ticks, it is easier to look at these “tick” time units instead of number of rounds. The number of ticks included in this set of games can be determined as:
Number of ticks = (end exp-start exp) /((1000 exp+30 exp*(generation-1))*average multiplier)

Again, values of generation greater than 16 count as 16. For this set of data:
Number of ticks = 8812580 exp/ (1330 exp * 2.23) = 2971.3 ticks

And the amount of gold lost to maintenance per tick is:
58816 gold / 2971.3 ticks = 19.79 gold/tick

To determine the rate of maintenance, that is the gold lost to maintenance each tick per gold worth of equip items, the following equation is used:
rate of maintenance per tick = (gold lost to maintenance per tick)/(equipment cost)

Using my values:
rate of maintenance per tick = 19.79 gold / 9686 gold = 0.0020

This rate of maintenance per tick is normalized for generation, average multiplier, and equipment cost so it is independent of character and player. We can determine the sustainable monetary value of equipment using:
average equipment cost = 50 gold * (average multiplier)/0.002

From the immediately preceding equation, it should be obvious that the average multiplier of a player is important to what equipment they can afford. Unfortunately, the development team has not included a multiplier tracker or even a win-to-loss tracker. However, an average multiplier can be approximated using the probability tree below:

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This probability tree shows all of the possible permutations of results for the first 5 round of play after joining a server. After the fifth round, there is no change in the tree as the maximum multiplier remains 5x and the tree branches into identical pairs indefinitely after the fifth round. The probability of each result can be determined by multiplying the chance of winning or losing at each branch needed to reach a particular result. For example, to reach the 4x multiplier in the fifth round requires a loss, then a win, a second win, and finally a third win. With a 60% win rate and 40% loss rate the probability the 4x multiplier in the fifth or following rounds  would be:

Probability of a 4x multiplier in the fifth or following rounds  = ( Chance to lose)*(Chance to win)*(Chance to win)*(Chance to win)
Probability of a 4x multiplier in the fifth or following rounds  = 0.4 * 0.6 * 0.6 * 0.6 = 0.0864

This process is repeated for all 16 permutations of the fifth and following rounds. The following figure shows the theoretical average multiplier as a function of win chance along with a 10 data points extracted from 100 round strings of my last 1000 recorded rounds.

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From this plot it can be seen that the average player with a win rate of about 50% have an average multiplier of about 2x and that experimental values for average multiplier for a given win rate are within 0.2x of the theoretically predicted  value.

The most significant result of this new maintenance study is that the rate of maintenance has been reduced as the breakage chance is now actually the 4% claimed by the development team, instead of the ~5.4% it was in February, and the average player can now continuously maintain an average equipment cost of about 50000 gold, instead of the about 38000 gold as was the case in February.

TL;DR version: The average player can now continuously wear an average equipment cost of about 50000 gold.

[Rhaelys]

As always, stellar work.

[Gorath]

Well done as always Walt.  You're still a pimp.

[Tears of Destiny]

Very sexy as always.

[SkyrayFox]

Great work, appreciate it.

[Freland]

Always a pleasure to read your analysis.

[Vibe]

Someone send him some virgins.

[Elerion]

Fantastic as always. Great job.

I am however curious: When you did the test in February that yielded 5.4% average chance of repair per tick, did you use a loadout with larger cost emphasis in certain slots?

The reason for my question is that I feel my cavalry build (cheap in all slots except a very expensive horse) appears to cost more on average than my similarly costed infantry build (much more evenly distributed cost between slots). This may of course be perception and/or bad luck due to the increased volatility, but it could also be due to a quirk in the repair calculation that for some reason hits unevenly distributed loadouts harder.

[Dezilagel]

Nice work, bin waiting for this for quite a while now 

+1

[Penitent]

Ok, now how to I use MY average multiplier (which seems to be lower than the universal average multiplier) and plug this in to see how much equipment I can have and still break even?

My average multiplier is probably like 1.5.  I'm always on the losing team.   No, I don't have the math to back that up.

NVM, I figured it out: 37,500 gold

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average equipment cost = 50 gold * (average multiplier)/0.002

THanks for making it easy for dummies!

[Oggrinsky]

That's awesome, thanks Walt.

[WaltF4]

I am however curious: When you did the test in February that yielded 5.4% average chance of repair per tick, did you use a loadout with larger cost emphasis in certain slots?

Test in February was done wearing:

Straw Hat                                0
Tunic over Mail                         3459
Ankle Boots                              143
Lordly Mail Mittens                    1376
Masterwork Awlpike                  4532
Heavy Board Shield                   4710
Total Cost                                14220


Test in May and June was done wearing:

Straw Hat                                 1
Studded Leather Coat                2679
Ankle Boots                              153
Lordly Mail Mittens                    1376
Masterwork Long Spear             5477
Total Cost                                9686

[Jacko]

Well.. how big of a buffer would need for those 50k worth of gear?

[Strider]

i cant beleive all the time u spent doing this.. GJ anyways. 

[WaltF4]

Well.. how big of a buffer would need for those 50k worth of gear?

You can calculated values for the expected amount of gold gain or gold loss in a given set of games. First, the probability of winning a given number of games in one "sitting" can be determined from the chance to win an individual game. I am going to define a sitting as a set of 30 games. In theory, any number of games could make up a "sitting" so my choice is somewhat arbitrary, but 30 games on the NA battle servers amounts to about 2 hours of game play and about ~100 clock ticks which is long enough for statistics to begin having some statistical significance but remains short enough to act as a practical unit of time. The probability to win a specific number of game in one sitting out of the total 30 games can be calculated using the probability mass function:
probability of winning k games out of n total games = ( n! / ( k! * (n-k)! ) ) * ( p^k * (1-p)^(n-k) )

Where:
k = number of games won in the sitting
n = total number of games in the sitting
p = probability to win individual games

And where the "!" symbol mean the factorial of the immediately preceding value. For this example I will use n=30 games (one sitting) and p=0.5 (average player winning half of their games.) Solving for values of k from 0 to 30 gives the following plot:

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Standard deviation (sigma) for the number of games won can be calculated as the square root of the variance:
sigma = sqrt( np(1 − p) )
sigma = sqrt( 30*0.5*(1-0.5) ) = 2.74 games

Since the distribution of the number of games won is approximately a normal distribution, and interval 4 sigma wide centered about the average number of games won (15 games in this case) contains the results expected to occur in  ~95%  of sittings. I have shaded the ~95% confidence interval in blue and green on the above plot.



Second, the probability of an item breaking a given number of times in a 30 game sitting can also be calculated using the probability mass function. In this case:
probability of the item breaking in k games out of n total games = ( n! / ( k! * (n-k)! ) ) * ( p^k * (1-p)^(n-k) )

Where:
k = number of game in which the item will break in the sitting
n = total number of games in the sitting
p = probability the item will break in an individual game

From my most recent study, the chance of an item breaking per tick is 4% and the average game on the NA battle server lasts 3.25 ticks, so each item has a 13% chance to break each game. Solving for values of k from  0 to 30 gives the following plot:

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As with the previous distribution, standard deviation for the number of games in which the item will break can be calculated as the square root of the variance:
sigma = sqrt( 30*0.13*(1-0.13) ) = 2.74 games

Again, the blue and green shaded region give the ~95% confidence interval on the above plot centered around the average result of 3.9 games with the item breaking.



From these two probability distributions, we can determine a worst case scenario in which many more games are lost than on average and in which items break much more often than on average.  From the distribution of the number of games won, we know that in more than 95% of sitting the average player will win more than 15 - 2 *2.74 = 9.24 game out of 30 total games, which amounts to a win rate on ~30%. Using the method to calculated the average multiplier from a given win rate outline in this thread, a 30% win rate yields an average multiplier of ~1.4x. Likewise, from distribution of the number of times a item breaks, we know that in more than 95% of sittings an item will break less than 3.9 +2*1.84 = 7.58  time in 30 games.

So, the gold gained in this worst case sitting is:
Gold gain per sitting = 50 gold * average multiplier * number of games * number of ticks per game
Gold gain per sitting = 50 gold * 1.4 * 30 games * 3.25 ticks/game = 6825 gold

And the gold loss to maintenance is:
Gold loss per sitting = 0.05 * number of times item breaks * equipment cost

For 50000 gold worth of equipment:
Gold loss per sitting = 0.05 * 7.58 * 50000 = 18950 gold

From this we can see that an average player wearing 50000 gold worth of equipment will lose 12125 gold in a 30 game sitting under what is nearly the worst case scenario possible. It should be noted that such a scenario would be expected to occur in less that 0.25% of sittings. A more common bad, but not quite worst case scenario where the player experiences game losses at a single standard deviation below average (41% win rate and 1.66x average multiplier) and breakage at a single standard deviation above average (breaks 5.7 time in 30 games) our player would lose 6158 gold in a 30 game sitting. This more likely scenario still only occurs in ~2.5% of 30 round sittings.



I think, as a general rule of thumb, having a gold buffer equal to the total cost of your worn equipment would be more than sufficient to cover brief shortfalls and other random fluctuations.