Dark Orbit - finally a "real world" application for math

In this thread I will cover some of the questions in Dark Orbit that can be answered with math. This first post will serve as a table of contents with links to the post containing the discussion.

New topics will be added on an irregular basis, so if you think there is something missing feel free to bring it up.
Same thing if you spot any errors, I can handle some constructive criticism.
One more thing: English is not my first language, please keep this in mind if some of the expressions I use may sound funny


1. Galaxy Gate Generator

1.1 When to use a multiplier
1.2 How many jumps to complete a gate (and on which level to use a multiplier)
1.3 Average cost for each gate
1.4 Why dont I get 13% gate parts from the GGG
1.5 What is the breakdown of parts for Alpha/Beta/Gamma gates
1.6 Can we do better than that
​

2. What is a good configuration

2.1 Introduction
2.2 Goliath vs. Goliath
​

3. Item Upgrading

3.1 Introduction and Summary (in progress)
3.2 Uridium Items
3.3 Credit Items
3.4 CBS base modules
3.5 CBS auxiliary modules
​

4. Item Chances

4.1 Introduction
4.2 LF4, Havoc, Hercules and Saturn​

 

1. Galaxy Gate Generator
1.1 When to use a multiplier?

Lets start with something simple.
The situation is: while throwing hard-earned ingame currency into the Galaxy Gate Generator (GGG) you got a multiplier. Now when should you use it to maximize your chances for a gate part.
Remember: if the random number generator (RNG) creates a part while one or more multipliers are activated, you get additional parts for this gate no matter if you already have this specific part. If on the other hand the RNG creates a part that you already have without a multiplier activated, you only get another multiplier.

The answer is simple: activate the multiplier immediately

Here is why:
When you got a multiplier, you have to guess how many jumps from now the RNG will create another gate part. Lets see how the chances are for the next, the next but one and so on jump.
First jump:
The chances to produce a gate part are 13%. This is exactly the chance to get a gate part (or more with an activated multiplier) on the next jump: 13%
Second jump:
The chances on this jump are 13% again, but to get to the second jump without producing a gate part or multiplier on the first jump the first jump must have produced something else. The probability for this was 100%-13%=87%. The overall probability for a gate part or multiplier on the second jump is thus 13%*(100%-13%)=11.31%. Significantly lower than for the first jump.
Every other jump:
With the same scheme the probability on each jump is reduced by another factor of 87%, making it less and less probable that a gate part or multiplier is produced after a high number of jumps (seen from the perspective before the first jump)

Mathematically, the probability can be described as a geometric distribution, the discrete counterpart to the exponential distribution.
Let p(N) be the probability that the next gate part or multiplier is produced after N jumps and q the probability for a gate part. Then the formula is



A figure created from this formula looks like this:



A while ago I tested this hypothesis with around 5000 consecutive jumps. Unfortunately, I dont have the data any more so you will have to take my word for it that the GGG in Darkorbit behaves like probability theory suggests it.

Edit: Found the old data again, added some new data I had to sample anyway:



The 99% interval is an estimator based on the sample size (1624 parts)
So although the probability for an interval of 1 is lower than the probability for an interval of 2 in my sample, this discrepancy is well within the expected amount of scatter due to the relatively small sample size.

I know that this does not convince everyone. Still there are some more reasons why using a multiplier on the next turn is the best choice.
But instead of bringing up more thought experiments i refer to some of my next posts where things will be discussed from a more practical point of view.

 

Galaxy Gate Cost Chart

It's already been done, but by all means it would be interesting to compare your results. Also I would have advised making the thread when you actually had something to show us ^^

 

Dude I am writing. I already have the results.
I know similar threads exist in the old forum, but I am not satisfied with some of the assumptions made there. And the sample size in the given thread was not enough to derive a probability density function from the simulation. The max and min values given there are almost entirely random.

 

1. Galaxy Gate Generator
1.2 How many jumps to complete a gate?
- and on which level to use a multiplier -

Summary
Since there was a request for a "condensed" representation of the results here we go:
A table with the percentiles of the distributions. In case you dont know, a percentile of 10 percent means that 10 percent of the results lie below this value. In other words: most of the time you will need some number of turns between the number in the left and right column of the table.



See how the mean value for each gate is lower with the x2 multiplier. This is one of the main results here: Use any multiplier you get instantly.
These results also disqualify the use of more than one jump at a time. Using 100 jumps at a time, you dont have control over the multipliers and they are almost always used on very high levels, making gates unnecessarily expensive as we will see in the next post.


How I got to this conclusion
Since we already know that it is best to use a multiplier immediately, we only have to find out on which Level (x2 to x6) we should use it to build a gate with minimal cost.
To answer this I performed a series of so-called Monte Carlo simulations to determine the probability distribution function for the number of jumps to complete a gate.
I will cover the single gates first because these require no additional assumptions.

Each of the simulations consists of 10^7 gates completed. The number of jumps to complete each gate was recorded and the probability density distribution was derived from these results.
Some additional measures were taken to reduce the statistical scatter of the results mainly for cosmetic reasons. The first moment of the distribution function (the mean value) is not affected by the statistical scatter to the same extent, so the results for this moment are very accurate. The estimated standard deviation for the mean value is way less than 0.5 jumps.

In the following figures N denotes the number of jumps, p(N) the probability density function, m is the level at which the multiplier was used and "mean" denotes the average amount of jumps needed to complete the gate.







It can clearly be seen that the mean values are lowest when the multiplier is used on level 2. In some rare events a gate can be completed with a lesser number of jumps when multipliers are used on higher levels. But in the long run, using it on level 2 is the cheapest way.

What about the "combined" gates Alpha Beta and Gamma?
Before performing the same simulations, we need to find out the breakdown of the parts for these 3 Gates. This was done in 1.5, and apparently it is an equal 33% distribution for all three gates.

Here are the results for the first three gates.
Only results for the cheapest (multiplier level 2) and the fastest but most expensive (multiplier level 6) are shown for clarity.
Completed gates are removed (played through) immediately because using the GGG with a completed gate in it is a total waste of ingame currency.
Each simulation uses a sample size of 10^11 spins instead of some constant number of completed gates.

 

Hi El_Burro, it is good to see a thread that focuses on true data.

With regard to the A,B,Y data, I am guessing that this data is, post the change in the .php that weighted Alpha, Beta ,Gamma gate yield respectively?

There use to be a breakdown in yield of A=45% B=30% Y=25% in the .php now that has been removed. Whether that has been further adjusted on the server side is questionable.

From your data, what is the -/+ 3 sigma or better still 6 sigma values for yields per 100 spins (%)?

As your charts show there is a skewed distribution as there is the potential for a value of 0 to potentially ∞.

Okapi32 how sad that you post "by all means", who are you? just a rose coloured glasses wearer, that posts part code as if it is prof of anything. What you show is, you have limited information and try to post it as fact. I would "advice" you take off your rose coloured glasses and post a level head view of the facts or views you have.

 

Oh stop crying Pacman lol, you always find a way to say something to me. I was just pointing out other results are available and I am interested to see how the two sets compare.
What is wrong with that?

Get over yourself.

 

First and foremost, its a game. All this energy being used on data percents etc. I am sure you have a few graphs on the odds of getting a LF4 vs a LF3 from the booty boxes out in the maps. Whats next, are you going to explain the mystery behind the cosmos?
If people would learn to manage their ammo and their accounts. Such crazy tables and graphs would be need to help explain away your lack of getting gates built or keeping your account up to speed. Ya I am sure you have tons of ammo, blah blah blah.

 

LMAO I'm always smiling

Okapi32 I would love to be able to agree with you, or not need to comment on your posts, but your posts attempt to project some misguided superiority compared to your knowledge;
and invariable imply you have access to ulimate code and have superior knowledge of code, that make your rose coloured glasses all seeing, but your rose colour glasses are distorted.

 

Whats sigma?

 

LMAO, it is a game;

how long have you been playing DO?
How many hours have you spent to obtain an account that can compete on a reasonable level ( no need to say if you bought the account ).

These days gate yields are a major factor in the ability for a player to compete regardless of how much one spends, assuming the yields are not manipulated as some would like to believe.

Any factual data can help all players, from newbies to veterans, decide how to best manage their account.

The thread is good and has already give some useful charts.

 

Sigma is a measure of deviation from the average. It is a calculation for a set of data that returns a value for a percentage of the total data.
Basically +/- 3 sigma is almost the min and max expected from a normal distribution of data.

 

Guys please don't let this thread become another back and forth battle of words like so many others. I for one am grateful to the OP for taking the time to show mathematically what I've been trying to explain to people for a long time. ie. use the multi on x2 as soon as you get it.

The data given here suggests this is the best way and shows the maths behind explaining why. There are actually new players coming into the game and if they have anything about them they are studying as much as they can from the forums to learn about the game. The information in this thread could help them increase their efficiency.

 

how about let him post stuff here and try not to spam this thread also and eventually getting it closed . ( u know who I'm talking about )
on topic. great data and keep it up

 

Last time I checked the breakdown was in fact more than 2 years ago.
I know people have been complaining about Alpha gates becoming more expensive. This could be explained with the change in breakdown from 40%/30%30% (or 45%/30%/25% as you suggest) to an equal 33%/33%/33%.
Did I read you correctly that these values used to be open readable in the source code but were removed a while ago?
My simulation was run with 40%/30%/30%. If there should be any evidence that this is far from reality I can re-run the simulations.
Somebody would have count the parts (and multipliers) created by the generator again.

Not sure if get that. The standard deviation (some call it sigma) for the mean values in my simulation is way lower than 0.5 spins. That is why i am confident enough to post mean values like 677.
But I suspect you are talking about the standard deviation of the probability density function. This value would be kind of misleading here, and the chart of the probability density function itself contains a higher amount of information than the standard deviation alone.
Additionally, because the distributions are not normal distributions, the usual perception of a standard deviation would be misleading. As you already noticed, the distributions are kind of skewed, partly due to the impossibility of very low values.
Subtracting 3 times the standard deviation from the mean value might even lead to negative amounts of spins in some cases.
I could extract a 99% confidence interval from the given distributions though.

We all have lots of ammunition, but still I dont want to pay more than necessary for my gates. So what is your point? If you are not too much into this kind stuff (math) I suggest you just ignore this thread.
To me this kind of analysis is more about solving the problem itself than about the relevance of the problem.

Offtopic: -Pacman2- and Okapi32 I would really appreciate if you could sort your dispute somewhere else, I dont feel like it has anything to do with the topic.

 

Hi El_Burro.
I am impressed at the time and effort this must have taken and I am sure that a lot of players WILL benefit from the data and charts you have posted. Many thanks for your time.

...... Couldn't have put that better myself. If you guys have nothing constructive to add to a post then could you please stop poking at each other. You both made valid comments before you started flinging muck at each other and pulled the thread away from the topic.

Harley

 

As far as the distribution of parts among the Alpha, Beta and Gamma gate is concerned lets have a look at the numbers.
I tried 3 different distributions: (33/33/33), (45/30/25) and (40/30/30).
To compare the results in a convenient way i came up with the term gate unit (GU) a while ago. One GU corresponds to one completed Alpha gate. Since everything (especially the rewards) are doubled/tripled for the Beta and the Gamma gate respectively, a Beta gate counts for two GU and a Gamma gate for 3 GU.
Now we can simply compare the number of turns needed to complete one GU.



First thing to mention is that if the distribution was really changed to (33/33/33), this change was to the players advantage. The number of turns per GU is lowest here.

The difference may seem small but the difference in costs is higher (I subtract the uridium reward from the costs to build a gate). So before we advance, we need some clarity on how the distribution for the first three gates really is.
If anyone wants to shed some light on this feel free to do so. We could also collect some data here if some of you post results like this:

Number of turns performed : xxx
Number of parts for gate Alpha : xxx
Number of parts for gate Beta : xxx
Number of parts for gate Gamma : xxx

The number of parts must also contain multipliers. See the text in the popup window which part would have been created.
If you created several Parts at the same time with an activated multiplier, this only counts as one part.

 

What i would like to know how many people actually play this game when on hope page they say 87m people registered lmao

 

Hi El_Burro, the current .php just give 13% yield for gate parts, so assuming no server side adjustment then the A,B,Y gate part distribution would be A 34/164 (21%) B 48/164 ( 29%) y 82/164 (50%)
However I see closer to A 33% B 33% Y 33% these days. So I would guess there is some server side adjustment.

As there are more gamma parts than beta / alpha and more beta than alpha, this would affect the distribution of completed gates.
The A 45% B, 30% Y, 25% refers to the weighting of part yield, rather than complete gate distribution, so would not be representative of number of completed gate shown in the gate ranks.
The original A,B,Y complete gate distribution was closer to 4 Alpha / 2 Beta / 1 Gamma.

The last reference I have to the .php having a breakdown of A 45% B, 30% Y, 25% was Nov 2011. I do not know when that was changed.
The change if we accept that the overall yield remained at 13% ( which I am not seeing statistically ) would be better for the players, as we would receive more Beta and Gamma gates so get better rewards.
Unless you are a WW chasing Kronos gates.

I have added some info to this post highlighted in lighter text.

 

Hi "ramnik although off topic here is a link to my > server stats thread < which is a bit out of date ( I will get round to posting the latest data eventually.)

I would say the average wk on wk active player numbers ( players adding 1 EP or more ) is closer to 250k ( on all 58 servers excluding the few area restricted servers ) rather than 80 mil.