Statistical analysis: how many games it takes to reach Rank 1

[CapHastings]

I did some statistical analysis on the 16 season ranked in order to evaluate the number of games needed on average to obtain rank 1. The results below shows that :

 

* In order to obtain rank 1, WR is crucial
* In all the cases you have to play a lot of games.

 

 

Assumptions :

* The analysis is based on evolutions between rank 12 (which is irrevocable) and rank 1. So you had to obtain rank 12 first, and this step is not included in the study.

* There are 43 stars to obtain in order to go from rank 12 to rank 1.

* The model is a Markov chain with transition probabilities :
    * p : probability to gain a star : this is the WR (the ranked one).
    * r : probability to stay in place , losing the game but being the best of his team.
    * q : probability to loose a star : this is 1-WR-r, losing and not be the top-scorer of the team.
    

Results :

    
The analytical solution is the following :

 

 

with
T(a,b) the average numbers of games needed to go from a stars to b stars. r is indirectly

 

The following graph shows that with a WR = 50% and r=1/7 (we are the top score in our team one seventh of the games) , we need 518 games on average to obtain rank 1. In comparison, an unicum player with WR=65% needs only 120 games. Btw an unicum player will probably be more often on the top on the score-sheet so he will grind even faster.

Fig 2

 

And if your WR is smaller than 50%, well, you can gain some ranks but it's not reasonable to aim rank 1 :

 

 

 

That's about the means. Hereafter are the plots of the distributions obtained with a Monte-Carlo simulation :

 

With WR=50% we need, on average 518 games to rank 1 but we see that it is possible to have to play 1000 games in order to obtain that.

 

 

WR=55%

 

WR=60%

Here is some quick and dirty  Python code for those interested .

And sorry for the presentation : I'm not used to this text editor...

 

 

def Gamesneeded(a,b,p,q,r):
    q=(1-r)*q
    qp=q/p
    return (1/(p-q))*(b-a-qp**(a+1)*((1-qp**(b-a))/(1-qp)))

a=0
b=43
p=.50
q=1-p
r=1/7

ps = np.linspace(0.45,.50,200)

games=[]
for p in ps:
    games.append(Gamesneeded(a,b,p,1-p, r))
plt.figure(figsize=(13,8))
plt.xlabel('Winrate')
plt.ylabel('#games to Rank1 ')
plt.plot(ps,games)

from random import random
import numpy as np
import matplotlib.pyplot as plt
a=0
b=42
p=.60

def session_gagnante(a,b,p):
    niveau=a
    nb_parties=0
    p_first=1/7
    q=1-p-p_first
    while niveau < b and nb_parties<10000:
        if random()>p:
            if random()>p_first:
                niveau = niveau-1
        else:
            niveau = niveau +1       
        nb_parties=nb_parties+1
        if niveau < 0:
            niveau = 0
    return nb_parties
            
    
def MonteCarlo():
    results=[]
    for x in range(5000):
        results.append(session_gagnante(a,b,p))
    nb_parties = np.mean(results)
    print("parties",nb_parties ,"heures",nb_parties/3,"jours",nb_parties/20)
    plt.figure(figsize=(8,5))
    plt.xlabel('#games to Rank1 ' + " with WR =" + str(p))
    plt.hist(results, bins='auto')
    
   
    
MonteCarlo()    

 

 

 

 

[Leo_Apollo11]

Hi all,

 

14 minutes ago, CapHastings said:

I did some statistical analysis on the 16 season ranked in order to evaluate the number of games needed on average to obtain rank 1.

 

Nice !

 

BTW, can yo do the similar for the "Ranked Sprint" ?

 

 

Leo "Apollo11"

[GulvkluderGuld]

Great work

 

So basically our 45% wr glue sniffers ranking out with 1k games would have to save a LOT of stars.

[loppantorkel]

I'll repeat my praise: Thanks! Great effort!

 

Just as a tangent. I checked the 23 players who needed the most games to reach Rank 1 and compared it to their Solo WR at the same tier and they are at approximately at 2,7% points higher in Random games compared to this season's Ranked. If people want to predict how many games they need to rank out, they might need to take this into account and not except their WR in Random games to carry over directly.

 

2 minutes ago, GulvkluderGuld said:

So basically our 45% wr glue sniffers ranking out with 1k games would have to save a LOT of stars.

I don't think any 45% did though. The worst Ranked WR I found was 47,5% and even these players are at least average in Random games, many of those needing a lot of battles are above average.

[thisismalacoda]

GJ, thank you.

 

To add just two numbers, with r=0 (i.e. without star saving), it would take 2150 games at 51% WR to gain the 43 stars and 1075 games at 52%.

[thisismalacoda]

Vor 11 Minuten, GulvkluderGuld sagte:

Great work

 

So basically our 45% wr glue sniffers ranking out with 1k games would have to save a LOT of stars.

To advance at all with 45%, you'd have to save your star in at least 20% of your losses. That's realistically not going to happen. 

[N00Boo7]

Is not the win rate, u can have low win rate but actually to save a lot of stars. I  ranked out season 12th in 263 battles with 55,51% win rate. This season i did in 222 battles but 53,15% win rate.

[loppantorkel]

10 minutes ago, N00Boo7 said:

Is not the win rate, u can have low win rate but actually to save a lot of starts. I  ranked out season 12th in 263 battles with 55,51% win rate. This season i did in 222 battles but 53,15% win rate.

Haven't they changed Ranked a bit over the last few seasons though? I'm not sure, but I think due to taking away some irrevocable ranks, they decreased the number of stars needed to rank up, I think. Not sure when these changes were made or if I'm totally accurate about this.

[N00Boo7]

Just now, loppantorkel said:

Haven't they changed Ranked a bit over the last few seasons though? I'm not sure, but I think due to taking away some irrevocable ranks, they decreased the number of stars needed to rank up, I think. Not sure when these changes were made or if I'm totally accurate about this.

Yes, i forgot this part, u are right.

[CapHastings]

Il y a 2 heures, Leo_Apollo11 a dit :

Hi all,

 

 

Nice !

 

BTW, can yo do the similar for the "Ranked Sprint" ?

 

 

Leo "Apollo11"

 

Yes, but we need to know what are the irrevocables levels. I didn't find this information for the incoming Sprint.

 

 

[thisismalacoda]

Gerade eben, CapHastings sagte:

 

Yes, but we need to know what are the irrevocables levels. I didn't find this information for the incoming Sprint.

 

 

'Traditionally' it's 10/9/8/5/3.

[loppantorkel]

18 minutes ago, CapHastings said:

 

Yes, but we need to know what are the irrevocables levels. I didn't find this information for the incoming Sprint.

Possibly a bit more difficult due to divisions being allowed. Might not change things, just make the results a bit less useful.

 

...if cvs are in, they might mess the data up a bit too....

 

edit. doesn't factor in I suppose.

[CapHastings]

Il y a 1 heure, thisismalacoda a dit :

'Traditionally' it's 10/9/8/5/3.

Ok, so there are 6 levels :

1 stars from 10 to 9
2 stars from 9 to 8
6 stars from 8 to 5
6 stars from 5 to 3
6 stars from 3 to 1

With 50% winrate we expect :

2 games to gain one star (reach rank 9)
5.66 games to win two stars (reach rank 8)
32.09 games to win six stars  ( needed 3 times : from 8 to 5,  from 5 to 3 and  from 3 to rank 1 )
So 2+5.66+3*32.1=104 games

and here is the graph in relation with the WR :

 

All those irrevocables ranks make the grind a lot easier. .

 

[CapHastings]

Il y a 1 heure, loppantorkel a dit :

Possibly a bit more difficult due to divisions being allowed. Might not change things, just make the results a bit less useful.

 

...if cvs are in, they might mess the data up a bit too....

Agree with that, IMO the effect of the inclusion of divisions is so huge that all this analysis is secondary. Especially if you play in a division.

[gabberworld]

your statistic is not 100% correct, good players with good team was able go to rank 1 with  less than 70 battles .

 

but even if you good and if mm fill drop  bad teams one by one, it may take forever for ranked  out.

 

also i think you need actually measure this by avg player experience not by win rate, for get more accurate results

[CapHastings]

Il y a 1 heure, gabberworld a dit :

your statistic is not 100% correct, good players with good team was able go to rank 1 with  less than 70 battles .

 

but even if you good and if mm fill drop  bad teams one by one, it may take forever for ranked  out

I guess that a very good player often will be the top scorer of his team, basically almost never loosing stars... I wish we had the "average position on the scoreboard" stats for the players...

[loppantorkel]

33 minutes ago, gabberworld said:

your statistic is not 100% correct, good players with good team was able go to rank 1 with  less than 70 battles .

 

but even if you good and if mm fill drop  bad teams one by one, it may take forever for ranked  out.

 

also i think you need actually measure this by avg player experience not by win rate, for get more accurate results

Avg player experience doesn't translate to ranking up as well as WR though. How would that be more accurate?

[gabberworld]

26 minutes ago, loppantorkel said:

Avg player experience doesn't translate to ranking up as well as WR though. How would that be more accurate?

winrate is the team victory  and loss , its not only yours. experience in other hand shows how good you actually are in game.

 

winrate will be accurate only if this game is 1vs1 player

[Birkebein]

Great work on the statistics. Your statistical analysis toolkit is more than up to analysing the data. 

 

I'm wondering though, does anyone know how the MM works in ranks?  Anecdotally, I seemed to charge ahead and got to rank 7 or 8, then my luck changed, or my sub-par skill level caught up with me, and winning became the exception, is the MM dynamic, does it recognise individual player performance and try to balance performance by creating balanced teams?  

 

My experience was that my win rate and progress tumbled after my team became the team with more no-clan players.  After a little over 60 games my WR was 60% plus, then the MM gave me the teams with team members who didn't belong to clans - I kept grinding for a while but my teams were always "worse".  The basic assumption I'm making is that players who don't belong to clans are worse than players who do.

 

If the MM is dynamic, then being able to save a star becomes the ultimate goal of playing ranks.

 

 

[gabberworld]

2 minutes ago, Birkebein said:

win rate and progress tumbled after my team became the team with more no-clan players.

 

funny that you say this, i notice this as well

[Ze_Reckless]

 

Assuming the worst case scenario that this player had warships premium account active (+65 % XP) he had 1781 average XP. His star retention rate should be 6/7 or 7/7 and with only 50 games luck is an additional (small) factor.

 

And then there are more determined players:

 

[loppantorkel]

18 minutes ago, Birkebein said:

Great work on the statistics. Your statistical analysis toolkit is more than up to analysing the data. 

 

I'm wondering though, does anyone know how the MM works in ranks?  Anecdotally, I seemed to charge ahead and got to rank 7 or 8, then my luck changed, or my sub-par skill level caught up with me, and winning became the exception, is the MM dynamic, does it recognise individual player performance and try to balance performance by creating balanced teams?  

 

My experience was that my win rate and progress tumbled after my team became the team with more no-clan players.  After a little over 60 games my WR was 60% plus, then the MM gave me the teams with team members who didn't belong to clans - I kept grinding for a while but my teams were always "worse".  The basic assumption I'm making is that players who don't belong to clans are worse than players who do.

 

If the MM is dynamic, then being able to save a star becomes the ultimate goal of playing ranks.

Maybe CapHastings will give you a different answer, but looking at your stats, you have a 48,8% WR solo randoms in tier 7. A very rough estimate - minus 2,7% for Ranked - would put you at 45,1% if you were to try rank out, unless you'd be able to negate the tougher opposition by playing an OP ship or so. Ending at Rank 10, down from 8, gave you an overall WR of 49% this season. Ranking out with an overall WR at between 45,1-48,8% would make for a tough and gruelling season and you can't expect to rank up unless you improve these numbers.

[gabberworld]

i'm guessing if user have avg experience near 2000 or over 2000 can ranked out with less than 70 battles

[GulvkluderGuld]

31 minutes ago, Birkebein said:

Great work on the statistics. Your statistical analysis toolkit is more than up to analysing the data. 

 

I'm wondering though, does anyone know how the MM works in ranks?  Anecdotally, I seemed to charge ahead and got to rank 7 or 8, then my luck changed, or my sub-par skill level caught up with me, and winning became the exception, is the MM dynamic, does it recognise individual player performance and try to balance performance by creating balanced teams?  

 

My experience was that my win rate and progress tumbled after my team became the team with more no-clan players.  After a little over 60 games my WR was 60% plus, then the MM gave me the teams with team members who didn't belong to clans - I kept grinding for a while but my teams were always "worse".  The basic assumption I'm making is that players who don't belong to clans are worse than players who do.

 

If the MM is dynamic, then being able to save a star becomes the ultimate goal of playing ranks.

Carrying as Rank 6 or 7 is harder than 9 or 10. Same for Rank 2 or 3.

Ranks actually mean something on average for players skill. Whatever it might seem like, the higher ranked players tend to play better.

As R6, getting matched with 6 rank 10 players usually means having to carry 6 braindead lemmings. As R10, getting matched with 5 R10 and one R6 player means you might have an ally carrying 5 braindead lemmings.

 

TL;DR: Usually the lowest rank players are of lower skill or unaware of the meta in the bracket due to having just moved up. The more of them on your team, the harder it is to carry and the more often you will lose.

 

[thisismalacoda]

Vor 24 Minuten, GulvkluderGuld sagte:

Carrying as Rank 6 or 7 is harder than 9 or 10. Same for Rank 2 or 3.

Ranks actually mean something on average for players skill. Whatever it might seem like, the higher ranked players tend to play better.

As R6, getting matched with 6 rank 10 players usually means having to carry 6 braindead lemmings. As R10, getting matched with 5 R10 and one R6 player means you might have an ally carrying 5 braindead lemmings.

 

Numbers seem to - at least - not confirm this. While Rank 10 - and especially Rank 11 and 12 players, who may or may not have spent time in the 6-10 bracket before dropping out - have lower WRs than Rank 6-9 players, there is no meaningful difference between Ranks 6, 7, 8 and 9. Actually Rank 2, 3, 4 and 5 players have even slightly lower WRs, but they will have played against stronger opposition.

 

 

Ranked Battle Season Stats in season16

rank	players	battles	battleRate	winRate	randWR
1	2546	494216	194.11	57.89	57.25
2	75	16609	221.45	55.51	54.89
3	402	79243	197.12	55.82	54.86
4	917	163531	178.33	55.74	54.22
5	2631	415071	157.76	55.66	53.71
6	2077	279292	134.47	55.94	52.89
7	2163	221305	102.31	56.52	52.48
8	3419	277211	81.08	56.57	52.04
9	7251	453146	62.49	55.95	51.10
10	21225	937538	44.17	54.44	50.02
11	11598	449868	38.79	51.84	49.18
12	8514	304002	35.71	48.81	48.34
13	3521	42478	12.06	57.15	49.31
14	5981	71208	11.91	49.75	48.61
15	9532	112371	11.79	43.88	47.87
16	7782	48154	6.19	46.25	48.01
17	6688	16947	2.53	54.92	48.04
18	5946	9969	1.68	0.41	47.15
total	102268	4392159	42.95	49.17	49.62