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Winrate for Mathematicians - a quantitative Analysis

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scrontch #1 Posted 15 February 2014 - 07:51 PM

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This post is about win rate as a reliable and objective measure of a player's skill and is the first post to my knowledge to include a quantitative analysis. (In the hope it will get "stickied")

 

WoT is about winning tank battles and win rate measures exactly that. It does so without any "a priori"-modelling like all other so called performance rating schemes. It is a blind judge that assesses a player's contribution to winning a battle, independent of what the player has actually done. It already includes assessment of any tactics you can think of (and even those you can't think of!) out-of-the-box.

 

So what's the problem people have with winrate? The problem is that winrate is a statistical measure. Roughly speaking, winrate gets significant (read meaningful) only after a lot of battles fought.

 

So how much is "a lot of" actually?

 

We assume that the skill of a player is strongly linked to what we call the "intrinsic winrate" of the player, which is given as the mean probability of winning a random battle when he participates, evened out over all tanks, maps, friendly and enemy team composition and any other random factors that might be involved.

Mathematically speaking, we must distinguish between this "intrinsic winrate" (linked to  skill) - which is unknown - and the player's "actual winrate" that will serve as an estimate for the former.
This estimate will get better and better the more battles fought.

 

We can then ask the following question:
How many battles do we need to fight, so that a given difference in actual winrate between two players cannot be the effect of pure randomness and thus must be linked to a real difference in intrinsic winrate (that is, the skill of winning).

 

If we don't take into account draws (which make up only for about 2% of battles), this is similar to asking the question whether a coin is biased (i.e. intrinsically different from a given "perfect coin") when we only have a given number of outcomes of flipping of the coin.

I.e. we can apply an equivalent reasoning as in  http://en.wikipedia....a_coin_is_fair.

Taking the formula given in http://en.wikipedia....rue_probability,
we obtain for the number n of battles to fight in order to ascertain that a winrate difference is significant

 

n = Z^2 / (4 * E^2) 

where Z is the level of confidence.

And E is the tolerance of error.

 

Taking E=0.01 (in order to "resolve" differences in winrate at the 1%-point level) and Z=2 (a usual scientific grade 95%+ confidence level), we obtain

n = 10000 

 

which in full words reads
Given two players (A and B) with win rates Wa and Wb (in the interval [0..1]), and Wb = Wa+0.01 (1%-point difference), then we need to have around 10k battles to ascertain for almost sure that B is the better player than A, i.e. their difference in winrates cannot be explained by sheer luck/randomness/MM.

 

(Note that if we allow an error of E=0.05, we go down to n=400.
So if B has Wb=55% compared to Wa=50%, we can be pretty sure after only 400 battles that B is doing better than A. But we shouldn't be any affirmative about a player C with Wc=53% compared to A or B.)

 

As a rule of thumb, let's retain

  • Do not even think about comparing win rates at around 100 battles.
  • Winrate starts to get meaningfull at around 1K battles, but you shouldn't be too confident when comparing 1%-point differences.
  • Around the 10k battles order-of-magnitude, winrate can be used to compare winning-skill with a scientific-grade confidence level down to a 1%-point resolution.

(This is true for overall winrate but also when comparing tank-wise, tier-wise etc.)


And as a corollary for the many stats sites out there...

  • Printing decimal fractions on winrates (like in "53.28%") is ridiculous, as it would require an astronomical number of battles for this resolution to become significant.

 

 


Some common misconceptions about winrate...


Is it possible to boost one's winrate?

 

... by driving in platoons or companies?

Short answer: No.

Because for each win of a company team, there is one loss of a company team.
The same is roughly true for platoons. It is irrelevant that platoon members probably cooperate better than random players. Because the competitors for win rate are not only the random players, but also the platoons from the other team! The match maker explicitely tries to distribute platoons evenly among both teams. This is not always perfect, but given a sufficient number of matches, there is a good chance that you end up with the same amount of platoons that have played against you as have played with you (including your own).
So platooning by itself alone cannot increase winrate for the same reason as company can't.

 

However, what is almost sure is that playing in a *good* platoon or *good* company will increase your win rate. (Where *good* means above-average intrinsic winrate of participating players). It is then much likely, that the platoon or company will actually score even better than the average win rate of the constituting players, due to synergy effects that arise amongst good players and the fact that such platoons/companies are more likely to face platoons/companies with lesser winrate than their own.

In other words, imagine that a company is made of 10 players with a 53% intrinsic winrate each. Then it is almost sure that the company will win significantly *more* than 53% of the time, because it will face many companies with less skilled players.
(This claim needs some thorough statistical verification though. It is based on the assumption that players of all winrate segments play companies at an equal frequency, which might not be that obvious)

So except from having some very skilled and altruist friends, the only way to increase your winrate in companies and platoons is to be already a good player to start with.


... by seal-clubbing on low tiers?

Yes, there are numerous examples of players farming winrate with this. However still there is something to be noted here: Seal-clubbing works only as long as the number of seal-clubbers is low compared to the number of seals. (I leave this obvious fact to figure out to the ambitious reader. Hint: What if there's also a seal-clubber on the enemy team? ;))


... by driving OP tanks?

Again yes, it is possible. OPness is actually *defined* by the fact that players score better winrates in these tanks than in the others they drive.
(See http://ftr.wot-news....nk-performance/)
But again, when those tanks are played a lot, people in those tanks will end up facing these tanks also in the enemy team, thus cancelling the effect out.
This btw is the reason that for example the KV-1S (by far most driven T6 heavy) has a relatively poor overall tank-wise winrate http://www.vbaddict....=won_lost_ratio , although that when comparing the tank's winrate player-wise, we can see that it is blatantly OP: http://ftr.wot-news....013/12/kvas.png
So the KV-1S is an example of an OP tank that cannot be used to effectively farm winrate for the simple reason that it is too abundant on the battlefield.


As a corollary, we can assert that,
whatever technique you will find to increase winrate, it's effectiveness will always be limited by the number of players using that technique. In that sense, winrate boosting is self-regulating to some extent.

 

One final word...

"That can't be true! [This and that experience i had that other day] clearly shows that ..."

No. Please stop it. The thoughts presented above are of a statistical kind. Don't even try to argue against a statistical argument with personal experience. You are off-topic in that case.
Not that i am immune to any criticism, but your arguments should be based on math and at least 10k of objective data points. (If you are making a more general point, then 1 million data points is preferable.)
Thank you.

 

 


Updates after discussion

 

2014-02-16
It has been argued that since WoT is way more complex than tossing a coin, it's results cannot be predicted by such a simple approximation and thus the reasoning must be flawed.

 

While the part about the complexity is obvious, the *results* of both games are the same: win (heads) or loose (tail) - neglecting draws. Thus the same statistical treatment as for any binomial distribution can apply.
That is the whole strength of the winrate argument, that it does *not* rely on any modelling of the underlying game. We analyse past yes/no results statistically and find ways to forge our confidence for predicting future yes/no results. We don't need to know the mechanics behind any individual yes/no result at all.

 

 

2014-02-16
It has been noted that winrate also depends on using premium/gold advantages and thus winrate measures "performance" rather than "skill".

 

This is a valid point. It is certainly true that winrate will reflect this in the long run. Feel free to call "performance" what i named "winning-skill".
Note that my intital argument is not "winrate equals skill". But rather: Given a difference in winrate, how many number of games you must have played so that the differences cannot be explained by randomness/MM. Anything other than randomness could be valid reasons then.
Though i still believe that "skill" is the most prominent non-random effect.

 

Update 2014-02-28

I have been given a set of real-life win/loss stats over nearly 2000 battles by SpankyMnky (thanks!). I want to seize the opportunity to get rid of some flawed perceptions people have when considering random effects.
The following graph shows the wins (1) and losses/draws (0) over time (pale blue bars in background).(click to show full size)


The green line is a moving average over 10 battles.
The dark blue one over 100 and the red one over 1K.

The graph nicely shows that although apparent randomness seems to be at full effect at the smaller scales, our player does hold a pretty stable above average winrate between 51 and 53% in the long run.

 

Some things worth pointing out:
The green 10-battles-average line goes down to 0.1 on several occasions and even touches 0 once.
That means 9+ losses-in-a-row streaks for our poor friend Spankymnky!

Still no reason to start a "MM is rigged thread", as that line also touches or passes the 0.9-line at even more occasions (That's 9+ wins in a row. The same people would probably *not* cry "rigged MM" in that case, huh?). It should become apparent to anyone that 10 (or 20 or 50) battle results are of virtually no use whatsoever to make any claim on a systematic bias. 

However Spanky's above average performance was probably at work in nearly all of those battles and this shows as a systematic offset of the 1K-line.

Again in this practical example, 1K battles comes out as a minimum order of magnitude to give reasonable and significant results. Remember this as the primary argument in any loss-streak whine-threads.

Thanks again to Spankymnky for providing the data!

 

Update 2014-03-19

For some more thoughts around winning/loosing streaks with mathematically precise analysis using binomial distribution, see ortega456's comment here .

 


Edited by scrontch, 19 March 2014 - 01:54 PM.


LordRoyh #2 Posted 15 February 2014 - 08:19 PM

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... by driving OP tanks?

Again yes, it is possible. OPness is actually *defined* by the fact that players score better winrates in these tanks than in the others they drive.
(See http://ftr.wot-news....nk-performance/)
But again, when those tanks are played a lot, people in those tanks will end up facing these tanks also in the enemy team, thus cancelling the effect out.

 

This doesn't make sense to me. One can only effect one tank choice in the game, his own. You are facing those good tanks regardless if you play in over performing tank or not.



Norrin_Radd #3 Posted 15 February 2014 - 08:36 PM

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View Postscrontch, on 15 February 2014 - 07:51 PM, said:

 

Some common misconceptions about winrate...


Is it possible to boost one's winrate?

 

... by driving in platoons or companies?

Short answer: No.

Because for each win of a company team, there is one loss of a company team.
The same is roughly true for platoons. It is irrelevant that platoon members probably cooperate better than random players. Because the competitors for win rate are not only the random players, but also the platoons from the other team! The match maker explicitely tries to distribute platoons evenly among both teams. This is not always perfect, but given a sufficient number of matches, there is a good chance that you end up with the same amount of platoons that have played against you as have played with you (including your own).
So platooning by itself alone cannot increase winrate for the same reason as company can't.

 

However, what is almost sure is that playing in a *good* platoon or *good* company will increase your win rate. (Where *good* means above-average intrinsic winrate of participating players). It is then much likely, that the platoon or company will actually score even better than the average win rate of the constituting players, due to synergy effects that arise amongst good players and the fact that such platoons/companies are more likely to face platoons/companies with lesser winrate than their own.

In other words, imagine that a company is made of 10 players with a 53% intrinsic winrate each. Then it is almost sure that the company will win significantly *more* than 53% of the time, because it will face many companies with less skilled players.
(This claim needs some thorough statistical verification though. It is based on the assumption that players of all winrate segments play companies at an equal frequency, which might not be that obvious)

So except from having some very skilled and altruist friends, the only way to increase your winrate in companies and platoons is to be already a good player to start with.


... by seal-clubbing on low tiers?

Yes, there are numerous examples of players farming winrate with this. However still there is something to be noted here: Seal-clubbing works only as long as the number of seal-clubbers is low compared to the number of seals. (I leave this obvious fact to figure out to the ambitious reader. Hint: What if there's also a seal-clubber on the enemy team? ;))


... by driving OP tanks?

Again yes, it is possible. OPness is actually *defined* by the fact that players score better winrates in these tanks than in the others they drive.
(See http://ftr.wot-news....nk-performance/)
But again, when those tanks are played a lot, people in those tanks will end up facing these tanks also in the enemy team, thus cancelling the effect out.
This btw is the reason that for example the KV-1S (by far most driven T6 heavy) has a relatively poor overall tank-wise winrate http://www.vbaddict....=won_lost_ratio , although that when comparing the tank's winrate player-wise, we can see that it is blatantly OP: http://ftr.wot-news....013/12/kvas.png
So the KV-1S is an example of an OP tank that cannot be used to effectively farm winrate for the simple reason that it is too abundant on the battlefield.


As a corollary, we can assert that,
whatever technique you will find to increase winrate, it's effectiveness will always be limited by the number of players using that technique. In that sense, winrate boosting is self-regulating to some extent.

 

One final word...

"That can't be true! [This and that experience i had that other day] clearly shows that ..."

No. Please stop it. The thoughts presented above are of a statistical kind. Don't even try to argue against a statistical argument with personal experience. You are off-topic in that case.
Not that i am immune to any criticism, but your arguments should be based on math and at least 10k of objective data points. (If you are making a more general point, then 1 million data points is preferable.)
Thank you.

 

OK your wrong here, to be honest scanned the top and when I came to this started laughing.

Platoons are 90% friends playing together, not always bad players but often this is the case. So the 10% (I am totally guessing the figures here) of organised platoons will you have to agree give you much better odds of winning. Same applies to companies and team battles. I play these with my clan and we will win about 90% against random teams and say 40-60% against other clans/organised teams.

Seal-Clubbing you are right it only works if there are many more seals than clubbers, which is the case this means that in most battles you will have a larger effect on the game than say playing T10 and when you meet a seal clubber on the other side you are still around 50%

OP Tanks so the KV-1S has a low win rate of 50.98% compared to the sever average of around 48% even saying 49% it is an advantage and also you still see people with a 45% WR playing them and the graph shows statistically that the better you are at playing the better you will perform with them. Also you took a KV-1S why not take a T57 which is considered OP, but is only OP in the hand of players with over a 52% WR

 

Since these points have totally shown that your understanding of statistics is poor at best, we can totally ignore the top anyway.

 

 



PartyTanker #4 Posted 15 February 2014 - 08:53 PM

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View Postscrontch, on 15 February 2014 - 07:51 PM, said:

(...)

n = Z^2 / (4 * E^2) 

where Z is the level of confidence.

And E is the tolerance of error.

 

(...)

 

I like how you dedicated a lot of time and thought on this post.

 

However, the crux of your argument is flawed. You wrongly assume that win/loss follows a binomial distribution. This is an oversimplification and it doesn't approximate the actual distribution. 

 

To be more specific, there are 30 players on each team. This seriously reduces the impact of a single player on the result of a match. (n goes down as a result). Furthermore, you design your random outcome as being 50/50 equal W/L, which is totally wrong. The presence of a player (even if he was the only one influencing the outcome - which he isn't) within a battle do not result in a either an  100% unavoidable victory/ 100% wipe loss (like the flip of a coin). There are many degrees of contributing/detracting to/from a win. More theoretical: the distribution of player effectivity is a lot flatter than the distribution of a coin flip. Thirdly, the relative effectiveness of most players (barring the absolute reds) is somewhat to the middle to the curve. All this severely reduce the variation of the distribution of the outcomes.

 

Basically, you are using entry level statistics to deal with a rather complicated problem. I like the effort, but I would expect a first year engineering student to design a car either. So, don't take it personally.

Now, if you WERE an engineering student, you should. The n=10000 should immediatly give away your logic is flawed, even without theoretical background. Always check if your (theoretical) result fits into the practical context. And given that most WR are more or less stable after 1k games, give or take 1% (again, barring some of the weird "fluctuations" we see in some players WR)., your result of n=10000 should immediatly hint you your formula is wrong.

 

Btw, we had a similar discussion on noobmeter where someone did the exact same reasoning, I'm giving you the brief summary of several pages of statistical arguing. So, don't take it personal, I like it when someone puts effort in their posts.


Edited by PartyTanker, 15 February 2014 - 09:08 PM.


Blue_Badger #5 Posted 15 February 2014 - 09:11 PM

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I do a lot of statistics at uni and essentially what the OP has done is prove that the only variable which influences w/r is you.

 

The other factors balance out or are part of an uncertainty factor. All statistics have an uncertainty, none are exact. A 55% is not necessarily worse than a 56%. +/- 1% is within the uncertainty margin. Not that I know what the margin is, its just a reasonable assumption. I don't know if it is even possible to work it out without access to the server statistics. 

 

Good post OP, +1. 



Woody1999 #6 Posted 15 February 2014 - 09:18 PM

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Very good post OP, brilliant effort. Now it's time to ask my maths teacher what the hell is going on. :-P

-

Can we get this topic pinned? I see lots of players getting use out of this.

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Woody



ClassicFrog #7 Posted 15 February 2014 - 09:38 PM

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Regarding platoons and companies, it depends on the skill of people platooning.

 

Bad players teaming up will destroy their wr even more while good players will boost it greatly.

 

If you are a good player you will win more platooning with other good players.

 

Same with company, good players banding together will roflstomp bad and poor skilled teams. Period.

 

 

So short answer (without disrespect), is yes you can and you are just plain clueless.

 


Edited by ClassicFrog, 15 February 2014 - 09:40 PM.


scrontch #8 Posted 15 February 2014 - 09:47 PM

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View PostClassicFrog, on 15 February 2014 - 09:38 PM, said:

Regarding platoons and companies, it depends on the skill of people platooning.

 

Bad players teaming up will destroy their wr even more while good players will boost it greatly.

 

If you are a good player you will win more platooning with other good players.

 

Same with company, good players banding together will roflstomp bad and poor skilled teams. Period.

 

 

So short answer (without disrespect), is yes you can and you are just plain clueless.

 

 

You didn't read past the "Short answer is No". Did you?

Cause what i'm saying further down is exactly what you are saying.

 



scrontch #9 Posted 15 February 2014 - 09:57 PM

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View PostPartyTanker, on 15 February 2014 - 08:53 PM, said:

The n=10000 should immediatly give away your logic is flawed, even without theoretical background. Always check if your (theoretical) result fits into the practical context. And given that most WR are more or less stable after 1k games, give or take 1% (again, barring some of the weird "fluctuations" we see in some players WR)., your result of n=10000 should immediatly hint you your formula is wrong.

 

 I don't quite get what you mean is so obvious with n=10000 being flawed. You seem to argue with what you call practical (I'd rather call it subjective) impression. Relying on one's feelings is risky in a statistical debate. (I'd rather like you showed me wrong with some more mathematical arguments. I might come back on your first argument later, since it needs a bit of more time to answer.) Note that the n=10000 is needed for having a 1%-point accuracy. I agree that you can get the rough direction at much less battles, as i mentioned in my post.

 



ClassicFrog #10 Posted 15 February 2014 - 09:59 PM

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View Postscrontch, on 15 February 2014 - 08:47 PM, said:

 

You didn't read past the "Short answer is No". Did you?

Cause what i'm saying further down is exactly what you are saying.

 

 

Unless your post was aimed towards utter tomatoes (that re not reading forums to begin with), the short answer is yes (and then you add a wall of text explaining that it is "yes", unless it's a platoon of windowlickers)



RichardNixon #11 Posted 15 February 2014 - 10:02 PM

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View PostPartyTanker, on 15 February 2014 - 08:53 PM, said:

However, the crux of your argument is flawed. You wrongly assume that win/loss follows a binomial distribution. This is an oversimplification and it doesn't approximate the actual distribution.

 

The only real problem with approximating game results to a binomial distribution is that humans don't actually play at a consistent level. Other than that it works fine: The reason that you have a 60% chance of winning each game may be extremely complex (map, teams, dispositions, RNG etc), but it's still a 60% chance, independent of previous games.

 

 



Furlock #12 Posted 15 February 2014 - 10:03 PM

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Nice post.

 

A really good player will show always on the long run.



RichardNixon #13 Posted 15 February 2014 - 10:10 PM

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View PostLordRoyh, on 15 February 2014 - 08:19 PM, said:

This doesn't make sense to me. One can only effect one tank choice in the game, his own. You are facing those good tanks regardless if you play in over performing tank or not.

 

A little-known feature of the matchmaker (obviously absent in Confrontation) is that it tries to match identical tanks across teams. Hence if you play a KV-1S, you're much more likely to be matched against another KV-1S than if you played an M6. This will dampen the influence of popular overpowered tanks to a degree.

 

However, this isn't the main reason why the average win-rate for the KV-1S is low. That's simply because it has a high proportion of new and/or bad players.



rammstein_n #14 Posted 15 February 2014 - 10:12 PM

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View PostFurlock, on 15 February 2014 - 10:03 PM, said:

Nice post.

 

A really good player will show always on the long run.

Not he fights in fight after fight with tomatos or siemka.



Zitzeron #15 Posted 15 February 2014 - 10:29 PM

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very nice but you haven't taken in consideration one but crucial factor, rigged mm that favors long term premium accounts, this makes all your contemplations just pointless since no matter how good you are rigged mm and rng for your shots and armor protection will make wr exactly the way it's meant to be by the algorithm

I've counted that I'm on the bottom 3 games of 4 and mostly with very bad players, so it's obvious that somebody else will be on top every 3 games of 4

I've never bought premium, playing on std account


Edited by Zitzeron, 15 February 2014 - 10:41 PM.


scrontch #16 Posted 15 February 2014 - 10:38 PM

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View PostZitzeron, on 15 February 2014 - 10:29 PM, said:

very nice but you haven't taken in consideration one but crucial factor, rigged mm for long term premium accounts, this makes all your contemplations just pointless since no matter how good you are rigged mm and rng for your shots and armor protection will make wr exactly the way it's meant to be by the algorithm

 

Rigged MM is a conspiracy theory.

http://www.youtube.c...h?v=UwGiwkPqmbw

(That being posted, i'm totally aware that it's futile to discuss with a conspiracy theorist.)

 



arnorwarrior #17 Posted 15 February 2014 - 10:43 PM

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View PostZitzeron, on 15 February 2014 - 10:29 PM, said:

very nice but you haven't taken in consideration one but crucial factor, rigged mm for long term premium accounts, this makes all your contemplations just pointless since no matter how good you are rigged mm and rng for your shots and armor protection will make wr exactly the way it's meant to be by the algorithm


Oh noes, the scarie algoritm is out to get ous!!:ohmy:

 

Do you seriously believe this?

 

 

 



TankMaster_Flash #18 Posted 15 February 2014 - 10:50 PM

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As partytanker pointed out, this is not real maths and the reasoning is flawed in several places.

BattleMetalChris #19 Posted 15 February 2014 - 10:51 PM

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View PostZitzeron, on 15 February 2014 - 09:29 PM, said:

very nice but you haven't taken in consideration one but crucial factor, rigged mm for long term premium accounts, this makes all your contemplations just pointless since no matter how good you are rigged mm and rng for your shots and armor protection will make wr exactly the way it's meant to be by the algorithm


 Hang on, what? So if you pay, you get rewarded with a worse experience? Care to explain the reasoning behind that?



Ulltussa #20 Posted 15 February 2014 - 10:51 PM

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Load of works for something most of us already know.

 

Prove that the mm is rigged instead.

 







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