**R = 10^(D/400)**. This means a player with 400 more rating points with his opponent is 10 times as likely to win the game. It does not matter if it is 200 vs. 600 or 500 vs. 900. With ELO, all that matters is the

__difference__of the ratings.

Started by
The Prof
, Jul 09 2013 06:14 AM

25 replies to this topic

Posted 09 July 2013 - 06:14 AM

For those who are interested, here’s an explanation about how the ELO rating system works at this site:

Let D = Your opponent’s rating minus your rating. (If you have a higher rating than your opponent, D will be negative)

Your approximate probability of winning a matchup is given by P = 1 / [ 1 + 10^(D/400)]

If you win the matchup, you will win 25*(1 – P) points. If you lose the matchup, you will lose 25*P points

If it’s a draw you win 25*(0.5 – P) points if you were the underdog, and you lose 25*(P – 0.5) points if you were the favorite.

For example, if you have a 330 rating and your opponent has a 250 rating, you should win P = 1 / [ 1 + 10^(-80/400)] = 61.3 % of the time.

If you beat this opponent, the your rating increases by 25*(1 - .613) = 9.67, or 10 points, rounded to the nearest whole number. If you lose, your rating decreases by 25*(.613) = 15.33, or 15 points. A draw works out to a loss of 2.83, or 3 points for you.

Instead of probabilities, we can also think in terms of odds. Let R = the odds against you winning. For example, if you have 25% chance of beating a particular opponent, then the odds against you are 3 to 1, so R = 3 in the case. R is related to P by the formula R = (1 – P) / P or P = 1 / (R + 1)

Now, there is a very nice formula that relates R and D: **R = 10^(D/400)**. This means a player with 400 more rating points with his opponent is 10 times as likely to win the game. It does not matter if it is 200 vs. 600 or 500 vs. 900. With ELO, all that matters is the __difference__ of the ratings.

One more thing: A precise definition of P is that it's your probability of winning plus one-half the probability of a draw. So in reality your probability of winning a game is slightly less than P.

- Fairway likes this

Posted 09 July 2013 - 12:27 PM

Yo, I bet all the weak lads, they know

that the P stands for pro

so if you think those peeps acting slow,...

then you will be treated with a piece jacking show

and that's why they don't need any tactical lessons from their sweet dad at home!

The D stands for dummy

and this geek brand is funny...

cuz' nine out of ten this dweep lacks the money

to play like a deepstacking sonny

so you can call him a fish like a sea mansion tunny!

I love the smell of Napalm in the morning

Posted 10 July 2013 - 05:35 AM

This table shows how the change in rating points after the outcome of a game depends on the difference in two player's ratings. The first column is how many points are gained by the winner AND lost by the loser if the lower rated player wins the matchup. The second column shows the same in the case where the higher rated player wins the matchup.

Rating Lower Higher

Difference Wins Wins

1 to 27 ........... 13 12

28 to 56 ......... 14 11

57 to 85 ......... 15 10

86 to 115 ....... 16 9

116 to 147 ..... 17 8

148 to 181 ..... 18 7

182 to 219 ..... 19 6

220 to 263 ..... 20 5

264 to 315 ..... 21 4

316 to 381 ..... 22 3

382 to 477 ..... 23 2

478 to 676 ..... 24 1

677 and over 25 0

The next table shows how many points are gained or lost following a draw, depending on the difference of the two player's ratings.

Rating Points

Difference for Draw

0 to 13 .......... 0

14 to 41 ........ 1

42 to 70 ........ 2

71 to 99 ........ 3

100 to 130 .... 4

131 to 164 .... 5

165 to 200 .... 6

201 to 240 .... 7

241 to 288 .... 8

289 to 346 .... 9

347 to 424 .... 10

425 to 552 .... 11

553 and over 12

EDIT by GLS 3-1-15: In case anyone is curious the term ELO comes from a man named Arpad ELO. Read more about him here: https://en.wikipedia...o_rating_system

Rating Lower Higher

Difference Wins Wins

1 to 27 ........... 13 12

28 to 56 ......... 14 11

57 to 85 ......... 15 10

86 to 115 ....... 16 9

116 to 147 ..... 17 8

148 to 181 ..... 18 7

182 to 219 ..... 19 6

220 to 263 ..... 20 5

264 to 315 ..... 21 4

316 to 381 ..... 22 3

382 to 477 ..... 23 2

478 to 676 ..... 24 1

677 and over 25 0

The next table shows how many points are gained or lost following a draw, depending on the difference of the two player's ratings.

Rating Points

Difference for Draw

0 to 13 .......... 0

14 to 41 ........ 1

42 to 70 ........ 2

71 to 99 ........ 3

100 to 130 .... 4

131 to 164 .... 5

165 to 200 .... 6

201 to 240 .... 7

241 to 288 .... 8

289 to 346 .... 9

347 to 424 .... 10

425 to 552 .... 11

553 and over 12

EDIT by GLS 3-1-15: In case anyone is curious the term ELO comes from a man named Arpad ELO. Read more about him here: https://en.wikipedia...o_rating_system

- leppie likes this

Posted 18 July 2013 - 02:21 PM

I think the formula should be influenced by the winning ratio. And this influence should be higher as the number of games played is higher, because it means way more to have a 100% ratio after 100 games than after 2 games. This way the players matching would be very accurate.

Posted 18 July 2013 - 09:12 PM

Yo, it's not fun to play the same players every day on the site.

It's in fact very lame that it might...

happen but if you climb up the ladder by beating many greats in a fight,

then there isn't any way to describe

how annoying it is to face the same players each day,

it's starting already to be a waste of your time!

But I guess that's the price you have to pay for the grind,

for decapitating his spy, devastating his mind

if you're facin' this guy for the ten thousandth time with the goal of a higher rate on the line!

So I guess the system doesn't need a debate, it is fine...

and I accept the repeating matches even if they're not changing the vibes

of this game and its pride,...

cuz' you won't gain points against a bronze spy who blatantly tries

to say you goodbye

when he's planning to blitz and send all of your troops in the ground in the state of Dubai !

You gotta beat equal players to rise out of the average foks and keep your faith in the stride...

Only then you'll be a die hard who wants to improve his game on this site!

I love the smell of Napalm in the morning

Posted 19 July 2013 - 05:42 PM

A player’s rating is influenced by three factors:

(1) Win ratio*, which determines how often you gain points versus lose points.

(2) The strength of your opponents relative to you, which determines how many points your gain for a win or lose for a loss.

(3) The number of games played, more of which will tend to increase your rating up to the point where it matches your true skill level. After that, your rating will stabilize unless you improve as a player.

*Technically the relevant statistic is win ratio in ranked games since you last had a 100 rating. Since no rating can drop below 100, new players are not penalized for losses. For example, a player who wins only 5 out of his first 20 games and then wins his next 8 out of 10 will have about the same rating as a player who started out winning 8 out of 10.

As an enhancement idea to address trickz’s point, I’d suggest that the search engine that creates matches between players looking for games be designed not to match a player with one of his last 10 previous opponents.

Posted 25 December 2013 - 08:55 PM

What are all of the levels as they are named?

I never pay attention to the names, except that there is a Bronze and a Silver category.

What are the second level of names.

For instance I have seen Silver Spy and Silver Miner.

What are the second words for?

Is Miner a grade higher than Spy?

What are all the grades names within Silver and Bronze is my main question

Posted 25 December 2013 - 09:20 PM

What are all of the levels as they are named?

I never pay attention to the names, except that there is a Bronze and a Silver category.

What are the second level of names.

For instance I have seen Silver Spy and Silver Miner.

What are the second words for?

Is Miner a grade higher than Spy?

What are all the grades names within Silver and Bronze is my main question

It is related to the in-game rankings..

So all "second words" are Spy, Scout, Miner, Sgt., Lt., Captain, Major, Colonel, General, Marshall.

Makes sense no? Didn't ya really noticed this?

What you lookin' at? You all a bunch of *bleeping* A holes. You know why? You don't have the guts to be what you wanna be. You need people like me. You need people like me so you can point your *bleeping* fingers and say, "That's the bad guy." So...what that make you? Good? You're not good. You just know how to hide--how to lie. Me, I don't have that problem. Me, I always tell the truth. Even when I lie. So say goodnight to the bad guy!

Posted 23 January 2014 - 08:44 PM

This post in my series explaining the ELO ranking system examines the degree to which results of past games affect one's current rating. In a recent thread, Nortrom correctly stated that “ELO is too much of a today's ranking, not a long time ranking” So, by how much does ELO favor new games over older ones? After running a number of simulations, I concluded that any given game counts about 3% less than your next game. Over time this accumulates, to the point where a win or loss 23 games ago counts only about half as much as does a game today. The math on this is .97^23=.496

You might be wondering how this happens, since you earn the same number of points for a victory against an opponent with a given rating difference with you whether the game was played today or three months ago. Well, suppose you just lost a couple games that you think you should have won. This means your rating is 50 points lower than what it could have been (Note here that all games, no matter the ratings of the players, result in a 25 point net difference between winning and losing). So then, say 40 games later, is it still true that your rating would be 50 points higher had you won those two games? No, it’s not. This is because during those 40 games you were getting slightly more points for each win and slightly fewer points deducted for each loss than would have otherwise occurred had you started that sequence of 40 games with a rating 50 points higher. In fact, after 40 games, the difference those two games has on your rating is now only about 15 points ( 50*.97^40=14.79).

So take this as a bit of comfort after a tough loss. Its impact gradually gets wiped away as future games are played. After about 100 games, that one game makes only 1 point difference to your rating, and after about 130 games it’s likely the effects will have vanished entirely.

Posted 23 January 2014 - 09:57 PM

As an enhancement idea to address trickz’s point, I’d suggest that the search engine that creates matches between players looking for games be designed not to match a player with one of his last 10 previous opponents.

I know this is an old post, but I would like to echo this notion. Often I'll end a game with someone, push to search for another opponent and I end up getting the exact same person I just finished with. I don't know how common that is, but it happens quite a bit to me.

GLS

The complete GS&F Rules can be found here: http://forum.strateg...rum-rules-2016/

Draw Refusal Rules, specifically, can be read here: http://forum.strateg...931#entry468931

Posted 23 January 2014 - 09:59 PM

This post in my series explaining the ELO ranking system examines the degree to which results of past games affect your current rating.

The Prof, I know this is a pretty basic question. But what does E.L.O. stand for exactly?

Thanks,

GLS

The complete GS&F Rules can be found here: http://forum.strateg...rum-rules-2016/

Draw Refusal Rules, specifically, can be read here: http://forum.strateg...931#entry468931

Posted 23 January 2014 - 10:55 PM

Hi The Prof,

I like very much your explanations and your maths. I have a question and I don't know if you can answer. My observation is that based on the today's ranking there are:

9 accounts above 900

32 accounts above 800

100 accounts above 700

320 accounts above 600

738 accounts above 500

1506 accounts above 400

3104 accounts above 300

6763 accounts above 200

so there is always a constant factor 3 between the number of accounts for each 100 points in ranking in the silver league and then about a factor 2 between 200 and 600 (except between 500 and 600 where the factor is 2.5). 32 =~9*3; 100=~32*3; 320=~100*3; 738= ~320*2.5; 1506=~738*2; 3104=~1506*2 and 6763=~3104*2.

Is there any mathematical explanation for this more or less constant factor?

Question 2: Let's consider that the 6763 accounts ranked above 200 are all different players of equal force (= for each game played there is an equal probability to win or to lose for each players) and experience and that all of them have randomly played 100 games. What should be the mathematical distribution of the ranking? How many players should have a rank above 1000 if any? Same question if each of the player would have played 200 games?

..answer only if you have fun, otherwise I will just continue my observations (3 months ago you could have found the factor 3 for each range of 100 ranking points between 200 and 1000 (there were no switch from 3 to 2)

My question 3 is linked to your post above:If I understand well what you say it means that if you continuously play with the same "level of experience" the more you play the more your ranking will increase independently from the level of experience of the other players. So you can increase your ranking not only if you improve your level of experience but also if you play more and more with the same level of experience, is that right?

Apocalypse

*And I will show wonders in the heavens and in the Earth: blood and fire and pillars of smoke. The Sun shall be turned into darkness and the Moon into blood, before the coming great and awesome Day of the Lord*

Posted 24 January 2014 - 03:29 AM

Apocalypse, those are very interesting questions. I'll try to give you a good answer later. For now, I'll reply to Gary's questions 'cause they're easier

Often I'll end a game with someone, push to search for another opponent and I end up getting the exact same person I just finished with. I don't know how common that is, but it happens quite a bit to me.

This happens frequently if you search for a new opponent immediately following the end of your game, since it is common for your opponent to do the same, and you will thus both be looking for a new game at the same time. Try waiting a few minutes after the conclusion of your game before you start another and you'll rarely get the same opponent.

The Prof, I know this is a pretty basic question. But what does E.L.O. stand for exactly?

ELO is named after it's creator, Arpad Elo. It was first used in 1960 for Chess ratings. The system used in Chess today, and the one used by Stratego.com are modified forms of Elo's original system.

Posted 24 January 2014 - 03:30 AM

A mid to high ranked player loses a lot of points when they lose to a player ranked under 300. All new nicknames start out on the low side but may ultimately become a 700 ranked player. So this loss should be weighed down over time. This is to say the level of competitiveness of your past losses and wins should be a portion of your own ranking. Meaning if you beat a 900 ranked player in one of their first 10 games on the site,when they were 235, your rating should rise even when you are not playing. But with the way things are now, you never recoup those lost points at all over time.

Since you can't pick your opponent, you don't know whether he has reached his plateau yet or not, but you should still gain pebbles of points as he climbs, and you should lose pebbles of points as he falls

Since you can't pick your opponent, you don't know whether he has reached his plateau yet or not, but you should still gain pebbles of points as he climbs, and you should lose pebbles of points as he falls

Posted 24 January 2014 - 04:58 AM

This is why I like the Kleier rankings system better Maribo. It auto-adjusts your rank post-match if you lost (or won) to a new opponent who later turns out to have a high ranking (or low ranking) after they have played more games in contrast to ELO which evidently (based on The Prof's explanation) is an immediate snapshot where the points lost (or gained) are never adjusted post-match as the rank-gaining (or rank-losing) opponent you lost to moves up (or down) the ladder.

In regards to Nortrom's comment, there is a constant called a smoothing constant or factor that can be increased or decreased to give more or less weight to previous matches. So perhaps an adjusted ELO system with a smaller smoothing constant can be implemented at this website.

A mid to high ranked player loses a lot of points when they lose to a player ranked under 300. All new nicknames start out on the low side but may ultimately become a 700 ranked player. So this loss should be weighed down over time. This is to say the level of competitiveness of your past losses and wins should be a portion of your own ranking. Meaning if you beat a 900 ranked player in one of their first 10 games on the site,when they were 235, your rating should rise even when you are not playing. But with the way things are now, you never recoup those lost points at all over time.

Since you can't pick your opponent, you don't know whether he has reached his plateau yet or not, but you should still gain pebbles of points as he climbs, and you should lose pebbles of points as he falls

Posted 24 January 2014 - 08:57 AM

A mid to high ranked player loses a lot of points when they lose to a player ranked under 300. All new nicknames start out on the low side but may ultimately become a 700 ranked player. So this loss should be weighed down over time. This is to say the level of competitiveness of your past losses and wins should be a portion of your own ranking. Meaning if you beat a 900 ranked player in one of their first 10 games on the site,when they were 235, your rating should rise even when you are not playing. But with the way things are now, you never recoup those lost points at all over time.

Since you can't pick your opponent, you don't know whether he has reached his plateau yet or not, but you should still gain pebbles of points as he climbs, and you should lose pebbles of points as he falls

I don't know if I'm in the minority here or not, but I don't think it's a good desire to want to get points off of what your opponent does after you play him. He may go on to bigger and better things, and that is to be lauded, but what rights does that give you to ask for claws to ride his coattails?

Are you slighted because he was a newbie who, after you played, actually got better? Or is it that you played a shark who was in a new account and not yet high again? In the latter case you might feel wronged, but you still have no valid claim on anything that person does. In the former case you can't even claim moral indignation. You played a newbie and got the commensurate reward at the time. Case closed.

Carrying on a bit further. . . in the latter case the website has an answer in the current rules. If we were to ban multiple accounts, that would end the shark/hustler problem quick!

But back to the points issue, I say don't look to get something for nothing.

GLS

The complete GS&F Rules can be found here: http://forum.strateg...rum-rules-2016/

Draw Refusal Rules, specifically, can be read here: http://forum.strateg...931#entry468931

Posted 25 January 2014 - 03:06 AM

Hi The Prof,

I like very much your explanations and your maths. I have a question and I don't know if you can answer. My observation is that based on the today's ranking there are:

9 accounts above 900

32 accounts above 800

100 accounts above 700

320 accounts above 600

738 accounts above 500

1506 accounts above 400

3104 accounts above 300

6763 accounts above 200

so there is always a constant factor 3 between the number of accounts for each 100 points in ranking in the silver league and then about a factor 2 between 200 and 600 (except between 500 and 600 where the factor is 2.5). 32 =~9*3; 100=~32*3; 320=~100*3; 738= ~320*2.5; 1506=~738*2; 3104=~1506*2 and 6763=~3104*2.

Is there any mathematical explanation for this more or less constant factor?

Question 2: Let's consider that the 6763 accounts ranked above 200 are all different players of equal force (= for each game played there is an equal probability to win or to lose for each players) and experience and that all of them have randomly played 100 games. What should be the mathematical distribution of the ranking? How many players should have a rank above 1000 if any? Same question if each of the player would have played 200 games?

..answer only if you have fun, otherwise I will just continue my observations (3 months ago you could have found the factor 3 for each range of 100 ranking points between 200 and 1000 (there were no switch from 3 to 2)

My question 3 is linked to your post above:If I understand well what you say it means that if you continuously play with the same "level of experience" the more you play the more your ranking will increase independently from the level of experience of the other players. So you can increase your ranking not only if you improve your level of experience but also if you play more and more with the same level of experience, is that right?

Apocalypse

First of all, it’s important to realize that we don’t have a pure data set here because of the proliferation of multiple accounts taking up space on the leader board. Also, when a good player starts a new account it takes at least about 75 games to get his/her ranking up to where it was, and in the meantime all his/her opponents are getting too few points for wins and more points deducted for losses than they deserve.

That said, that the ratio of the number of players in groups of 100 points is close to constant is a striking observation. It could be due to the fact that a player who is rated 100 points higher than his opponent has exactly the same probability of winning, no matter whether the match-up is 900 vs. 800, or 300 vs. 200, or 573 vs. 473. This is a fact built into the ELO system. An easy metric to remember is that for every 120 point difference, a player is twice as likely to beat his/her opponent. Thus, if you increase by 240 points, you will 4 times more likely to win against a given opponent than you were, and a 360 point increase means you would beat your former self 8 times for every 1 loss. A linear increase in rating corresponds to an exponential increase in skill. I don't know for sure why the ratio decreases to about 2 for bronze league players. It could be due to the problems I mentioned earlier with the leader board or it could be a natural phenomenon. Has anybody studied this effect in Chess, which also uses ELO and probably has much more data available?

On your question #2, when you say all have equal force, I’m not sure if you mean they should all start with the same ranking. But even if they start with their current rankings, after 100 games whether someone started with a high or low ranking will not make much difference. The distribution of the 6763 players can be approximated by a normal distribution with mean 334 (this is the approximate average rating of the 6763 players) and a standard deviation of 62.5 So we’d expect the 6763 player to be distributed as follows:

__Ratings__

Under 146: 9 players

147 to 209: 145 players

210 to 271: 919 players

272 to 396: 4617 players

397 to 459: 919 players

460 to 521: 145 players

522 to 584: 9 players

Over 585: Not likely to have even 1

The reason it turns out like this with the players being so bunched together around the mean and without any players with very high ratings is because of the assumption that all the players have equal skill levels. Now, this approximation actually has more dispersion than what you would see in reality, because anytime a player’s rating is much above the mean, he/she will be more likely to be matched with opponents with lower rating and thus receive fewer points for a win (even though the players have the same skill) and so there will be an overall regression toward the mean. Even if 200 or more games were played, although the standard deviation would be larger, the ELO effect of pushing high or low ranked players toward the mean would restrain anyone from getting a very high ranking. Because of this, it is quite unlikely you’d ever see a 1000 rating no matter how many games were played.

On question #3, once you have played enough games so that your rating corresponds to your true skill level, you cannot increase your rating (except for normal up and down fluctuations) by simply playing more games. Your rating is just as likely to decrease as it is to increase by playing more if we assume your skill level remains constant.

Posted 25 January 2014 - 09:24 AM

Thank you very much. very interesting. Just one question: How did you get to the 120 points difference to be 2x more likely to win?

As you I think that if the change of factor from 3 to 2 around 600 points is probably linked to the fact that lots of players may decide to open a second account once they are high in the bronze ranking, so there should be a higher concentration of unused accounts in the bronze marshall area.

I'm just thinking to another good question, ... answer only if you have fun brainstorming your math skills: If Satan-NL has 1080 points, is it possible to "calculate" how much time more skilled he is compared to the average of the players ranked above 200 points?

Have a good day

Apocalypse

Posted 25 January 2014 - 06:24 PM

Sure, no problem. In the first post in the thread I gave a player's probability of winning P = 1 / [ 1 + 10^(D/400)]. Here the number 400 is a constant used in nearly all ELO systems. From this, you can calculate that if D, the difference between your opponent and you is 120, then P = .3339, or basically one-third. The player who is 120 above his opponent therefore has a two-thirds chance of winning and so is 2 times more likely to win.

An easy way to find this, and also to answer your second question is the formula I gave at the end of my original post:

**R = 10^(D/400) **Here, R is the number of times more likely you are to win than your opponent if you have D points more than them. Try it with D = 120, and you get R = 2. Now, suppose SATAN-NL plays against a player with 334 points.

D would equal 746 and R = 10^(746/400) = 73.28. Thus SATAN-NL should beat the "average" player 73 out of 74 times!

Posted 25 January 2014 - 08:08 PM

Thank you really very much for taking the time to answer all my phd questions. I will contradict you: in a preceeding post you wrote that the ELO system selects randomly the games between 2 players but makes always sure that the ranking difference is never more than 677 points. So Satan-NL will never play against a player having 334 points,having a rank at 1080 his minimum that would be randomly selected is 403 points, so he cannot win 73 times out of 74 but somewhat less.

You say that the ELO system is using the constant of 400 in its formula. Where is this constant coming from and what is the rationale behind it?

So another question: if I'm now about 740 points in ranking and that is the ceiling corresponding to my skills, and I get a game against Satan-NL, what is my probability to win?

...again if you don't like making these maths don't answer, I will just make my practical experiences.

Another question: If the level corresponding to my skills is 740 and if in a month I play twice as much games as Satan-NL who is ranked 1080, how long do I need to be ranked higher than him? => if I play twice as much as him it means I play against randomly selected players not necessarily against him, so my ranking should increase correct?

Apocalypse

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