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entolla.com - Ranking System.


Introduction:

This page describes the working details of the Player rating system,
explaining "points", "levels", and how they are computed.

Contents: Overview   Summary   FAQ   Rough details   Greater details

Overview:

The Pentolla ranking system has these purposes:

Separate ratings are maintained for each type of two-player game:
C2 (4 colors, 2 each) and 2-colors (1 each).

Each player who has played at least 15 rated games has an associated rating (number). These numbers (for players who have played in the last 90 days) can be seen at pentolla.com/viewRanks.

In addition, players are separated into levels based on their rating, and each player's level is displayed in the online game to assist with finding opponents. (Hover your mouse over player names in the lobby.) The levels are assigned after 25 rated games.

The ratings range from 0 to greater than 1000. Higher ratings indicate stronger players, within the limitations of the rating system. Players should not make too much of small differences between players of similar ratings.

The levels are as follows: (see the viewRanks page for the numbers of points required for each level).

Note that because of rounding, a point value displayed as being right on the boundary between two levels might actually be a fraction above or below it, as the case may be.
Note that these point levels may change from time to time in order to maintain roughly the same percentage of players within each level.

As a result of each rated game played, points are awarded or deducted from each player's total. The two main factors determining how many points "change hands", so to speak, are the margin of victory and the ratings of the two players.

Larger margins of victory generally result in larger awards, but the number is reduced if the winning player has a much higher rating than the losing player, making it difficult for players to rise in the rankings simply by beating beginners all the time.

To summarize:

Important note:
In some cases, the player who wins a game loses points, which the other player gains.

That can occur for two reasons: To compensate (a little) for the advantage of playing first, the rating algorithm is slightly biased in favor of the second player. Therefore a first-player who wins a very close game may lose a small number of rating points.

Based on the difference in ratings, there is an "expected margin of victory" in the game, which can be fairly high if there is a big difference in the players' ratings. Winning the game, but by less than that, can result in a small loss of points.

For those who are interested, there is a more detailed explanation of the workings of the system below.

A sampling of how different game results change players' points is available at ranksamples.txt page.

FAQ:

Does one player always win the same number of points that the other player loses?

Usually that is the case, but not always.

These two numbers can be different if one of the two players is still in the "provisional" level - which means they have yet to play a full 25 games on the site. In that case, the provisional player's rank is still somewhat volatile, and that player will typically have a larger change in rating than the other player.

Also, sometimes the point changes will appear to differ by one, but in fact one fraction has been rounded up, and the other down. For example, if one player has gained 2.5 points and the other lost 2.5 points, and of their previous scores had fractions which resulted in different rounding.

Why can you lose points for a win?

The most likely reason is that the rating system assumes that higher rated players will be better than their opponents! This means that a higher rated player has to do more than just win in order to increase his/her rating, he/she has to win by a big enough margin. Usually this margin isn't huge, but if the rating difference between the two players is extreme, the higher ranked player will have a lot of work to do. Among other benefits, this helps make sure that strong players can't simply beat up on beginners all the time to keep increasing their rating.

It is also possible that the player who played first won the game, but still lost points because of the komi. The player who moves first has an advantage, and the rating system attempts to compensate for this, by giving the player who moves second a small bonus, called the komi. The Pentolla rating system gives a komi of 4 points for C2, and 3 points for 2-colour. If the player who moved first won the game by less than the komi, the rating system will still count it as a loss, and that player might lose points.

Why are games weighted more heavily for players with closer ratings?

There are a few reasons for this. For one thing, when two players have vastly different ratings, the outcome of the game is typically more certain. For example, if a very highly rated player plays against a low rated player, it should be quite common for the player with the high rating to win, but that doesn't necessarily mean that his rating should go up. In general, it's (usually) harder for the rating system to decide whether players need to gain or lose points when their ratings start further apart, so in these cases the system compensates by changing both ratings a little less.

This has a number of positive effects:

  1. It encourages players to play games against other players with a similar skill level.
  2. It helps avoid a situation where players are discouraged from playing others much below their rank (which is often a good learning experience for the lower rated player).
  3. It helps the system more quickly differentiate between two players of very similar levels.
  4. It prevents good players from making new (second) accounts and significantly and unfairly affecting the ratings of their opponents.

But if I beat a player with a much higher rating, shouldn't I gain lots of points?

Yes! This does happen, but it isn't caused by the weighting factor, it's taken care of by the other part of the formula. That being said, players with rating differences of more than 400 can't effect each others rank.

Hey, I played a game, but my points didn't change! What happened?

One possibility is that your rating was more than 400 points different from your opponent's. That's expected behaviour.

Another possibility is that your points did change, but changed by less than 1 point, so you didn't see the difference. Our servers keep track of your rating very accurately to a number of decimal places, but we don't display all of them. This is mostly likely the case if the outcome of your game was very close to what would be "expected" based on your rating and your opponent's. For more information about what differences are "expected", have a look at the table at the end of the Ranking System Guide.

One last possibility is that there was a communication error between the front and back ends of our server. This is very rare, but it can happen. Don't worry, your game is not lost in this case, we still have the record of the game, and your rating will be updated a little later on.

Why do players with rating differences of more than 400 not effect each others ratings?

More than anything else, this is to help prevent some extreme cheating. It's also true that the system pentolla.com uses to determine how likely one player is to beat another player based on their rating difference starts to break down at such high differences, and not having this cutoff would have some pretty strange consequences.

Players with rating differences of more than 400 are still encouraged to play against each other - in fact, the 400-point cutoff means that two players with very different ratings can freely play together without the player with the higher rating having to worry about winning by a huge margin in order to not lose points. This is often a good learning experience for the player with the lower rating.

There's this high ranked player, but he's not on the list. What gives?

The list displayed only shows players who played a game (any game of either type, as long as it was recorded) in the last 90 days.

However, that player's data is retained and can still be viewed via the stats pages. If a player returns from an absence and plays more games, he will reappear on the list, with his ranking points intact, but of course his position among the players may have changed, if others around him moved up or down.

Why does Player X have an icon, but does not appear on the list ?

The two game types are tracked separately, so if someone has played many, say, C2 games, but very few 2-colors games, they may only have a C2 icon. In such a case, searching for them on the 2-colors list will not be successful, but they will appear on the C2 list, with all available information shown.

Why is the number of games on the ratings list different from the stats pages?

The ratings list is based only on Rated games. Unrated games are included in the stats, so the number of games shown there might be higher. The stats page offers an option to see only Rated games.

Why do players have ratings after 15 games, but need to play 25 games to get a colour?

This has to do with the "confidence" of the rating system. The system starts keeping track of everything right from a player's first game, but after just a handful of games, the ratings don't have much meaning. At this point, the system has a very low confidence that any number that could be assigned would be correct. After 15 games, the system has a better sense of things, and is confident enough to assign a tentative rating number.

However, ratings will still be somewhat volatile, and mistakes are more common. By not assigning players a level (colour) until after 25 games, the system is simply drawing attention to the fact that although it is able to assign a rating, the confidence in that rating is lower than usual. After 25 games, the system is much more confident that the rating it assigned is accurate (enough), and then players get assigned colour levels.

How well does the system rank players?

Quite well, we believe.

This Forum Post illustrates the general behavior:

  • Players tend to have winning records against others ranked lower, and
  • losing records against others ranked higher.

Which leads us to believe that in most cases players are ranked "about right".

Rough Details:

The rating difference between two players affects the change in rating in two main ways: a weighting factor and an expected winning value.

Weighting factor:

The weighting factor reflects how close the two players' ratings are. The closer they are, the greater the weighting factor, and the greater the resulting changes to their ratings will be. This has a few effects:

  1. It encourages players to play games against other players with a similar skill level.
  2. It helps avoid a situation where players are discouraged from playing others much below their rating (which is often a good learning experience for the lower rated player).
  3. It helps the system more quickly differentiate between two players of very similar levels.
  4. It prevents good players from making new (second) accounts and significantly and unfairly affecting the ratings of their opponents.

Expected winning value:

Based on the rating difference of the two players, and who played first, there is an expected margin of victory for the game. The expected winning value will be higher for the player moving first, and for the player with the higher rating. The system is symmetric, so the expected winning value for one player is the negative of the expected winning value of the other player.

For example, if it is expected that a game between Alice and Bob will have Alice winning by 5 points, her expected winning value is +5 and his is -5. If Alice wins by more than 5, her rating will increase, and in that case Bob will have lost by more than 5, so his rating will decrease. However, if Alice wins by less than 5 (or loses), Bob will have lost by less than 5 (or won), and Alice's rating will decrease, and Bob's will increase.

For examples of these expected winning values, see the end of the next section.

The ratings change by comparing the expected winning value to the actual winning value, and multiplying the difference by the weighting factor.

Nitty Gritty Details (Warning: Mathematics Ahead)

Note: In the Rough Details section, reference was made to an expected winning value, when in actuality the system calculates a winning probability (assuming some distribution of how often players will win against players of different skill levels). The nature of using probabilities means that the calculations will inherently be dealing with numbers between 0 and 1. The result is that the ranks which are most natural to use are typically between -1 and 1 (though not always). We post-process these numbers afterward by shifting them by 2 (so that no one has a negative rank), and by blowing them up (by a factor of 200) to make them easier to understand. The following details will refer to the unprocessed numbers.

The way the system works is that it makes 3 calculations: The weighting factor, the winning probability, and the winning value (the value the win is worth - a number between 0 and 1 so that it can be compared to the winning probability). Then it computes the new rating of a player by taking the difference between the winning value and winning probability, multiplying that by the weighting factor, as well as a damping factor (which is 0.5), and adding the result to the old rating:

newRating =
oldRating + (winningValue - winningProbability) * weightingFactor * dampingFactor

Weighting Factor Calculation:

There is a "maximum" score factor as the base (0.25 for C2, 0.15 for 2C). From this maximum, the system subtracts a constant (0.0375 for C2, 0.025 for 2C) multiplied by the square of the rating difference between the two players.
weightingFactor = maxFactor - speedConstant * (rankDifference)2

This means that players who have the same rank get the maximum weighting factor, and it decreases quadratically (continuously) up to the point where players have a rating difference of 2. Games between players with a rating difference of 2 or more are manually set to have a weighting factor of 0, which means the game has no effect on the rating of either player.

There are also some special cases:

  1. Games AGAINST players with less than 25 games are weighted less - in particular, if your opponent has less than 25 games, the weighting factor calculated above is further multiplied by their number of games divided by 25.

    weightingFactor = (maxFactor - speedConstant * (rankDifference)^2)*(opponentsNumberOfGames/25)

    This means that new players who are much better or worse than a typical beginner to Pentolla won't be able to have a large impact on the ratings of others.

  2. Games BY players with less than 25 games get a bonus (0.1) added to their weighting factor, (unless they are playing someone with a rank 2.0 or more different from theirs, in which case the game still has 0 weight).

Winning Probability Calculation:

The player who has the lower rank has a winning probability of:

p = 1/(1+e(F * RankDifference))
F is a constant which affects the slope of the function. It is currently set to be 1 for C2 and 0.75 for 2C.

The player with the higher rank has a winning probability of (1 - p).

Note that if two players have the same rating, than this number is 0.5 (for each player), and is always greater than 0.5 for the player with the higher rating, and less than 0.5 for the player with the lower rating. This does not take into account the first-player advantage, which is handled separately (below).

Winning Value Calculation:

Before calculating the actual value of the win, the system does a few things: First it gives the second player a komi (some free additional points for going second) - the komi is 4 for C2 and 3 for 2-colour. Second, it calculates a cutoff for the game score - in other words it determines a "maximum winning score", and doesn't pay attention to score differentials greater than this maximum. This maximum is dependent on the rank differential of the two players. It is calculated as follows:

There is a base cutoff (20 for C2, 10 for 2-colour), which is the MINIMUM cutoff value. Then, it adds to this value a scaling constant (12.5 for C2, 5 for 2-colour) multiplied by the rating difference of the two players.

cutoff = cutoffMinimum + rankDifference * constant

Note: The cutoffs are larger between players with larger rank differentials because in those cases, it is expected that the higher ranked player will be very likely to win, and so the value of the win becomes more of the question. Accordingly, the winning probability is also higher in such cases, so the higher ranked player needs to win by more in order to gain points.

After the cutoff is calculated, the score differential is set to be the cutoff if it was above this value (or negative the cutoff if it was below this value), and then the winning value is determined by adding the score differential (which now includes the komi) to the cutoff and then dividing by twice the cutoff.

winningValue = (scoreDifferential + cutoff)/(2*cutoff)

This means that when a game is won by the cutoff (or more) the value of the win is 1 (the maximum), if the game is a tie the value of the win is 0.5, and anything in between is the linear extrapolation.

A simple example: In C2, if Alice plays as Blue (first move) against Bob, and she is rated 120 points higher than he is, then if Alice wins by 13 or more, she will gain points, and Bob will lose points. Conversely, if Bob comes within 12 of Alice (in other words he loses by 12 or less, or ties, or wins), he will gain points, and Alice will lose.

See the table for further examples of the margins of victory needed to gain points.

2-colors Margin of Victory Needed
to Gain Rating Points
Rating Difference with Violet with Orange
399 13 7
360 12 6
300 10 4
240 9 3
180 7 1
120 6 0
60 5 -1
0 3 -3
-60 2 -4
-120 1 -5
-180 0 -6
-240 -2 -8
-300 -3 -9
-360 -5 -11
-399 -6 -12
C2 Margin of Victory Needed
to Gain Rating Points
Rating Difference with Blue with Yellow
399 39 31
360 35 27
300 29 21
240 23 15
180 18 10
120 13 5
60 8 0
0 4 -4
-60 1 -7
-120 -4 -12
-180 -9 -17
-240 -14 -22
-300 -20 -28
-360 -26 -34
-399 -30 -38