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How rating works?

Introduction

In the following section first we roughly describe the basic rating assumptions and its operation and later we present the detailed equations used for the rating calculation.

Assumptions

Our rating system is based on the previous results achieved by players and to some extend can be used to predict their future results. We have used statistical methods to calculate the ratings properly, but we have also tried to design the rating in a way that will meet the demands of the players' community as well as possible.

We present the basic assumptions for the ranking calculation:

the fact that you play with skillful or weak players, either new or regular ones should not have a serious impact on the rating ,
the more often you play the more stable your rating is,
new players get the medium rating as cause we do not know how good their rating should be. So after a few games some players will be rated lower and others will be rated higher than the medium value,
when a player rated higher wins with a player rated lower, he receives fewer rating points than he can lose when he loses the game with a lower rated player.
a higher rating does not involve the player rated higher always achieving a better result than a players rated lower. It involves the fact that the bigger the player's difference is, the more likely to win the player rated higher is,
the player's ratings are changeable. The statistical data analysis proves that the rating fluctuates (in case of our rating system the fluctuation should not exceed +-150 points for regular players). Why do these fluctuations occur? Mostly because people do not always play at the same skill level. Sometimes they are tired, sometimes they are not in mood and sometimes they are in such a good mood that they can win with anyone.
we wish to penalize players who cheat. We need your feedback to protect you better from people that do not play fair.

Mathematical rules

Browsing through available rating systems we've been looking for a proper solution that has strong mathematical foundations and can generate plausible ratings. Probably the most popular rating system (know as ELO system) was designed by Dr. Arpad Elo for chess. That system was adopted by FIDE (Federation Internationale des Echecs) in 1970 and since then its variants have been adopted by most chess associations all over the world. It is also used as a rating system for other games.

ELO system is pretty good, but there are some problems with it. In 2001 Professor Mark E. Glickman designed a new rating system (the so-called Glicko system). Basically, it resembles ELO system - the main difference is that it introduces the so-called rating deviation that describes the uncertainty in a rating. The Glicko system prevents some pathological situations that happen when a player plays with a new player who has an unknown rating. We've decided to use a slightly modified version of Glicko system for our games:

We penalize players that cheat.

The rating in our games is calculated in the following way:

GR: The Glicko rating used in our games

Step a) For each player we convert the Glicko rating (in our games we use two ranges for rating - some games (pool, darts etc.) have rating in range 0-10000, other (chess, checkers etc.) in range 0 -3000 - we present equations for ratings in range 0-10000, equation for rating in range 0-3000 are analogical) to internal Glicko rating (range from 0-3000 with the center at 1500).

IGR: the internal Glicko rating

IGR = 1500 + 1500 * (GR-5000)/10000

Step b) In the Glicko rating we use a rating deviation - we recalculate initial rating deviation:

IGRD: the internal Glicko rating deviation

If no games were played before:

IGRD = 150 (in original Glicko term 350 was used - we find that constant improper for some reasons).

Otherwise:

T: time from last game played by player (in days)

IGRD = min(150,sqrt(IGRD* IGRD+63.2*T)

Step c) We calculate the new internal Glicko rating.

NIGRA: the new internal Glicko rating
NIGRDA: the new internal Glicko rating deviation
NIGRB: the new internal Glicko rating
NIGRDB: the new internal Glicko rating deviation

NIGRA = IGRA + QA/(1/ IGRDA2 + 1/DA2) * QRDB * (SA,B - EA,B)
NIGRDA = sqrt(1/(1/ IGRDA2 + 1/DA2))
NIGRB = IGRB + QB/(1/ IGRDB2 + 1/DB2) * QRDA * (SB,A - EB,A)
NIGRDB = sqrt(1/(1/ IGRDB2 + 1/DB2))

Where:

PI: pi constant (3.14...)

SX,Y: the result of the game: 1 when X wins, 0 if Y wins, 0.5 if a draw

Q = ln(10) / 400

QRD = 1/sqrt(1+3*(Q*IGRD/PI)2)

EX,Y = 1/(1+10-QRDy*(IGRx-IGRy)/400)

DX = 1/sqrt(QX*QX*QRD Y *QRD Y * EX,Y * (1-EX,Y))

Step d) We convert the new internal Glicko rating back to the new Glicko rating:

NGR = 5000 + 5000 * (NIGR-1500)/3000

And that rating is displayed in the game.

Cheating

What do we consider as cheating? When someone does not finish the rated game (intentionally or by network malfunction), he is penalized. Unfortunately in that way people with poor Internet connections are penalized additionally, but we are forced to do it, because some people simulate network malfunction in order to quit games that they prefer to not finish. In order to minimize the impact of network malfunctions we do not consider first disconnections as cheating - if however disconnections happen more often, we start to penalize players and to decrease their ratings. It is achieved by storing the exact time the disconnection occurred in the database and granting the player so called bombs that affect the rating.

Exact rules:

If no games were played before, a player has a bomb value (B) set to 0.
If a player is cheating (or at least we observe that he/she does not finish games) and B value is larger than 20, we modify the player's rating as if he/she lost the game with the player that has the lowest rating possible.
If he the player is cheating, we increase the value of B by 10 points, but B cannot be larger than 50.
If the player finishes one game without cheating, B is decreased by 1. So if someone had problems with the connection (B increased by 10) and then played 10 games without cheating (10 x B decreased by 1), his B will return to its previous value.

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