online games played

# The Ranking System of the Online Game

As many of you often ask about the ranking system used in our online games, here is a detailed description, including the formulae for the mathematically inclined reader. Read on, and you will know it all!

## Foreword

There are many ranking systems in sports (tennis, football,...) and in games (chess, bridge, backgammon, etc.). All these systems have their limits, and are often the subject of heated debate inside the federations. So you need to be aware that there is no perfect system.

The reason is that the goal of a ranking system is to sort the players by their "proficiency". This is a very subjective notion, that varies in time depending on the player progress or fatigue, and on the amount of chance of the game. Even with chess, the best ranked players do not systematically beat the lesser ranked players.

The Gauss Curve

## Principles

Let's consider two player games (duels). There can be three possible outcomes of a game: player A wins, player B wins, or stalemate. Any system must cope only with these three possibilities (even though some try to include the fact that the victory was more or less easy, but this is not our case here).

Let's suppose that each player plays at a constant level of proficiency (this is wrong of course, but it can be considered valid during a limited time). The trick is to "converge" the player toward the place he belongs to by having him play several games.

Everybody agrees that there are few good players, few bad ones, and many average players. This is a classic "bell curve" distribution (called a Gauss curve). **A good ranking system will have to show such a distribution.**

## The ELO system

The ELO system was developed for the International Chess Federation by mathematician Arpad Elo in 1978. It was based on the following ideas:

- The scores of the two players are compared before the duel. Let's call A the one with the best score, and B the other.
- If A wins, it is considered normal: he was better ranked. A will win only a few points (very few if B was far from him, more points if B was closer). Likewise, B will lose very few points.
- If B wins, then it means that the original ranks were incorrect. B will win many points, and A will lose many. The number of points won/lost will increase as A and B were far apart.

This way, after many games with various players, A and B will end up "at their place". After that, if they always beat players who are below them, and are always beaten by better ranked players, then nothing happens, their ranks are stabilized.

Comparison with the theorical curve on 10/2/2003

## Details

- Players start at the middle of the "bell curve" (for example, 1,500 points). They are considered as average players at the beginning.
- A player cannot win or lose more than 32 points in a game. It is slightly different in Ticket to Ride online: the amount of points a player can win or lose depends on the number of players in the game (8 for a 2-players game, 16 for a 3-players game, 24 for a 4-players game and 32 for a 5-players game).
- The number of won and lost points is the same. This is a very important notion. This way, the sum of the scores remains constant. This prevents a drift toward higher or lower scores, whatever number of players we have.

## Limits and Issues of the ELO System

"- What? A beginner enters with 1,500 points, and I have played dozen of games and I have 1,450 points! This is unfair!". True, this is an issue. It makes little sense to mix players with an unstable score and players with a stabilized score. That's why it was decided to create two ranking boards:

- A Provisional Players Ranking, for players with less than 20 games
- An Established Players Ranking, for players who played 20 or more games. These are considered having a stabilized rank.

"- OK, but I am well ranked, and so I don't want to play with beginners because of the risk to see my score drop dramatically if they have an excellent hand!" Formulas were modified to make sure that experienced players would play with beginners without second thoughts. The impact of a defeat is very limited, especially if the beginner player has played only a few games.

"- I am at the top of the ranking, but I keep going up and down! What is happening?" Yes, this is inevitable. When you are in the top list, almost all the other players have lower ranks. So if you lose, you do down immediately. This is amplified by the fact that only few players have a good score. The other players are likely to have scores that are far from yours, which amplifies your fall. On the other hand, if you are in the average group, you will find that your rank and score changes only slowly.

"- I am on the Provisional Ranking, and my scores have huge variations: as much as 100 points after my very first game!" This is normal, beginning players use accelerated convergence formulas, which behave gradually like ELO as their number of games increases. Their scores is then highly unstable at the very beginning.

## Great, maths! I just love it...

For the mathematically inclined reader, here are the formulas! They come from another online game, Hardwood Hearts.

Let's call rA the starting score of player A, and nA his number of games. Same thing with rB and nB.

Let's compute the new score of player A.

We need to define two values, s1 and s2, as follows:

- s1 = 1 if A wins, -1 if A loses, and 0 for a stalemate
- s2 = 1 if A wins, 0 if A loses, and 0.5 for a stalemate

If A and B are both on Provisional Ranking:

If A is on the Provisional Ranking, and B on the Established one:

If A is on the Established Ranking, and B on the Provisional one:

If A and B are both on the Established Ranking, we use straight ELO:

The new score of B is computed the same way.

## Computation for 4 players

There are 4 players in a Gang of Four game. Let's name A the winning player, B the second one, C the third one and D the last one. We consider that there were 6 duels: A won against B, C and D. B won against C and D. C won against D. We compute independently the new scores for each duel, and then we average the values for each player.

This way, the second player will go up a little bit, because he still won 2 duels on 3. This gives some interest to the game in case the first player is far in front of the others. It is still worth fighting for the second place, and even for the third to limit the damage.