The Math Behind the Formula

We’re happy you have decided to learn about the ranking theory and the underlying formula for the scoring on all of our ranking websites. Our goal is to make this information accessible to everyone, so we will start off by presenting the theory through examples and finish by discussing the intricate details of the mathematics.

Brief Introduction: Who Is Best? A Relative Measure

Humans have been inventing games and ways to compete against each other throughout recordable history – so it is only natural to want to know – who is best?

It simply doesn’t suffice to list the skills we have mastered, the moves we know or the tricks we have invented. We want to see how they hold up relative to others because that is what motivates us to invest the time required for continued improvement. Sure, it’s fun to play, it’s satisfactory to improve but it is exhilarating, challenging and rewarding to compete and win.

Considerations: History, Diversity

So you have been practicing for a while and getting accustomed to the game and are ready to test the waters of competition in one of our clubs. Since you have never played against any of the players, there is no way to know how you rank because, as we just saw, ranking is a relative measure.

It can only be assumed that you are an average player, so you will start you off with an average score. Of course if you win, your score will improve and likewise if you lose, it will decrease – but it’s not that simple. If you beat a player with a higher score, your score will increase more than if you beat a player with a lower score. Let’s see why.

Example 1

Consideration 1: One Game

On your first game, if you beat the worst player in the club, it is logical to assume that you are a better player, but how much better?

Conclusion 1: Game History Matters

If you beat the worst player 10 times in a row, you are decidedly better, but if you win 5 out 10 times, you are really about the same. By establishing a game history with that player, your score and relative rank will become more accurate. Hence it is clear that you have to establish a pattern and a history to have a more accurate score and a more accurate ranking.

Consideration 2: Multiple Players

If you beat the worst player 10 times in a row, you are decidedly better than that particular player, but how do you rank in relation to the other members who also have beaten and established that they too are better than the worst player?

Conclusion 2: Diversity Matters

You will have to play another member of the club to get an idea of how much higher you rank above the worst player. If you beat the person in 5th place, it is fair to assume that you could also beat the players who rank below the 5th place member, but what about the players in the first 4 positions? It is clear that it is important to play against players of different ranks levels in order to get a more accurate score because diversity matters.

Consideration 3: History vs. Diversity

If you play 10 games against one player, your score will be more accurate in relation to that player only (History Only). Your score relative to the other members of the club is still unknown (Lack of Diversity). If you play 10 games, each one against a different player your relative score in the club will be more accurate within the club (Diversity). However, you have only played each member once (Lack of History) and have not established a pattern with that particular player.

Conclusion 3: You can’t have one without the other

As a final conclusion for this section, it is important to establish History and Diversity together in order to have an accurate score.

Score vs. Rank

The score is the numerical value that you have earned by playing games. The Rank indicates how this score compares to others in your club. These are both relative numbers with no inherent meaning or indication of skill level. Let’s look at some examples to see why this is the case.

Consideration 4:

Now let’s consider the situation in which there are two clubs that play separately from each other. In one club, the players might be professional players who are very equally matched and so their scores will never diverge much from the initial 2000 point average that they were awarded when they signed up with our rankings websites. This is because they loose roughly the same number of points that they gain when they win, and thus keep handing points back and forth when everyone is equally matched.

In the second club on the other hand, let’s say that the top ranking player is much better than the worst player and can win 100% of the time. If they play each other often, the best player’s score will keep increasing and the worst player’s score will keep decreasing. The top ranking player might have as many as 3000 points and the worst player have 1500.

Conclusion 4

There is still no way to know who will win if a player from one club plays the player of the other club. Now consider if the players scored at 2000 are professionals, while the members of the other club are not, it likely that the professionals would decisively beat the player with a score of 3000. Clearly the scores and ranks are relative and do not indicate the actual skill and the rules of history and diversity follow for inter-club play as well.

The Universal Club

The universal club is the comparison of each players score against all of the other players. As seen from the previous considerations and conclusion, the universal ranking has very little meaning. It is important only in as much as it provides a way for two players who do not have a common club to play against each other. If clubs intermingle more often, it will come to have more accuracy. The more that clubs play each other and intermingle, the more accurate the universal score will be. It is important to realize that the universal club has little meaning, but still fun to see how people’s scores compare.

Coming Soon: Frequency – Unranked Players

One thing that we wanted to take into consideration is the question of frequency. We found early on that some less enthusiastic players would play just enough to get a good ranking and then stop playing. By not playing any games, they simply maintain their status and are difficult to challenge. To address this situation, or the situation in which a player leaves a club, there will soon be an “unranked” value. If you do not play enough games with a club, you will drop into the unranked category, whether you were in first or last place. Our goal at Winito is to have game prosperity, not score prosperity.

Development of the System

Our goal was to have an intelligent statistical tool to analyze scores. We aren’t the first to tackle the challenge of ranking. In fact, there is a very good system called the ELO system, which is the statistical rating system most widely used for comparing chess players. It is this statistical comparison itself that convinced us of its worthiness because it is only from a statistical analysis of a player’s history that one may get a reasonable understanding of that player’s skill set. Clearly this is only a relative measure of a player’s skill and it relates only to the group from which the statistics were gathered. It cannot be applied to any player outside that group – no different from real life.

Our theory is not a replacement of ELO, but an extension of it. Arpad Elo a master-level chess player invented a statistical system to measure relative skill set details could be found here. Link to http://en.wikipedia.org/wiki/Elo_rating_system

Explanation of the ELO Theory

According to the theory, a player starts with some “seed” score. This is an initial score assigned to the player when the player first joins the system. Ranks of players in the group are sorted according the score.

Performance can't be measured absolutely; it can only be inferred from wins and losses. Ratings therefore have meaning only relative to other ratings. Therefore, both the average and the spread of ratings can be arbitrarily chosen. Élo suggested scaling ratings so that a difference of 200 rating points in chess would mean that the stronger player has an expected score of approximately 0.75, and the USCF initially aimed for an average club player to have a rating of 1500.

A player's expected score is his probability of winning plus half his probability of drawing. Thus an expected score of 0.75 could represent a 75% chance of winning, 25% chance of losing, and 0% chance of drawing. On the other extreme it could represent a 50% chance of winning, 0% chance of losing, and 50% chance of drawing. The probability of drawing, as opposed to having a decisive result, is not specified in the ELO system. Instead a draw is considered half a win and half a loss.

If Player A has true strength RA and Player B has true strength RB, the exact formula (using the logistic curve) for the expected score of Player A is:

Similarly the expected score for Player B is:

Note: EA + EB = 1. In practice, since the true strength of each player is unknown, the expected scores are calculated using the player's current ratings.

NOTE:

Here 400 is the rating interval scale, these could vary from game to game and is a measure to scale the score difference in the formula.

When a player's actual tournament scores exceed his expected scores, the ELO system takes this as evidence that player's rating is too low, and needs to be adjusted upward. Similarly when a player's actual tournament scores fall short of his expected scores, that player's rating is adjusted downward. Élo's original suggestion, which is still widely used, was a simple linear adjustment proportional to the amount by which a player over performed or underperformed his expected score. The maximum possible adjustment per game (sometimes called the K-value) was set at K=16 for masters and K=32 for weaker players.

Supposing Player A was expected to score EA points but actually scored SA points. The formula for updating his rating is

This update can be performed after each game or each tournament, or after any suitable rating period. An example may help clarify. Suppose Player A has a rating of 1613, and plays in a five-round tournament and here are the results:

- Loses to a player rated 1609
- Draws with a player rated 1477
- Defeats a player rated 1388
- Defeats a player rated 1586
- Loses to a player rated 1720

The score is computed as follows 0 + 0.5 + 1 + 1 + 0 = 2.5. His expected score, calculated according the formula above, was (0.506 + 0.686 + 0.785 + 0.539 + 0.351) = 2.867. Therefore his new rating is (1613 + 32*(2.5 - 2.867)) = 1601.

Note that while two wins, two losses, and one draw may seem like a par score, it is worse than expected for Player A because his opponents were lower rated on average. Therefore he is slightly penalized. If he had scored two wins, one loss, and two draws, for a total score of three points, that would have been slightly better than expected, and his new rating would have been (1613 + 32*(3 - 2.867)) = 1617.

Why Our Ranking System Is Different

This updating procedure is at the core of the ratings used by FIDE, USCF, Yahoo! Games, the ICC, and FICS. Each organization has taken a different route to dealing with the uncertainty inherent in the ratings, particularly the ratings of newcomers, as well as the problem of ratings inflation/deflation. New players are assigned provisional ratings, which are adjusted more drastically than established ratings, and various methods (none completely successful) have been devised to inject points into the rating system so that ratings from different eras are roughly comparable.

Some Problems and Solutions:

- Initial Score: How do we determine the initial score of the player and how credible is the rank of that player at that time?
- As we know from the above formula that we use a constant value K, but what is the right value of K? And should it differ from sport to sport? We know that if K is too low, it is harder to win points and if K is too high the system becomes too sensitive and hence the rank fluctuates a lot. Good values of K are discussed in the cited papers.
- There exist situations when players become greedy about their scores and hence selectively play players who are weak to gain points and hence increase the distance between themselves and trailing players.
- What happens if a player is not playing anymore, of very infrequently?

For example:

Player X has a score 2100, Player Y has a score 1700, Player Z has a score 1400.

If player X is doing well already and if he keeps playing weaker players, he will keep winning more and more points and in turn maintain his rank. Similarly, Player Z wanting to improve his rank would keep playing Player Y because he is more likely to beat him and improve his score and avoid Player X completely. Players Y and Z could choose to block out Player X and cause him to be at a stand still until one of them catches.

Coming Soon!

This tendency of the algorithm to promote score prosperity makes it impractical. To tackle this problem, we will be adding a variance on the score calculator depending on the frequency of games played with different players using graph theory and details will be revealed soon.

One more aspect of problems is when players try to protect their rankings. Good players, who already have reached a score of 2100 in our example, stop playing other players in scare of losing points and hence still stay at a good rank without being actively involved.

A good score or a grand master score in chess is around 2800 as of April 2006.

At TableTennisRankings.com we initial your scores to 2000 and after a 30 day sampling (trial) period the score tends to stabilize to a more realistic score of a player. The initial scores can be changed by the administrator of the club if you have already been using the ELO rating system and have scores that are already determined, or for other reasons he may have.

K value problem:

Now considering the K value problem, the question is what value is too sensitive and what value is not sensitive enough (what value causes the score to fluctuate a lot after every game and what value causes less fluctuation)?

In a competitive game especially where the scores and ranks are involved, players with lower score would wish that the K value is high so that when they win a game, their score fluctuates to a better score at a much faster rate and conversely would decrease the score of the higher ranking player at a faster rate. Similarly the statement holds true for players with high scores who would prefer that the K value to be low so that if they lose, they do not lose a lot of points. A good compromise is to have 3 K Values, each of them being applicable on in a certain score range. For example:

For scores < 1800, K value would be 32

For scores >1800 and < 2200, K value would be 24

For scores > 2200, K value would be 16.

The higher K values causes more fluctuation, so there is an incentive for players ranking under 1800 to play those who are ranked in a lower K value range. At the same time, the person in the lower K value range is not discouraged from playing against the player in the higher K value range.

This is the logical median point that is best for everyone. This way the top players have the incentive to keep playing, but can’t increase their score beyond reach, while at the same time encourage lower ranked players to keep playing everyone in the club, not matter what their score.

Conclusion - Custom K Values

At TableTennisRankings.com ranking system allows the administrator of each club to adjust the K values for 3 intervals according to what he feels is fair and the most representative of his club’s actual ranking. In this way the algorithm can cater to the needs of many different sports and the individuality of each club.

We do encourage you to read further by studying the documents linked in this article and in your own research. We would love to hear your thoughts and contributions on the subject so that we can improve the theory and continue the arduous task of sports ranking and probability analysis.

References

http://en.wikipedia.org/wiki/Elo_rating_system

http://www.chessbase.com/newsdetail.asp?newsid=562 for information on how the K value factors into it.

http://chess.about.com/od/organizations/ss/aa04104_5.htm

ICC K-factor help http://www.chessclub.com/help/k-factor