Zeitgefühl

Unglaublich, dass es Zeromagic unterdessen bald zweieinhalb Jahre gibt! Das sollte eigentlich eine spürbar lange Zeitspanne sein, aber irgendwie fühlt es sich für mich nicht so an. Einer meiner frühesten Einträge lautete damals Magic ist krank – meine Vermutungen damals kamen zwar noch mehr aus dem Bauch heraus (das war lange Zeit, bevor Ingo Muhs uns darüber aufklärte, dass Wizards Turnierspieler zu Gunsten von Casual-Spielern absichtlich und gezielt vernachlässigte), aber das Thema ist immer noch aktuell.

Wenn man dieses Hobby so lange betreibt wie ich, besitzt man eben eine ganz andere Perspektive. Knapp ein Jahr ist es her, da hat Tobi auf PlanetMTG einen Artikel von mir als “Pischner Classic” veröffentlicht, der damals auch gerade erst eineinhalb Jahre alt war (er war ungefähr zur selben Zeit wie Zeromagic entstanden) – DAS war also bereits ein Klassiker?

Noch einmal ein Jahr füher (also vor ca. dreieinhalb Jahren) hatte ich begonnen für Magic Universe zu schreiben – ist DAS jetzt lange her? Oder vielleicht mein (damaliger) Abschied vom Planeten (das war Ende 2004)?

Überhaupt, PlanetMTG gibt es ja auch erst seit Anfang 2002. Wisst Ihr noch, dass ich bereits für dessen Vorgängerseite, McElder, geschrieben habe?

Und selbst das war erst der Beginn meiner Karriere als Schreiber deutschsprachiger Magic-Artikel. Ja, ich gehöre noch zu derjenigen Generation Magic-Spieler, die an den Dojo Turnierreports geschickt hat! (Und mindestens einmal habe ich auch für Mindripper, die damals von Scott Johns betreute Seite, einen englischsprachigen Artikel geschrieben.) Die meisten dieser Stücke sind unterdessen verloren gegangen, auch wenn ich zuletzt gerade wieder doch noch ein paar meiner ganz alten Turnierberichte im Dojo-Archiv lokalisieren konnte.

Gerade erst – und damit meine ich auch bereits wieder “vor gut zwei Jahren”! – habe ich auf Zeromagic einen uralten Dojo-Beitrag von mir noch einmal abgedruckt. Jetzt allerdings bin ich bei der Durchsicht meines Ordners mit Magic-Artikeln (und der ist verdammt dick!) auf ein ähnlich prähistorisches Dokument gestoßen, das ich heute mit Euch teilen möchte. Seinen Unterhaltungswert zieht es sicherlich eher weniger aus dem extrem trockenen Inhalt… (Obwohl – der Atog zumindest sollte sich jetzt eine Schüssel Popcorn holen und sich amüsieren!) Passend zum endgültigen Ableben des Composite Ratings präsentiere ich Euch:

X-Rated: What does your rating mean to you?

Introduction

Every Magic player who participates in a sanctioned tournament gets rated. This rating is meant to approximate the level of playing skill she has achieved and thus to allow comparisons to other players. However, many players do not exactly understand how these ratings are calculated. They might also find it hard to connect their performance in a tournament with the change in their rating that follows it.

In this article I will try to give an overview over the ratings system used by the DCI, as well as a rough-and-dirty method to estimate how your ratings will change after a tournament. While I will try to avoid serious blunders, this text is not meant to be absolutely mathematically exact. My primary goal is to give the average tournament player who wonders about the ratings system a few clues.

Finding your Rating

(You should skip this chapter if you are familiar with the rating pages and rating categories.)

If you don’t know where to look up your rating(s): Go to the Magic Homepage at http://www.wizards.com. You will find a link dubbed “Personal Stats”. Follow that link and you will be asked to enter your DCI PIN (Personal Identification Number). Doing so will enable you to look at your ratings not only from Magic, but all games which have DCI-sanctioned tournaments (for example, Pokemon). Since this is an article for Magic players, I will just comment on your Magic ratings. Among those, you get rated in several categories as an individual, and might also be rated together with other players as a team if you ever played in a team tournament. I will concentrate on your individual stats, though.

You will find yourself rated in one or more of the following categories: “Constructed”, “Limited”, “Vintage” and “Composite” (which is a special case). The first three are akin to Magic disciplines: “Constructed” encompasses Standard (still called type 2 by many), Extended and Block Constructed tournaments. In “Limited” fit all sanctioned limited formats, meaning Sealed Deck, Booster Draft and Rochester Draft. “Vintage” contains those older constructed formats which are no longer relevant for professional play: Vintage and Vintage Restricted (perhaps still better known as type 1 and type 1.5). If you do not have a rating in any one of those three disciplines, you did not play a sanctioned match of that kind (or it is not yet processed by the DCI).

Whenever you play in a tournament which might, no matter how longwinded, eventually lead you to a Nationals Championship, a Grand Prix or a Pro Tour, you play either in the “Constructed” discipline, or in “Limited”. While “Vintage” tournaments are still held from time to time, they are not part of the regular circuit. So you essentially possess two important ratings which have no connection with each other, although they are calculated in exactly the same way.

Your “Composite” Rating is just the average of your “Constructed” and your “Limited” rating. It never changes on its own, but is just used as a measure of how well you do in Magic overall. If you do not possess a rating in “Constructed” or “Limited”, it is assumed to be 1600, which is the starting point for every Magic player.

If you are rated in a category where you have played at least 10 sanctioned matches, you will also get RANKED there. This means you will appear on a ratings list of all players worldwide, as well as on more local lists for your continent, your country and maybe your region or state, and can compare your position to that of other players. If you are ranked, there’ll be a direct link to your place in the list. You can also get to all ranking lists from a link of Wizards’ Magic page (“Ratings and Rankings”).

One last word about your Personal Stats: When you have accessed this area, you will be offered the opportunity to participate in the Player Reward Program. That is absolutely free, and offers you not only a chance to get free promotional cards, but also allows you to look at your “ratings history” where you can see how your rating changed from match to match, with all your opponents listed. You will need access to your ratings history if you ever want to appeal a match you played (if you think it was misreported – a not too unusual occurence). Anyway I cannot think of a single reason NOT to participate in the Player Reward Program.

The Zero-Sum Game

When you look at the worldwide ranking lists, you will find nearly 100,000 players rated (as of 09/02/2001) in the composite list. Since this list contains all players which are EITHER ranked in constructed OR in Limited, those lists are smaller: About 80,000 and 60,000 players, respectively. But there are even more players in the DCI database: All players which are unranked due to the fact that they played less than 10 sanctioned DCI matches in any given format have to be accounted for as well. I will not try to make more than a wild guess (it would be possible to make a far more educated one by analyzing the ranking lists – see further down), but I’d bet this means at least a few thousand players more.

When you look at all those rating scores, you should realize that they are 1600 points too high. ALL OF THEM. The simple reason is that the DCI lets every Magic player start with a rating of 1600. This is a COMPLETELY ARBITRARY number. It has nothing to do with the calculation of ratings whatsoever. These points are added once – at the beginning of your Magic career – and will make your score exactly 1600 points higher forever. Since this is done with every player’s score, it evens out. The reason WHY it is done is probably of a completely psychological nature: If you started at zero (the natural starting point from at least a mathematical perspective) and lost points in your first tournament (which is normal) you would have a NEGATIVE score. Mathematically, this is completely o.k.. It just shows that you’re below average. However, it might be demotivating. I guess the DCI chose the 1600-point-bonus to make sure that even the least succesful players still have a 4-digit rating score.

But when one wants to analyze rating scores, it is important to understand that there is something like a rating of ZERO (1600, namely), “POSITIVE” ratings (higher than 1600) and “NEGATIVE” ratings (those less than 1600). Rather than subtracting 1600 from all rating scores whenever they come up, I will just use those terms. So if you have a score of 1550, you have a negative rating – you are 50 points in the red.

The rating scores in constructed range from above 2200 to 1275, those in limited from near 2200 to below 1350 (the artificial composite list evens things out, of course). You will notice that in both lists a lot more players seem to be above zero (1600, remember) than below. Also, the highest positive scores are more extreme than the worst negative ones.

This is peculiar, because the ratings in each format are a zero-sum game: Whenever a player gains points, another player loses exactly the same amount. This means: If you add up all players’ ratings, you should get at zero (all scores should average at exactly 1600). Why don’t they?

They do (barring any calculation mistakes). Don’t forget all those UNRANKED players, which are part of the system, too. There a large number of negative scores is hiding. A few positive ones are there, too, no doubt, but it makes sense to assume that most newer players are in the reds. So it is important to remember that the average score of all Magic players is indeed 1600. This is not only true for constructed, limited and vintage (which are all calculated exactly the same way), but also for composite ratings, since they average out scores whose average is 1600.

So, if you analyze the worldwide rankings list, you might be able to estimate how many rating points are missing there, and thus make a more educated guess at how many unranked players might be in the DCI database. I won’t.

One more complication, however, does exist: Just because the average score of all players in a format is 1600, it doesn’t mean that all players you actually play against have that average score. Even though you can safely discount the infinitesimal imbalance your own score causes since you don’t play against yourself, there is not only the fact that different tournaments are attended by different crowds of players: An additional imbalance is caused by those players which are no longer competing. While they are still part of the system, their points (positive or negative) are no loger circulating. I guess that this mechanism is INCREASING the active players’ average for the following reason: I assume that most players who chose to retire do this after a few tournaments at most, when they realize that tournament Magic (or their performance) does not meet their expectations. But this is only guessing, and even if I am right, I really cannot say how large this influence is.

How rating changes are calculated

Now that I have made a few basic observations on the ratings system, it is time to break down how those ratings are calculated EXACTLY. The formula can be found in the DCI Universal Tournament Rules, Appendix A: DCI Rating and Ranking Systems.

I will, for the sake of Simplicity, start with the straightforward formula. This formula, however, contains a term that will have to be explained afterwards in a second formula, which is a bit more complex.

Player’s New Rating = Player’s Old Rating + (K-Value * (Scoring Points – Player’s Win Probability))

Let’s work this formula “backwards”: Your new rating differs from your old one by a certain amount (the term behind the plus-sign) which is either positive (increasing your rating) or negative (decreasing). This obviously makes sense, since you can win or lose points. It is also possible (although unlikely) that your score stays exactly the same. In this case, 0 is added to your old rating.

Looking at this amount, we find it is a product: You multiply the k-value of the match (which is the same as the k-value of the tournament) with the term in brackets – I will jump a bit ahead and tell you that this is a value between -1 and +1. This means, your rating CANNOT change by more than the k-value of the tournament each match. So no matter who you play, if the k-value of the tournament is 16, your score will not change by more than 16 each match (and most of the time a good deal less).

The k-value of a tournament is always 8, 16, 24, 32, 40 or 48. The average tournament is meant to have a k-value of 16. So if this isn’t a special tournament, and the organizer is not deliberately taking steps to make this tournament more important by increasing its k-value (just ask him), you can assume the k-value is 16. If you play in a premiere event, however, this changes. Premiere events have fixed k-values: Prerelease Tournaments have also 16, but Pro Tour Qualifiers for example have 32, while the World Championships have 48. Friday Night Magic, a program which is meant to encourage more casual play in a sanctioned setting, has a k-value of 8. The official list of premiere events with their appropriate k-values is a bit complicated to find: On Wizards’ Magic homepage there is a link “For Organizers”. Follow that link. Follow the text to the paragraph “Resources”. There is a link “Description of Magic: the Gathering premiere events”.

Back to the formula: We have seen that the k-value just serves to vary how many points you gain/lose in different tournaments (all other things being equal). So, if you lost 6 points in a k-16 match, you should console yourself with the knowledge that, if that had been a PTQ-match, you would’ve lost 12 points.

Now that we know that the k-value is just enhancing the amount your rating changes, we get to the heart of the formula: “Scoring Points – Player’s Win Probability”. Scoring Points are easy to explain: If you lose, you get none (0), if you win, you simply get one (1). If you have a draw, it makes sense to give you half that points (0.5). So far, everything is still extremely simple. The crux is that last term: Your “win probability”. Now what does that mean?

Of course, this cannot be your REAL win probability (which can not be known at all). Instead, this is a term which tries to APPROXIMATE your win probability by comparing your rating score to your opponent’s. How it does that we will see when we get to the second formula (which defines your “win probabilities” against players with certain rating scores). For the moment, we will just state some aspects of this “win probability”:

1. It has a value between 0 and 1 (like real probabilities). If you have ALMOST no chance to win a match (there always is a chance!), it is ALMOST 0. If you will NEARLY CERTAINLY win, it is ALMOST 1 (which can also be expressed at 100%) . If you have the same chance to win as your opponent, it is 0.5 (the same as 50%). Note that for the sake of elegance, this “probability” does not care for the possibility of a draw. Since it is not a real probability, but just an artificial construct meant to solve a mathematical problem, this is just fine.

(Now that you know which values your “win probability” can assume, it should also be clear to you why the term “Scoring Points – Player’s Win Probability” always gives a value between -1 and +1.)

2. It depends on the difference between your rating and your opponent’s. If both are the same, it is exactly 0.5. If yours is higher, it is higher than that, if yours is lower, it is lower. Also, the greater the difference between both ratings is, the closer it is to 1 (or 0, respectively). But since it can never reach 1 (or 0), it cannot grow proportionally to the difference, but must instead converge against it. This means that, the more both ratings differ, the less the additional difference contributes to the “win probability”. For example, if your rating is 200 points higher than your opponent’s, it makes a heck of a difference compared to playing an opponent with the same rating as yours. But if your rating is 500 or 700 points higher than your opponent’s is much less important.

3. It is symmetrical. This means: If you have a 30% chance of winning (meaning a “win probability” of 0.3), your opponent must have a 70% chance (a “win probability” of 0.7): The chance that EITHER you OR your opponent wins is 100% (1), so your opponent’s “win probability” must be 1 minus your “win probability”. (Note once again that this artificial “probability” does not care for the possibility of a draw.) In other words: The same formula that gives your “win probability” when you enter both ratings, must also give the corresponding “win probability” for your opponent.

So the best you can do at the moment is just assume that this “win probability” somehow has these aspects. Now let us get back to the formula:

Player’s New Rating = Player’s Old Rating + (K-Value * (Scoring Points – Player’s Win Probability))

Let’s have an example: You play against an opponent with the same rating as you ( We’ll just assume this is the very first tournament for both of you, so you both start at 1600). Your “win probability” thus is 0.5. Let’s say this is a normal 16-k tournament.

If you win, you get 1 scoring points. 1-0.5=0.5, so you win half the k.value: You win 8 points! Your new rating will be 1608.
If you lose, you get 0 scoring points. 0-0.5=-0.5, so you LOSE half the k-value: You lose 8 points. Your new rating will be 1592.
If you draw, you get 0.5 scoring points. 0.5-0.5=0, so you neither win nor lose any points. You rating stays at 1600.

Not that difficult, is it? Now for another example: This time your opponent’s rating is higher – so much higher that his win “probability” is 90% (0.9)! So, obviously, yours is 0.1. If this still is a 16-k tournament, that means:

If you win, you win 90% of the k-value – 14.4 points – but if you lose, you lose only 10% of it – 1.6 points! Even if you only draw, you still win 6.4 points! (It should be obvious from the formula that in the case of a draw, the lower rated player wins points, since his “win probability” is lower than the score points he gets for a draw.)

Now look at this match from your opponent’s perspective: Even if he wins, he only gains 1.6 points, but a loss means his rating drops 14.4 points, and even a draw costs him 6.4 points. This is exactly why good players hate to be paired against players with low ratings…

A few interesting things can be seen from those examples:

First, we note that rating scores are not integers, even though they appear that way in the ranking lists. This is because the increments by which they change are normally decimals. When the DCI is displaying rating scores, they simply cut off the decimals, but in their database they keep those scores with several decimals for more exact calculation.

Second, whenever a player wins points, his opponent loses exactly the same amount of points (I already stated that in the chapter “The Zero-Sum Game”).

Third, the difference between a won and a lost match is exactly the match’s k-value. That means, after you lost a match in a Pro Tour Qualifier, you know that your rating would be exactly 32 points higher if you’d won.

It is very important to understand that your rating is re-calculated after each and every match. Actually, the concept of tournaments does not contribute anything at all to the ratings system except to give a number of matches the same k-value. As far as your rating is concerned, your whole Magic career is a series of non-related individual matches. But exactly that this is NOT the case in real life makes hard-and-fast guesses about the impact of your tournament performances on your rating a bit harder. More on this later.

The most interesting aspect of this system, however, is that it is self-correcting. This is because your “win probabilities” against other players, which determine the amount of points you gain/lose, are based on your rating, but those gained/lost points in turn obviously CHANGE your rating and thus your future “win probabilities”. A player with a low rating who starts to win a lot will thus win a lot of points at the beginning of his winning streak, but later, when his rating adapts, will win less and less. The whole idea revolves around an (mathematically) ideal system, in which all players have a fixed playing skill which determines their “real” win probabilities. If this were the case, it would not matter with which ratings those player started: Given enough matches, the self-correcting system would bring their ratings (and thus their “win probabilities”) close to their “correct” values (they cannot meet exactly, since every match played cannot help but change their ratings).

To illustrate this with an example: Let’s say you and your friend are both absolutely equally skilled Magic players, and therefore always tend to split your matches fifty-fifty. But for some reason, your friend has a much higher rating than you. Then you start playing sanctioned matches against each other. You win as many matches as she does, but since she gains less from a win than you do, each of your wins is offsetting several of her wins, so you come out ahead, closing the ratings difference. Slowly, but steadily both of your ratings will near the average of your initial ratings.

Determining your “win probability”

O.k, now I will finally get to explain how your “win probability” is calculated. Let’s recapitulate the conditions to be fulfilled by the formula: Input is the difference between your rating and your opponent’s. If this difference is 0, the output should be 0.5. If the difference is positive, the output should grow with it and converge against 1; in the negative, the output should decrease and converge against 0. Also, it must be symmetrical, meaning that your “win probability” and your opponent’s “win probability” should add up to exactly 1 (so the output for any given difference, plus the output for the NEGATIVE of that difference, must be 1).

A formula that fulfills this condition is:

WP = 1 / (x^((OR-PR)/y) +1)

WP: your “win probability”; OR: Opponent’s rating; PR: your rating;
x: any number greater than 1; y: any number except 0

This looks far more awkward than it actually is, due to the lack of proper mathematical notation. Sorry for that, but in my defense I can say that even in the Universal Floor Rules they do not use proper notation.

It is just the reciprocal of a sum: The sum of 1 and x to the power of “something”. This “something” is just the difference of your opponent’s rating and yours (yes, your opponent’s rating comes first), divided by y.

Actually, it is a relatively simple formula, although raising a number to a power which is not an integer might feel awkward if your math skills do not lead you far beyond basic arithmetical operations. I can’t help you with that: I’m not a math wiz myself, and anyway it is a bit beyond the scope of this article.

However, with just a little knowledge of fractions and powers you can easily verify that this formula does indeed fulfill the aforementioned conditions (everything except the symmetry is nearly self-evident, and the symmetry only needs one or two lines of basic calculations – in proper notation, that is. Sorry again.)

Maybe it becomes clearer if we use real numbers instead of x and y. So let’s try the following:

WP = 1 / (9((OR-PR)/400) +1)

This is ALMOST the formula used by the DCI. I use 9 instead of 10 for the sake of a more comprehensible example. As far as I know, this is also the formula used for chess ratings. The basic idea is that a 200-points ratings difference means a 75% “win probability”. We can easily verify that: Since -200 divided by 400 is -0.5 (negative because your opponent’s rating comes first), the sum in the denominator is 9 to the power of -0.5 plus 1. Since 9^-0.5 is the same as the reciprocal of the square root of nine (school was long ago, wasn’t it?), that means the denominator is 1/3 + 1, or 4/3 (fractions…). The reciprocal of 4/3 obviously is 3/4, and that is the same as 0.75, or 75%.

Still with me? Then let’s look at this example from your opponent’s perspective: For him, the denominator is 9^0.5 +1 (since now your rating comes first). 9^0.5 is the square root of nine, so the denominator is 3+1 = 4, and the reciprocal is 0.25 = 25%. Everything as it should be.

Just because it is even simpler, let’s try an opponent with the same rating as you. Your ratings difference is 0, so the denominator is 9^0 +1. Since 9^0=1 (EVERYTHING except 0 raised to the power of 0 is 1), the denominator is 2 – your “win probability” is 50%, as is your opponent’s.

If you got that, you might like to verify that, with this formula, your “win probability” against a player with a rating 400 points lower is exactly 90%. Have fun.

But what are those numbers needed for? Why do we use 9 and 400, and not 2 and no divisor to the ratings difference at all? Those two numbers are parameters which define how large a ratings difference makes for how high a “win probability”. If there was no divisor, a ratings difference of just 1 point would make for a 90% win probability. This would obviously lead to nonsensically drastic rating changes. Those parameters are there to make sure that the self-correcting aspect of the system neither works too fast nor too slow. 400 seems to be a good choice.

The 9 would be there to fix 75% and 90% “win probabilities” at easy-to-remember rating differences (200 and 400). However, for some reason unknown to me, the DCI uses not a 9, but a 10, leading to the following official formula:

Win Probability = 1/ ( 10^((Opponent’s Rating – Player’s Rating)/400) + 1)

Sadly, this does not lead to easy-to-remember correlations between “win probabilities” and rating differences. This formula is not much different from the former: 75% are reached at ca. 191 points difference, and 90% at ca. 382. But this is just awkward to keep in mind. Nonetheless, here is a table of ratings differences and their correlating “win probabilities”:

Your rating higher by: Your win probability: Your rating lower by: Your win probability:
0 50.0% 0 50.0%
25 53.6% 25 46.4%
50 57.1% 50 42.9%
75 60.6% 75 39.4%
100 64.0% 100 36.0%
125 67.3% 125 32.7%
150 70.3% 150 29.7%
175 73.3% 175 26.7%
200 76.0% 200 24.0%
225 78.5% 225 21.5%
250 80.8% 250 19.2%
300 84.9% 300 15.1%
350 88.2% 350 11.8%
400 90.9% 400 9.1%
450 93.0% 450 7.0%
500 94.7% 500 5.3%
600 96.9% 600 3.1%
700 98.3% 700 1.7%
800 99.0% 800 1.0%
900 99.4% 900 0.6%
1000 99.7% 1000 0.3%

Hmmm… didn’t know how awkward it is to paste such a table from Excel into an e-mail. Maybe there is an easier way, but since I don’t know it, I will cut down a bit on tables for the rest of this article. However, with this table as a guideline you should now be able to roughly calculate your rating changes by yourself, given that you can at least approximate your opponent’s rating. But since your rating is recalculated after every single match, you will have do the same after a tournament if you want to anticipate your new rating. And you probably don’t want to do that. Isn’t there an easier way? There is, but you sacrifice exactness for simplicity. If you think that may be worth it, read on.

Connecting tournament performance with rating change

Among all rating-related questions, the one most often asked after a tournament is: Did I gain or lose points, and how many? If you take the time to look up your opponents’ ratings and then calculate your rating changes match by match, you will get satisfyingly exact results. But if a rough estimate will do, it takes much less time to assume that you played against a number of opponents with an AVERAGE RATING. So, instead of calculating changes after matches against players with ratings 1500, 1650, 1800 and 1900 you just assume that all of them had a rating of 1712.5 (or simply 1715, since you’re estimating anyways). While this simplification has its problems, it has one less than many players think: It is completely irrelevant if you won against the high-rated player and lost against the newbie or vice versa. In one case, you won more points and lost more, in the other you won less and lost less. It cancels exactly out.

However, the ORDER in which you won or lost is indeed important. This is because the self-correcting system takes your previous results already into account. This means that newer results always have a bigger impact than older ones. For example, if you played in a Grand Prix, didn’t make second day there and then played a number of side events, and did quite differently in those tournaments, the order in which those tournaments get processed could make a significant difference to your rating (and the DCI seems to process such bumps of tournaments in quite a random order).

Another problem with the average-rating approach is that you’re less interested in the absolute ratings scores of your opponents, but in their difference to yours. But that depends on YOUR rating as well, and that is changing over the course of a tournament!

Finally, the biggest problem is the tournament structure itself. Tournaments are normally either held in swiss system or in single elimination. If you do well in such a tournament, you will get paired against other players who do well as well (nice sentence, isn’t it?). This means that their rating when they meet you is probably a bit higher than the one they started the tournament with. How much of a factor this is depends on the kind of tournament: In a Friday Night Magic tournament with 8 players, it is largely irrelevant. But if you meet someone in the 5:0 bracket on the Pro Tour, she will likely already have got more than hundred points in this tournament! The flipside is doing really badly in a swiss-style tournament (you cannot do really badly in a single elimination event) – here the problem is similar. The more rounds a tournament has, and the more extreme your results are, the bigger is the resulting inexactness of this method.

But nonetheless the average-rating method gives acceptable results. So let’s have an example: You get into a typical, 16-k 4-rounds swiss system local tournament. Your old rating is, say, 1650. You play a first-timer in the first round and win, then win against a more casual player who ususally ends up with a negative score, manage to get a draw against an excellent player who just participated with reasonable success in a Pro tour, and then lose to the local hero who somehow manages to win this tournament every second time or so. Now what might your rating be after this tournament?

The newbie obviously has a rating of 1600 – that is 50 points below yours. The casual player will be somewhere in the lower half of the 1500s – let’s say 120 points below you. The pro player came back from the tour with a rating of maybe 1960 or so – 300 points above you. The local hero has achieved a score of nearly 1900, let’s guess: Maybe 220 points above you. So you add up the differences in your head: -50 + (-120) +300 +220 = 350. Divided by 4, this gives an average of nearly 90 points above you (since we’re estimating, we do not have to be TOO exact). Now let’s look at the table above: This ratings difference gives you an average “win probability” of something in between 35 and 40%. Let’s say 37.5%, since this is exactly 3/8 and thus easy to do in your head. Now add the score points you got in this tournament (do NOT mix them up with tournament match points, though): Two wins, a draw and a loss mean a total of 2.5 score points. (We can simply add them up since we chose to ignore the fact that ratings are recalculated after each match.) We also have to subtract the win probability 4 times (once for each match): 4 times 3/8 is 3/2, or 1.5. 2.5 minus 1.5 is, well, 1. So, all in all, we can estimate that you won 1 time the k-value of the tournament, which is 16. Your new rating thus will be ca. 1666, or 1665 to choose a more convenient number.

This whole calculation only takes half a minute and can be done in your head (assuming you can look up or did memorize the “win probability” table), so on the way back from the tournament you can tell your buddies how many points they can expect to have won/lost. And then, with this finally out of the way, you can continue talking about how mana-screwed all of you were…

The End

And this is, finally, the end of this article. When I began it, I had a lot more things in mind, and a few more tables prepared, but I just didn’t realize HOW FUCKING LONG this baby would get. If you made it this far, you were obviously interested in this topic, so I do not have to apologize to you. Anyway, I hope you found this interesting. Estimating how our ratings will change a few days before we can actually see the change in our personal stats is not really an important skill for us Magic players (unless we catch a serious calculations mistake that way), but somehow we are inclined to do it anyhow. I hope this article helps you to do it.

Andreas Pischner
Team Istari

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3 Comments on “Zeitgefühl”

  1. Magnus Says:

    “(EVERYTHING except 0 raised to the power of 0 is 1)”. Hoffentlich wurde das damals nicht so abgedruckt, ansonsten muss man sich ueber die heutigen Pisa Ergebnisse ja nicht wundern…;). Sicher ist Dir das mittlerweile klar, aber da es nicht explizit als Erratum erwaehnt wird: Die Aussage ist nicht nur ganz allgemein falsch (sie gilt nur fuer Elemente aus dem reeelen Zahlenkoerper und einigen exotischen Sonderfaellen), sondern auch noch speziell, da 0^0=1, wie die zehnte Klasse es lehrt. Aber die ist ja lange her, wie im Artikel richtig angemerkt.

  2. SDF Says:

    Synopse bitte, es gibt berufstätige Leute… achso, das heißt ja: tl dr. Erster Satz ist übrigens ernst gemeint.

  3. Mercurius Says:

    Ich bin ja normalerweise ein absoluter Fan deiner Einträge (zumindest dann, wenn es sich um SPIELrelevanten Inhalt) und lasse mich auch von der Länge nicht abschrecken, aber auch ich bitte zumindest um eine kurze Inhaltsangabe, da ich mich für einen dermaßen langen Absatz über Rating leider nicht begeistern kann.

    P.S.: Das Pauper Turnier wird vielleicht Realität.


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