# The number of levels in chess

Consider a weak chess player who has just learned how the pieces move and who will lose to a Scholar's Mate. (We can say that he's male and call him "Mr. Newcomer".) Let's define Mr. Newcomer's level to be 0. Moreover, let's say that player A is a level above player B if A's expected score against B is 0.9 (for example, 9 wins and 1 losses in 10 games). Given these definitions, at which level would you find a player like Magnus Carlsen (who is currently the player with the rightest rating in the world)?

I have a feeling that you can give a quite exact answer to this question if you know Carlsen's rating (which is 2872 at the time of writng) and Mr. Newcomer's rating, and if understand the rating system better than I do...

The choice of 0.9 in the above definition of a "level" is pretty arbitrary; if you think a number like 0.8 or 0.7 makes more sense, just pick that number instead.

The rating system already kind of allows you to calculate this since you can convert any rating difference into an expected score using the formula: expected = 1/(1 + 10^(diff/400)) so a rating difference of 382 points should give you the .9 expected score.

Therefore Magnus Carlsen at 2872 should score 90% (which could be 9 wins and one loss out of 10 games or 8 wins and 2 draws) against a player rated 2490. and that 2490 player should score the same against a player rated 2108 and so on.

So you could keep subtracting 382 from Carlsen's rating until you got to a rating of about 100 which is probably what the worst chess player could be. if you subtract 7 times you get down to 198, so I guess you could say Carlsen is 7 of your levels above the weakes beginner.

Now a rating difference of 200 translates to about a .75 expected score and that happens to be the rating difference that the USCF uses for different classes. For example: Master = 2200 - 2399 / Expert = 2000 - 2199 / Class A = 1800 -1999 / Class B - 1600 - 1799 / Class c - 1400 - 1599 etc..

so you can kind of think of each class as a "level" like you describe where the middle of each class has about a .75 expected score against the middle of the next lower class. So you could say that using that criteria, Carlsen is about 14 "levels" or class above the weakest player.

• `Master = 2200 - 2399` what if someone is 2400+? the only reason I can think of why it isnt just "2200+" is if there is another level above, am I wrong? Mar 5, 2013 at 19:43
• Well that is a simplification to say master and grandmaster are just rating levels since they are titles that have to be earned by achieving a number of performance norms but you can think of 2200 - 2399 as master and 2400+ as grandmaster. Titles like IM and GM are awarded by FIDE and titles like FM are awarded by national federations. Since above I was talking about the USCF which doesn't award titles higher than FM I didn't include 2400+ Mar 5, 2013 at 23:36
• By the way, where I said FM above, I mean NM since FM is awarded by FIDE as well. Mar 5, 2013 at 23:45
• @ajax333221 the USCF has a new-ish title system. Basically, they call anyone 2000-2199 a Candidate Master, 2200-2399 is a National Master, and 2400+ is Senior Master. There used to be a title of Life Master for maintaining a rating of 2200+ for 300 games (this would also result in a rating floor of 2200). Now this title is called Original Life Master to avoid confusion.
– Andrew
Mar 6, 2013 at 22:25