Monday, 5 December 2011

Rating vs. Time on the Clock

In my previous article, An Important Discovery, I speculated that for human chess players, each doubling of the time of the clock results in a gain of about 200 Elo rating points in playing strength.  After reading the resulting discussion (in comments to the previous article and on Chess Tempo), it occurred to me that I had the data to test this hypothesis for problem solving.  I sorted the results for my first pass through Coakley into ascending order of solution time, and calculated:

(1).  The total time taken on all the problems that I had solved (correctly or incorrectly) within each of these solution times.

(2).  The time taken on all the problems that I had not solved within each of these solution times (i.e. the solution time multiplied by the number of unsolved problems).

(3).  The average time taken per problem for each of these solution times, i.e. (1) + (2) divided by the total number of problems.

(4).  My score, as measured by the fraction of the problems that I solved correctly within each of these solution times (i.e. the number of problems solved correctly divided by the total number of problems).

The graph below plots my score against the average time taken per problem (in seconds):

I made almost linear progress here.  I calculated the Elo point rating difference -400 * log(1/score - 1) for each point on the graph above (see my earlier article Rating Points Revisited).  The graph below plots these rating point differences against the average time taken per problem:

From this graph we get:

Seconds   Points    Increase
   6       -242        -
  12        -62       180
  24        170       232

An increase of 200 Elo rating points for each doubling of the total solution time taken appears to be about right, judging from this example.


  1. Have a look at the linear progress of Kasparove as I have reconstructed from his book "High play"

    Age Rating Fact
    05 1300 Learned the game
    06 1400
    07 1500 Started with lessons
    08 1600
    09 1700
    10 1800 Botwinnik-school
    11 2000
    12 2100
    13 2200 Candidate master,1st tourn.abroad
    14 2300 Master
    15 2450 First victory over a grandmaster
    16 2545
    17 2595 2nd grandmaster result
    18 2625 Grandmaster

  2. I was discussing rating against time on the clock in the article, of course, but that is another aspect of rating vs. time. The race is not to the swift. 100 Elo points per year will make you the strongest player ever - if you can keep it up!

  3. Though it is still hard for me to believe that a doubling in time results in a 200 elo gain, I start to believe it. I am suspicious, but since I have no better data, it is my nature to rather stick to the statistics than what I feel by "stomach".

    The subject is of major interest, because it leads to the thought: If I cant double the time - what if I can double my speed in solving puzzles? It should also result in a 200 elo gain.

  4. Perhaps it is easier to believe if you think in terms of halving the time that you need to find the solution.

  5. Bright knight:
    true, halfing the time would result in a much more poor result. Here I could imagine a 200 elo drop.

    @Tempo: For the 100 elo gain per year:
    I believe much more is possible. With the right training and the will power to do 1-2 CT hours of training (which Kasparov didnt have at his disposal).
    I can show you my elo gain in chesscube (board email me in CT, I then give you my chess cube login). It was around 200 within 2-3 months (from around 1800 to ~2000), with an performance of around 2150 elo within the last 20 games. My CT Blitz graph looks also promissing. Yes, it is possible to improve even as an A-class player. But it might only work for me (I dont think so, but could be). Uri Blass started doing now easy fork puzzles, too. I am curious if it works for him, too.

  6. I have not shown the numbers for very short time limits, but I got less than 200 points when I doubled times that were so short that I could only solve a few problems. If you can already solve most of the problems in the time available, doubling it is not going to help much either. The number of points gained by a doubling in speed depends on the distribution in difficulty of the problems. Practical chess contains tactics at a very wide range of difficulty, so 200 points looks possible, but may be optimistic.

  7. I started yesterday an experiment: I put my board view of the CT puzzles upside-down. My solving speed dropped a lot. This indicates, that we can much better calculate what we can threaten to our opponents, but that we hardly see, what the opponent could do to us. I dont dare to use my blitz rating for this experiment, though. But if one of you want to test it: It is really amazing how much worse we get if the view you have to find the tactic from top to bottom (at least most of the time). I would expect a drop by 200 elo... (since the time you need to solve a puzzle almost doubles).
    The full conclusion about this I have not derived, but somehow (again) I have the feeling that it gives us an important clue. I just cant fully grasp why.
    Hm. This is a bit getting astray from the topic here.

  8. My problem books all have White at the bottom, and most of them have Black to move problems. I find it harder with Black at the bottom, even if it is Black to move. Ideally, the order of problems should be different for each pass, and the positions should be flipped from White to Move to Black to move. Flipping from the Kingside to the Queenside can also be done automatically if the problem does not involve castling.

  9. I think : the increase of 200 Elo rating points for each doubling of the total solution time taken is to high/ to optimistic. You calculate the score of a problem with 0 if you did not solve it at the "given time". Thats to low. if you would have known that you have o solve that problem in the "given" time, then you would have guessed the sulution and you would have been sometimes/often right. The "average" score would have been >0. I think you did show : the increase rating points for each doubling of the total solution time taken is less than ~200.
    Uri Blass value of ~120 is the best estimate...

  10. If I had spent the whole minute on all the problems that would have increased my success rate at that time limit. (I got a proportion of my faster solutions wrong, and that proportion would have been lower if I had used the remaining time to check and recheck.)

    I had imposed a time limit of 30 seconds per move, my success rate at that time would have increased because I would have been able to guess some of the answers.

    It is not clear which of these two improvements would have been the greater.

    I could run an experiment in which I divided the Coakley problems into two sets: those from the even numbered pages, and those from the odd numbered pages. I could impose a time limit of 30 seconds on the first set, and a minute on the second set, and in both cases use all the time available and guess if needs be.

    In a real game, a time limit is not applied to each move individually. A better model is that of a fixed average time per move. My experimental protocol equates quite well to this. I stopped to clock as soon as I thought that I had found the right answer. I did sometimes guess when I got to one minute, but I often did not have a clue. My experiment corresponds well to a game in which every move is Coakley puzzle, and I have to maintain an average of 25 seconds per move. That may not be realistic, but it is clear what I am measuring.

    As I understand it, Uri Blass’s 120 was derived using Chess Tempo problem ratings. However, since Chess Tempo rating do not have any sound statistical basis, I do not believe that we can place any weight on this value. In principle, we could use the methods of my later article:

    We can also do direct measurements for practical chess in time handicap games, as discussed earlier.