It seems to me that the article overlooks one glaringly obvious issue: that the two blunders may not be independent events.<p>In this case, it seems quite likely that the second player's blunder was made much more likely by the fact that the first player had just blundered. To be more specific, white moved the king which appeared (at first glance) to prevent black from using a check threat to attack white's rook. The blunder was in not realizing that the check threat could still be used to attack white's rook, albeit in a more complicated fashion.<p>Black responded to this with another "blunder" -- failing to attack the rook and moving elsewhere instead. But this blunder was NOT independent of the first -- it is quite likely (I believe) that black saw the move and assumed white had successfully prevented the attack on the rook. He assumed that such a top-level player would never make such a mistake, and that caused him to not look closely enough at it. The first blunder helped cause the second.<p>(Thanks to stolio for linking to the game analysis I used here.)<p>One could test this hypothesis of mine using the same data set. Instead of looking just at single errors, look at error pairs (one error occurring in the move following another error). If the probability of a blunder is significantly higher on the move immediately after a blunder than it is at any other time, then my hypothesis (that the events are not independent, but correlated) is supported.
Using "number of pawns of evaluation lost" as a proxy for the severity of the blunder has some fundamental problems. The main one is that the relationship between evaluation in "pawns" and expected result (expected value of the game result, from 0 to 1) is not linear. (It couldn't be, since one of them maxes out at one.) It's actually more of a sigmoid curve.<p>This means that a player may easily make a horrific "3-pawn blunder" reducing his evaluation from +8 to +5, but in fact all he's done is reduce his chance of winning from 99% to 98%. Actually, the +5 move may even be better in practice, in that it might lead to a sure safe win rather than a tricky blowout.<p>Even if you changed the definition of blunder from "reduces the evaluation by n pawns" to "reduces the expected result by x", I would have an issue in that it ignores any of the human aspects of blunders. If someone drops a pawn outright for no reason (eval change -1), that is a blunder because it was so trivial to avoid. But if someone, even a grandmaster, makes a move that causes a large drop in eval due to allowing a sacrifice that no human could calculate all the ramifications of, because as far as he (and probably his opponent) could humanly calculate it didn't lose, it is hard to call that a blunder. (Conversely, failing to see some immensely complicated non-forcing winning move may be unfortunate but it's not a blunder.) But that's more a cavil with terminology than a methodological error; the study is still measuring something interesting, just not quite what I think it is claiming to measure.
> Due to cost limitations we had to limit crafty to 2 seconds of analysis time per move<p>A grandmaster with standard time controls could defeat a 2-second limited Crafty. So how do you know you're finding true blunders, and not simply positions that the engine evaluates incorrectly?
One thing I know from playing serious Bridge: An expert player makes far fewer mistakes than the average player. He does not make zero mistakes.<p>I read one tournament report, where an expert player revoked.<p>When an expert plays good or average players, he does not need to be brilliant to win. He just has to play competently, and wait for his opponents to make mistakes.
See also the papers by IM & PhD Kenneth Regan on "Intrinsic Chess Ratings" such as
<a href="http://www.cse.buffalo.edu/~regan/papers/pdf/ReHa11c.pdf" rel="nofollow">http://www.cse.buffalo.edu/~regan/papers/pdf/ReHa11c.pdf</a> .
Here's commentary on the Carlsen/Anand double-blunder: <a href="http://youtu.be/6K86f27uuP0?t=14m36s" rel="nofollow">http://youtu.be/6K86f27uuP0?t=14m36s</a><p>It's not easy to see.
These results surprised me. I expected a much wider gap in correct move % between a 1500-player and grandmaster. It'd be interesting to see if the slope of the graph is steeper for minor blunders that reduce the evaluation by less than a pawn. These are the more subtle positional errors - weakening a square, not maximizing piece activity, wrecking your pawn structure, etc. Amateur games are filled with these mistakes, but they are much rarer in GM games, and I'd expect the difference to be more than just a few percentage points. But Crafty's not the right engine for this job. You'd want something with a more sophisticated evaluation function, like Stockfish (several-hundred ELO stronger than Crafty).
Grandmasters blunder more often than this. I would venture to say that what correlates with blunders more so than rating is time. Error rate goes way up in Blitz and Rapid.<p>IMO the more interesting thing about chess skill at the top is how much way way better GMs are than everyone else.<p>To me, ratings at the top feel more like an exponential scale than a linear one. For example, I have beaten International Masters at chess lots of times but have never once beaten a GM.<p>If I studied or cared (which I don't), I think maybe it would be possible to squeeze out a lucky win once in awhile. Aspiring to be a punching bag isn't a very appealing notion though, so you can understand my lack of motivation. GMs are crazy good.
OK, so they've analyzed 4.9 million moves.
How many double blunders did they find in the set?
If the hypothesis of independent events is true there should be about (4.9e6 / 10,000) = 490 doubles in the data set.
An obvious way to test accuracy of the model is to compare that to the actual number.<p>Why hasn't that comparison been done/mentioned?
What exactly is blunder? Watch this game: <a href="http://www.chessgames.com/perl/chessgame?gid=1032537" rel="nofollow">http://www.chessgames.com/perl/chessgame?gid=1032537</a><p>Tal sacrificed horse and queen.
Not relevant to chess, but in Japanese there is a saying:
saru mo kikara ochiru. Not sure if I spaced that correctly but it amounts to "Even monkeys fall from trees"