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**QUESTION:** Can you give us an example to show how **complexity** is calculated?

**ANSWER:** The basic complexity score calculated is the difference between the green and red boxes below (1.14 - 0.90 = **0.24**).
We first determine that Ba7 is the best move at iteration 14 (and save the score at that iteration), then we find the score for Ba7 just before it become the best move.
Notice that if Ba7 had always been the "best" move, then complexity would be zero.

ITERATION | SCORE | BEST MOVE | SCORE FOR ULTIMATE BEST MOVE | ULTIMATE BEST MOVE |

10 | +1.02 | Nb6 | +0.90 | Ba7 |

11 | +1.02 | Nb6 | +0.90 | Ba7 |

12 | +1.04 | Nb6 | +0.90 | Ba7 |

13 | +1.04 | Nb6 | +0.90 | Ba7 |

14 | +1.14 | Ba7 | +1.14 | Ba7 |

15 | +1.16 | Ba7 | +1.16 | Ba7 |

16 | +1.21 | Ba7 | +1.21 | Ba7 |

**QUESTION:** Will you extend this project and include other great players such as Rubinstein?

**ANSWER:** I almost certainly will continue adding a few players every year.
DeLaBourdanais, McDonnell, Anderssen, Rubinstein, Pillsbury, Schlechter, Keres, Korchnoi are leading candidates for 2009.

**QUESTION:** Does the **Complexity Table** adjust correctly for both grandmaster draws and games against weak opposition?

**ANSWER:**
Yes, the complexity cross-reference table (hopefully) compensates for grandmaster draws
by converting "raw" scores (which exclude "blunders") to the numbers you see in the
"Percent Better Than the Average Grandmaster Moves" column.
And the "Blunders per 1000 moves" column is a completely separate measure that
is **assumed** to be unrelated to complexity, although obviously this can't completely
be true (a blunder is a rare thing in a grandmaster draw). Trying to explain
why one Champion blundered 10 or 20 times more frequently than another is
the $64,000 question. I think it may have to do with determination or the
will-to-win, and how one copes with the terrible tension & nervousness that most
players feel when playing tournament chess. I think the greatest champions
had periods when they would not allow themselves to make a mistake: Fischer (1968-1972)
and Kramnik in his title matches are two clear examples. It took Anand most of his career to
eradicate these blunders, but he seems to have done so in the last couple of years.

**QUESTION:** How reliable do you consider these numbers?

**ANSWER:** The longer the time period being considered, the better the rankings seem to be.

**QUESTION:** For rankings over a multi-year period, how do you "weight" the games from different years?

**ANSWER:**
Each move in each time period was weighted equally. If you look at
Fischer's data for his 1-year & 2-year periods,
you will find that he played 504 qualifying moves in 1968
and only 20 qualifying moves in 1969, so his data for 1968-1969 is, essentially, for just the one-year period of 1968.
That is why I prefer to look at the 5-year & 10-year data.

**QUESTION:** I noticed that in your example of games between Kasparov and Karpov, the complexity measures
for each player were very different. I find this result surprising, since I would have thought that on average
the complexity measure across a series of evenly matched games would be approximately equal for both sides. But apparently this is not the case, i.e., presumably Kasparov and Karpov typically reach positions that are objectively equal, but more difficult to play from Kasparov's side. Am I interpreting this correctly?

**ANSWERS**
Complexity, as I have designed it, is **higher** for "aggressive" players who must justify their aggression by
finding deeper, "non-obvious" moves & combinations and **lower** for "passive" players who are content to
keep the position balanced in which there are often several perfectly
acceptable "positional" moves that do not radically disturb the balance.
**Note:** when a player is thrown on the defensive, his move choice is usually greatly
restricted since he is often "forced" to make a particular (often fairly obvious) move or immediately lose the game.
This fact helps explain why the losing side almost always has a lower complexity score than the winning side
(since a losing player usually must play more defense than offense).

To answer your query about why Kasparov's complexity scores are significantly higher than
Karpov's despite an overall 73-71 score, **Second,** I counted the draws in which one side had significantly more complexity: the result, Kasparov over Karpov by 30-17.
To illustrate how draws can be lopsidedly-complex, look at games #4 and #11 from the 1990 match. In #4, Kasparov seizes the
initiative with (the dubious) 23.Re6, eventually ends up 2 pawns down, and is presented with a complex
task to draw. In #11, Kasparov as Black plays the King's Indian Defense and, by move 15, has already sacked the Exchange!
The game winds up on move 24 with Kasparov, **behind by 2 rooks,** forcing a perpetual check.
I don't know if there were other draws as wild as these two games, but I suspect that Kasparov was generally the aggressor, even in the draws.

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