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- - By lkaufman (*****) Date 2007-12-03 19:08
Today a puzzling game was played at the World Cup between two of the World's best chessplayers. [Event "World Cup"]
[Site "Khanty-Mansiysk RUS"]
[Date "2007.12.03"]
[Round "4.1"]
[White "Aronian, L."]
[Black "Jakovenko, D."]
[Result "1/2-1/2"]
[ECO "D53"]
[WhiteElo "2741"]
[BlackElo "2710"]
[PlyCount "47"]
[EventDate "2007.11.24"]

1. d4 Nf6 2. c4 e6 3. Nf3 d5 4. Nc3 Be7 5. Bg5 O-O 6. cxd5 Nxd5 7. Bxe7 Qxe7 8.
Rc1 c6 9. Qc2 Nxc3 10. Qxc3 b6 11. g3 Bb7 12. Bg2 c5 13. O-O Nd7 14. dxc5 Rfc8
15. c6 Rxc6 16. Qxc6 Bxc6 17. Rxc6 h6 18. Rc2 Rd8 19. Rfc1 Qf6 20. b4 Ne5 21.
a3 Nxf3+ 22. Bxf3 g6 23. Kg2 Qe5 24. h4 1/2-1/2

     The positionafter move 15 was quite even and drawish (a bit better for White according to Rybka). Then White simply exchanged queen for rook and bishop. Presumably this was a winning attempt, since White was not worse and could probably play any normal move and expect a draw offer to be accepted. But it is hard for me to believe that White is actually better after the queen sac, and more to the point, Rybka thinks that White has just blundered badly. In the final position Rybka thinks that White is really lost, having no compensation for the substantial material deficit.
     So, two questions. Why did White play this "sac"? More importantly, if we accept the judgement of the players that the game was equal at the end, how can I make Rybka "understand" that Black is not much, much better at this point? Our queen value is at least close to optimal in general; what are the factors that could offset what Rybka perceives as a large material deficit (roughly a pawn and a half). This is one of the largest discrepancies between GM evaluation and Rybka evaluation of a quiet position that I have ever seen (excluding special reduced endgames and problem-like positions).
Parent - - By Fulcrum2000 (****) Date 2007-12-03 19:41
Maybe it isn't a draw (didn't look at the game yet). Did you play out the game with Rybka, maybe it's an easy win for black after all, but both GM's overlooked it?.
Parent - - By lkaufman (*****) Date 2007-12-03 19:59
     It's just a boring, symmetrical type position with a queen vs. rook and bishop material imbalance. If Black is in fact winning (or close to it), it certainly won't be anything tactical. Rybka just values a queen as about 1.5 pawns more than rook and bishop, it's as simple as that. Probably due to the pawn symmetry there's no way for Black to win, but it would be impossible to prove this.
Parent - - By Hetman (*****) Date 2007-12-08 18:10
I am not expert in that. My post is not opinion. I am wondering rather. E. Lasker WCCH and matemathician) has explain such a concepction concerning evaluation. He started with valuating pawns in the openning the central pawn was 1, the 'a' pawn was 1/2 :-). Then he continued that the squares on the board have their value either but there were no details. He has written just that it is not the theme for that book.
Might be the controlling the squares around opponent King shall get additional value or controlling the break through squares as well.
Parent - - By lkaufman (*****) Date 2007-12-08 18:29
Of course Rybka knows that controlling squares near the opponent's king is valuable. I wrote a whole article for Chess Life a few years ago talking about the value of pawns on different files in the opening, as well as the pieces, based on computer studies. I concluded that Lasker had the right idea in general about the pawns but overstated the differences between the pawns and made some glaring errors regarding the pieces.
Parent - - By Hetman (*****) Date 2007-12-09 18:09
It sounds interesting. Is that article avaiable somewhere in internet ?

Lasker did not have the computers and databases, the chess were played other way so might be his evaluations were ok for the time when he wrote ?
I think that the evaluations will be behaving somehow dynamic with the chess development.

Parent - - By lkaufman (*****) Date 2007-12-09 22:27
No, although a previous article on "material imbalances" can be found on Dan Heisman's website. In my opinion, Lasker's values would have been considered bizarre even by other top players of his time. He may have been a genius, but I cannot help but think that he was just not a "numbers" person.
Parent - - By insipid (**) Date 2007-12-10 00:22
That seems like an odd thing to say about someone who was also a famous mathematician in his day.  Lasker even has a theorem named after him.  I would imagine him being rather good with numbers.
Parent - - By Banned for Life (Gold) Date 2007-12-10 00:44
Lasker was certainly a great mathematician, and his work is still relevant, but this was more about abstract concepts than banal arithmetic. Larry might well be right about him not being a numbers guy.

Parent - - By noctiferus (***) Date 2007-12-10 19:16
about noether rings: wow, at that time, he was really on the front of (abstract algebra) research :)
Parent - - By turbojuice1122 (Gold) Date 2007-12-10 22:12
Wow, I hadn't ever heard of Noetherian rings, and I took abstract algebra.  Wikipedia has a good article, though.
Parent - By noctiferus (***) Date 2007-12-10 22:17
Yes, I had to go there in order to remind it: 40 years are a long time :(
Parent - - By lkaufman (*****) Date 2007-12-10 01:50
In my high school, there was one student who was better than I was at advanced mathematics, but he was awful at simple arithmetic and numbers. He went on to become a university math professor. As for Lasker, I can only tell you that the values he published in his book imply various relationships that are obviously wrong and were obviously wrong even then.
Parent - By turbojuice1122 (Gold) Date 2007-12-10 04:35
I can confirm this with another identical example: the guy at my high school who was generally seen as my equal in math (except he was a freshman when I was a senior!) was better at the more advanced stuff, but was no match in mental calculation.  However, I think he's now a medical doctor...what a waste... :-D
Parent - - By Hetman (*****) Date 2007-12-10 18:28
That maths abilities, are they rule or the exception of the rule ?

It will be interesting to know the values adequate for thouse times.
Parent - - By BB (****) Date 2007-12-10 18:54
That maths abilities, are they rule or the exception of the rule ?

Anderssen was a professor of mathematics. Steinitz started math studies, but then left to play chess. Euwe taught math at a Lyceum. Petrosian's doctorate might have been for mathematics ("Chess Logic").
Parent - - By insipid (**) Date 2007-12-11 05:10
There is no doubt, really, that mathematical ability is correlated with chess ability.  It is interesting to talk about exactly why that might be.  Math in itself is abstract, in fact any number is abstract.  But of course this is a big digression now, so I will let it be. :)  I am still not willing to concede that anyone with the resume of Lasker in Mathematics can be said to not be a "numbers" person, although I will yield to Mr. Kaufman regarding his numerical chess theories.  You don't reach the abstract heights of a Lasker or an Einstein (they were friends) without being very firmly capable in the "numbers" which come before.
Parent - By turbojuice1122 (Gold) Date 2007-12-11 10:38
Actually, it's not a problem at all: in graduate classes in physics and astronomy, many of my professors were brilliant, and very few were "numbers" people; for that matter, even my mathematics professors weren't all that much, at least not to the extent that they were abstract thinkers.

in fact any number is abstract

Numbers have nowhere near the level of abstraction as such things as mathematical proofs, mentally manipulating three-dimensional objects, etc.  You can easily be good at one without being good at the other, and I've known many people on both sides of the coin.
Parent - By Hetman (*****) Date 2007-12-11 21:26
That is true but... Alechin was a lawyer, Tal reporter, Tarrasch medicine doctor, Phildor composer, Fine psychologist.
Parent - - By turbojuice1122 (Gold) Date 2007-12-03 20:13
I'm pretty sure that this is a drawn position: in order for black to make any progress whatsoever, he has to be able to activate his rook into white's position.  This is impossible: since black only has two fighting pieces, he cannot move that rook to the c-file, moving it to the b- or a-files after clearing pawns is unhelpful, moving it to other ranks on the d-file is either impossible or unhelpful, and a kingside pawn storm is risky given white's artillery, and even if white can't penetrate the weaknesses created by going so, black would gain nothing from such an action: he simply has only two pieces, and these cannot win against white's three pieces.  I think that in order for black to be able to win this position, that rook would have to be a queen instead, and even then the task seems nontrivial (though in that case, a kingside pawn storm would work--smash through the pawns, bring the queens into play, and when white defends, then go to the other side of the board to eliminate pawns, and then march the black pawns, but I'm not sure this would be completely possible).
Parent - By Roland Rösler (****) Date 2007-12-03 23:24
I´m sure, your estimation is right. But I can imagine, that Rybka with White can loose this against Dmitry (maybe with a little help by engine). Material (particulary pawns) has to be reduced for a black winning line and black can give one pawn for this plan. I don´t believe, that Rybka know that. Material reduction is normally good for the weaker side.
Parent - - By Fulcrum2000 (****) Date 2007-12-03 22:29
D*mn, computer trouble. Can someone post a FEN of the endposition?.
Parent - - By Uly (Gold) Date 2007-12-03 23:55
3r2k1/p4p2/1p2p1pp/4q3/1P5P/P4BP1/2R1PPK1/2R5 b - h3 0 24
Parent - By turbojuice1122 (Gold) Date 2007-12-04 00:17
Thanks (I wanted to post this awhile ago, but my chess program is doing other stuff); this appears to be the final position of the game in question, the position that I believe to be objectively drawn.
Parent - - By Roland Rösler (****) Date 2007-12-03 20:36
Maybe Dmitry has seen the game from Levon the day before (tie-break, rapid chess):

[Event "World Cup"]
[Site "Khanty-Mansiysk RUS"]
[Date "2007.12.02"]
[Round "3.3"]
[White "Inarkiev, E."]
[Black "Aronian, L."]
[Result "0-1"]
[ECO "D43"]
[WhiteElo "2674"]
[BlackElo "2741"]
[PlyCount "96"]
[EventDate "2007.11.24"]

1. d4 d5 2. Nf3 Nf6 3. c4 c6 4. Nc3 e6 5. Bg5 h6 6. Bh4 dxc4 7. e4 g5 8. Bg3 b5
9. Be2 Bb7 10. O-O Nbd7 11. Ne5 Bg7 12. Nxd7 Nxd7 13. Bd6 a6 14. Re1 c5 15.
dxc5 Rc8 16. a4 b4 17. c6 Bxc6 18. Nd5 Ne5 19. Be7 Qd7 20. Nf6+ Kxe7 21. Nxd7
Bxd7 22. Qd2 a5 23. Rad1 Rhd8 24. Qd6+ Ke8 25. Qb6 Ra8 26. Rd6 Rdb8 27. Qc5 Rc8
28. Qb6 Rcb8 29. Qe3 Rc8 30. h4 Ke7 31. Rb6 g4 32. f4 gxf3 33. gxf3 Bxa4 34. f4
Nd7 35. e5 Nxb6 36. Qxb6 Kf8 37. Bh5 c3 38. f5 cxb2 39. Qd6+ Kg8 40. Bxf7+ Kh8
41. f6 Rc1 42. fxg7+ Kxg7 43. Qe7 Rxe1+ 44. Kh2 Rh1+ 45. Kg3 Rg1+ 46. Kh2 b1=Q
47. Be8+ Kh8 48. Qf6+ Rg7 0-1

Levon don´t like Queens, I think :-) (have a look to move 19 of black). Maybe a bluff from Levon, but I´m not sure. Inarkiev thinks he is better (Rybka too) and avoid the repetition (move 29). But I think, Levon feels fine after 29. Qe3 (for many reasons). But he will never play this against Anand, Kramnik or Topalov. Chess is a game and only the real gambler can win :-).
Parent - - By lkaufman (*****) Date 2007-12-03 22:17
     I think that this queen sacrifice was played because the alternatives seemed worse. I don't believe he would have played differently against Anand or Kramnik. You can't choose a wild unclear opening and then expect to play simple chess. You may be right that his estimate of the queen's value is lower than that of others, but such differences in evaluation between grandmasters are usually very small. Anyway his opponent in today's game apparently agreed with him, as he accepted the draw.
Parent - - By Carl Bicknell (*****) Date 2007-12-03 23:10
I'm about 2100 elo by the way so take what I say with a pinch of salt.

However, I've given quite a lot of thought to queen sacs in general and I always notice the same themes coming up again and again: the side that offers the queen sac always rids himself of any target the enemy queen can latch onto.

In this game it's the same. Assuming a pair of rooks come off there is no way for black to attack anything since the only weak point is a3 and that is easily defended by the white rook.

If you're looking for a "test position" then there are a lot of queen sacs in the Benko where white plays Nb5 and black plays Qb5 and then Ra1. His pawn structure is one long chain - with no weaknesses.
Parent - By Carl Bicknell (*****) Date 2007-12-03 23:13
Just to clarify: I don't think the problem is with the queen value alone - it must be to do with recognising certain pawn structures and the queen value
Parent - - By lkaufman (*****) Date 2007-12-04 00:18
First of all if the rooks are traded the a3 weakness is not automatically safe, as a kingside pawn advance may force the rook to abandon it. And if the rooks are not traded Black can play for attack. I had Rybka play it out (at a very fast speed) both as is and with a pair of rooks off, and in both cases the queen won quite quickly. So it seems to me that either the judgement of both top GMs was very bad (Aronian for playing the queen sac, Jakovenko for agreeing to a draw), or else they are assuming a level of defense better than Rybka can show at bullet speed on a quad. Maybe Rybka is basically right about the advantage, but the symmetrical pawn structure increases the draw margin to more than this advantage. Anyway I am very puzzled about why Black didn't at least try to play on.
Parent - By Roland Rösler (****) Date 2007-12-04 00:46
and in both cases the queen won quite quickly
I´m sure that white looses, because Rybka trade pawns on the king side and white king was without coverage. I´m sure, Aronian can avoid this with white.
Parent - - By Carl Bicknell (*****) Date 2007-12-04 07:47
If the white rook goes to the third rank are you saying that a kingside pawn advance will get the rook to abandon the ENTIRE 3rd rank? I find that hard to believe.
Parent - By lkaufman (*****) Date 2007-12-04 15:05
I don't say it's forced, but it's also not obvious that after a kingside advance that the rook will always have a safe place to move to on the third rank, keeping Zugzwang in mind. The point is that all the winning chances seem to be with Black, even if a draw is the proper result. 
Parent - - By Vasik Rajlich (Silver) Date 2007-12-04 13:21
Indeed, this is a bizarre game. I can't imagine that black doesn't at least have a nominal advantage. Maybe white can defend everything without problems, but this is not obvious to me. Again, some strong player's comments would be interesting.

Anyway, it does seem that Rybka is overvaluing black a little bit here. As werewolf says, queens need loose pieces, exposed kings, etc. It's hard to think of rules for this. Maybe the symmetrical pawns help the weaker side with an imbalance like this one.

Parent - By turbojuice1122 (Gold) Date 2007-12-04 13:47
Could it simply be that each of white's pieces and areas is relatively straightforward to defend because in spite of black's material advantage, he still needs more pieces attacking an area than that defend the area, and this doesn't seem possible?
Parent - - By rivaldo (***) Date 2007-12-03 23:41
hi Larry,

my impression of the game during live relay was, that aronians opening attempt didn't work out or he even mixed something up. already after the simple 12...c5 the game is VERY drawish. a player of jakovenkos strength will hardly lose such a position. so aronian decided by playing 15.c6 to generate the only possible imbalance in the game with no risk of losing. in fact he can't lose the resulting position after 17.Rxc6. symmetric pawns and a strong bishop garantee a draw.

now you want to make the computer understand the position. well chess is a complex game and without complex algorithms this will be a hard task.
that's my opinion. we could formulate a human algorithm for this position, but will that really help you?
Parent - By lkaufman (*****) Date 2007-12-04 00:20
It will help if the rule is sufficiently general and clear. I don't see why Aronian can assume no risk of losing, though. He should draw, but it seems to me that Black has whatever chances there are.
Parent - - By tano-urayoan (****) Date 2007-12-04 05:45
The reasons seems unknown to us. We must ask  both players. This is pure speculation in my part, maybe Aronian was "feeling" lucky because the day before he just won a game against Inarkiev where he lost the queen for some passed pawns. He seems to feels ok in imbalance positions. For Jakovenko maybe he saw the lost of Inarkiev and didnt want to risk.
This brings me my other point we are getting close to finals the nerves of the players must be very high so maybe Jakovenko choose a safe approach, we will see if this hunts him later.

Mr Kaufman another interesting game was between Akopian and Shirov were black altough a pawn down had good positional initiative, he even won the game. Do you think that game could help you in your job of evaluation or Rybka has already that kind of knowledge.
Parent - By lkaufman (*****) Date 2007-12-04 06:08
In the Shirov game White had five (!) isolated pawns vs. none, so the compensation for the pawn was obvious and of course Rybka knows about isolated pawns, though of course the penalty could always be adjusted. As for the Aronian game, I don't believe such emotions had much to do with their decisions; at this level players must be very objective, there is no tolerance for making decisions on emotional grounds.
Parent - - By Permanent Brain (*****) Date 2007-12-04 20:41 Edited 2007-12-04 20:47
For comparison, Shirov played a surprising(?) knight sacrifice in today's World Cup game against Akopian - in the 12th move of a Najdorf, so I am not sure if it could have been preparation:

r3kb1r/1pqb1p2/p1nppp2/7p/3NPP2/2NQ4/PPP1B1PP/2KR3R w kq - 0 12


After 12...exf5, White has the d5 square for the knight but Shirov preferred to immediatly trade it for a bishop, so Black lost his bishop pair advantage. Except for that, White gets only one pawn... but seven moves later he had 3 pawns for a knight. I am not sure though how much of that was forced or not. For example, it looks like Black could have replied (12...exf5 12.Nd5) 13...Qd8 to keep the bishop pair, and go for O-O. Rybka evaluates the position after this alternative 13...Qd8 as = (14.exf5), in other words she sees a lot of compensation then. But the evals after 13...Qa5 are also not big.

[Event "World Cup"]
[Site "Khanty-Mansiysk RUS"]
[Date "2007.12.04"]
[Round "4.2"]
[White "Shirov, A."]
[Black "Akopian, Vl"]
[Result "1/2-1/2"]
[ECO "B96"]
[WhiteElo "2739"]
[BlackElo "2713"]
[PlyCount "72"]

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Bg5 e6 7. f4 Qc7 8. Bxf6
gxf6 9. Be2 Nc6 10. Qd3 h5 11. O-O-O Bd7 12. Nf5 exf5 13. Nd5 Qa5 14. Nxf6+ Kd8
15. Nxd7 Kxd7 16. Qb3 Be7 17. Qxb7+ Qc7 18. Qxc7+ Kxc7 19. exf5 h4 20. g4 hxg3
21. hxg3 a5 22. Bh5 f6 23. g4 d5 24. Rxd5 Bd6 25. Rf1 Nb4 26. Rd4 Nxa2+ 27. Kb1
Nb4 28. Re1 Nc6 29. Rc4 Rab8 30. c3 Rb6 31. Re8 Rxe8 32. Bxe8 Be7 33. b4 axb4
34. Rxc6+ Rxc6 35. Bxc6 Kxc6 36. cxb4 Kd5 1/2-1/2
Parent - - By Vasik Rajlich (Silver) Date 2007-12-06 10:33
Rybka's analysis here seems about right. These types of moves are probably more of a practical weapon, it's not clear that we'd want Rybka to play this without book preparation in a tournament game.


New game
r3kb1r/1pqb1p2/p1np1p2/3N1p1p/4PP2/3Q4/PPP1B1PP/2KR3R b kq - 0 1

Analysis by Rybka 2.3.2g7 mp 32-bit bench :

2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.e5 Qb4 7.exd6 Bxd6 8.Qxf7+ Ne7 9.g3
  =/+  (-0.31)   Depth: 10   00:00:00  0kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.e5 Qb4 7.exd6 Bxd6 8.Qxf7+ Ne7 9.g3
  =/+  (-0.31)   Depth: 11   00:00:00  0kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.e5 Qb4 7.exd6 Bxd6 8.Qxf7+ Ne7 9.g3
  =/+  (-0.27)   Depth: 12   00:00:01  70kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.e5 Qb4 7.exd6 Bxd6 8.Qxf7+ Ne7 9.g3
  =  (-0.24)   Depth: 13   00:00:05  348kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.e5 Qb4 7.exd6 Bxd6 8.Qxf7+ Ne7 9.g3
  =/+  (-0.30)   Depth: 14   00:00:12  879kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.Qxb7+ Qc7 7.Qxc7+ Kxc7 8.exf5 Bf6 9.c3
  =/+  (-0.33)   Depth: 15   00:00:26  1822kN
2...Qa5 3.Nxf6+ Kd8 4.Qb3 Be7 5.Nxd7 Kxd7 6.Qxb7+ Qc7 7.Qxc7+ Kxc7 8.exf5 Bf6 9.c3
  =/+  (-0.29)   Depth: 16   00:00:52  3587kN

(Rajlich,  06.12.2007)
Parent - - By Felix Kling (Gold) Date 2007-12-06 11:47
Mmh, those kind of moves are interesting for an attacking player when analysing a position, if a piece sacrifice leads to an only slightly worse position, the engine could tell the GUI that there is a maybe objectively not that good but interesting sacrifice in the position (sth. for the GUI?). Also those sacrifices could get a slight bonus in a "antihuman" mode, since it's quite difficult to play against a attack for a human (besides of the difficulties with material imbalances).
Parent - - By Banned for Life (Gold) Date 2007-12-06 16:10
Perhaps at some point in the future an option could be included to give extra weight to sacrifices or material imbalance? Both of these may be more desirable in certain circumstances, for instance when only a win will do.

Parent - By tano-urayoan (****) Date 2007-12-06 18:41 Edited 2007-12-06 19:07
There are example of games that when a player needed a win he just played quiet positional lines for example a game between Lasker-Capablanca,(Capablanca with a tie wins the tournament) Lasker choose the exchange variation of the Spanish and won. Or Kasparov-Karpov, Kasparov needed to win in the last game to tie the match and retain the chess crown, he choose an English opening and quietly outmaneuverd Karpov. Ibermax in the final game of the last Freestyle tournament choose an English opening and also outplayed his opponent Zack S in a nice positional style.
Parent - - By lkaufman (*****) Date 2007-12-06 23:08
Actually I'm not at all sure that material imbalance increases the likelihood of a decisive result. Just look at the game at the top of this thread -- despite (or because of) the queen for rook and bishop imbalance, the players simply agreed to a draw. Apparently neither player believed he had serious chances to win.
Parent - By turbojuice1122 (Gold) Date 2007-12-06 23:44
In general, this is probably correct, but in the case of Rybka, an engine that plays the resulting positions far better even than any other engine, even Zappa, it would be advantageous at this point for Rybka games to have such a thing.  However, if there ever comes an engine that is better than Rybka in these types of positions, then such a thing would have a negative impact.
Parent - - By Banned for Life (Gold) Date 2007-12-07 00:47
I think its an interesting and maybe even an important question:

Are positions with material imbalances more, less, or equally likely to be decisive?

The proper way to answer this question would be via a database query. You know as well as I do that pointing to 1 game (where the players may or may not have made the right choice) can't be considered the last word on this question.

I'm still betting on more likely...

Parent - - By lkaufman (*****) Date 2007-12-07 01:20
I can easily get answers from a database query, but the answer will depend on how the question  is phrased. If material imbalance includes simply being a pawn or more up (with or without compensation) then surely it's more decisive. I'm assuming by material imbalance you mean that each side has more of at least one type of chessman. Now of course bishop for knight imbalance is vastly more common than any other, and is probably more likely to lead to a decisive result than otherwise, but it does not fit many people's idea of material imbalance, I think (due to the equal nominal point value). If we limit ourselves to other imbalances and stipulate that the overall material balance is close, then we are mainly talking about rook vs. minor and pawn(s), two minors vs. rook and pawn(s), and various queen for piece imbalances. In general, I think that the various queen imbalances are more drawish than normal, because if the queen side is behind the queen can often give perpetual, and if the queen side is ahead the pieces can often make a fortress. As for the rook vs. minor(s), I'm not at all sure. So it's all a matter of semantics, really. If you pose the question in a rigorous way I (or a reader) can check it out. 
Parent - - By Banned for Life (Gold) Date 2007-12-08 14:20
I was referring to positions where one side might have a rook for a minor piece and a pawn or two, but the evaluation was still roughly equal. Normalizing for how far along the game is (obviously very difficult to do), would such positions be more or less likely to end up as draws?

Parent - - By lkaufman (*****) Date 2007-12-08 16:29
I checked my database (average rating of players 2400 and above) and your theory is clearly correct -- imbalances involving the Exchange make for less draws. I considered only positions with the queens off the board, thus avoiding opening theory-based positions and partially normalizing for game state. For positions with identical material on both sides 54% draws. For positions with identical material except one side up a pawn 42.5% draws. For positions with identical material except for a bishop for knight imbalance 48%. For positions with rook for knight or bishop and two pawns (knight or bishop came out the same) 38%. For positions with rook and pawn or two pawns for bishop and knight 33%. So an imbalance of the Exchange for pawns signifi9cantly increases the chance of a decisive result, and the imbalance of two minors for rook and pawns does so to an even greater extent.
Parent - - By turbojuice1122 (Gold) Date 2007-12-08 16:42
Have you done something similar with computers on this?  I would believe that in spite of the fact that a higher percentage of human games at high levels are drawn compared with computer games, there would be a higher percentage of computer draws compared with human draws in such positions because here one side has "motivation" to search for draws in its eval, and computers can be quite efficient at doing this when it's necessary.
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