In a pair of recent posts, I introduced some important references for the theory of material imbalances in chess: Kaufman's Material Imbalances and Notes on Material Imbalances. I ended that second post with a reference to Soltis's 'Rethinking the Chess Pieces'. The primary value of Soltis's work is in its numerous examples of unusual material that appeared in master play, where he documented the positions and analyzed their continuations. To put his examples into context, he made nearly two dozen references to Kaufman's earlier work. These provide avenues for further investigation.
p.17 - In 'Rethinking', as are the other page references below.
Jose Capablanca acknowledged the transitory nature of piece values when he wrote, in 'A Primer of Chess', that a Knight becomes weaker as pieces are traded off but a Rook becomes stronger. Timoshchenko quantified this, saying computer programs should be adjusted so that a Knight increases in value 3 to 5 percent after a trade of Rooks and decreases 5 to 10 percent after a swap of Queens.
American IM Larry Kaufman, in his own database survey of 300,000 games, concluded Capa's observation about Rooks and Knights was true about Pawn exchanges, not just piece exchanges. Soviet-era trainers gave this a different slant: Bishops increase in value compared with other pieces as the game goes on, they said. Even a casual look at the board can tell you that.
If we can't trust the chart [1/3/5/9], what can we rely on? Today's masters try to answer this in either of two ways. Some argue that the chart is skewed and the numbers just have to be tweaked. Kaufman, for example, said a Queen should be seen as worth 9.75 units. A Knight is worth 3.25 but that should be increased by 1/16th, he said, for each of his Pawns more than five that is traded.
Even if this is mathematically correct it is virtually useless when you are playing a game of chess. The other approach, taken by some top grandmasters, is that charts, rules, and guidelines do not matter - only calculation does. But calculation has to be based on some element of evaluation. Chess is more than visualizing a position four moves ahead.
I suspect we can get better answers than this in judging and appraising material - and to start we need to examine what goes into a piece's value.
Kaufman's database statistics indicated a normal Rook Pawn is 0.15 less valuable than other Pawns. It simply improves in value after it shifts one file toward the center.
p.37 - Timoshchenko's work is referenced in my 'Notes' post.
Timoshchenko's database survey indicated that the average Bishop is equal to the average Knight when there are· four Pawns each on the board. If there are more Pawns, the Knight is better. Fewer Pawns, the Bishop is better. (Kaufman's survey said the break-even point was five, not four.)
Leaving positional considerations aside for the moment, Kaufman's statistical analysis of 300,000 games offered this conclusion: Every "even" exchange is apt to favor one side or the other, sometimes by a substantial amount.
This sounds paradoxical but is consistent with what we know about the way piece values change in the course of the game. They change because trades of pieces and Pawns will improve or decrease the value of the material that remains on the board. We can readily see that in certain endings, such as with Rooks and Bishops of opposite color. The presence of Rooks gives the superior side more winning ideas, such as mate or an Exchange sacrifice.
Kaufman's observation also applies to many early and late middlegame situations. For example, a player with the two Bishops generally benefits by trades of Rooks. This can be explained by saying that the trade cuts down Rook counterplay, clears the path for his King and allows the Bishops to dominate.
p.110 - 'Tournament chess is not played with fractions', unless the player is an engine.
Some imbalances are fairly "even" but rare, such as Queen -vs.- three minor pieces. The Exchange (Rook -vs.- [minor] piece) is not close to even but increasingly common today and that makes it an appropriate place to start. What would make it "even"? Tarrasch offered the most widely accepted equation when he put the difference between Rook and minor piece at 1 1/2 Pawns. He said this "holds more for the endgame, not for the opening and the first part of the middlegame."
That is the most widely accepted formula today but the debate has been going on for generations. Sarratt, Staunton and later Capablanca, felt two Pawns was just about right. Petrosian said one Pawn. Steinitz said a Rook was slightly superior to a Knight and two Pawns but slightly worse than a Bishop and two Pawns. Purdy said proper compensation depended on the total number of Pawns on the board. Kaufman's database survey concluded that the Exchange was worth one and three quarter Pawns -- and this was reduced to one and a quarter if the player with the minor piece held the two Bishops.
But tournament chess is not played with fractions. Most players translate Tarrasch into a guideline: One Pawn may be sufficient compensation, two Pawns almost always is.
In endings a trade of Rooks is likely to benefit the player with the Exchange. But as noted in the diagrams on p.60 and p. 112 [both endgames] there are exceptions here too. Kaufman said the value of the Exchange becomes more than two Pawns when Queens and a pair of Rooks are off the board.
Trading minor pieces is likely to help the player with the Rook. But there is a major exception: when it is a Bishop-for-Knight trade that gives the opponent the two Bishops.
p.148 - The reference to Neishtadt isn't sourced. Neither is the reference to Spielmann, although his name appears as often as Kaufman's.
Perhaps the most accurate conclusion is that the Rook and piece need one and a half Pawns to balance the Queen. "In order to draw a fine line between profit and loss it's necessary to cut a Pawn in two," Neishtadt wrote.
Spielmann felt this was true if the minor piece was a Knight. If it was a Bishop, one Pawn was enough, he said. Kaufman's database indicated the difference between the two pieces was marginal.
This is a crucial point that can't be emphasized enough: extra pieces usually help the Queen more than the Rooks. Kaufman's survey found that the chart-based claim of Queen+Pawn being equal to two Rooks "is only true with no minor pieces on the board. With two or more minors each, the Queen needs no Pawn to equal the Rooks."
The addition of other material helps the Queen so much that Kaufman claimed that if you begin a game with one player removing his Queen and the other his Rooks, the Queen would have a big edge. (Staunton and Sarratt would surely have agreed.) This is something the reader can test for himself.
There is no better imbalance than Queen -vs.- pieces to illustrate Kaufman's conclusion about trades - "every 'even' exchange is apt to favor one side or the other, sometimes by a substantial amount." Trades will generally hurt the Queen as we saw in the second diagram on p. 162. But there are exceptions, based on the availability of targets and the degree of piece cooperation.
This provides support for another generalization, suggested by Kaufman. Based on his database survey, he found that the two Bishops were worth almost the same as a tempo. So, if you have to decide whether to spend a tempo on retreating a Bishop, or trading it off, "it's a tough call," he wrote in Chess Life.
Sacrifices such as 9...Nd4 are sound when two conditions are met - (a) the opponent's Pawns are fixed on one color and (b) he must give up his good Bishop for a Knight. The result is that the sacrificer obtains the two Bishops and his opponent has a bad Bishop. The two factors, added together, are worth a Pawn. (This has been confirmed by Kaufman's database survey.)
"Once a player no longer has both his Bishops, Knights and Bishops become practically equal," Purdy wrote. Kaufman's database survey came to the same conclusion: "An unpaired Bishop and Knight" -- such as in positions in which the minor pieces are Bishop+Knight -vs.- two Knights or Bishop -vs.- Knight -- "are of equal value."
But, Kaufman added, a pair of Bishops are worth on average an extra half Pawn. Timoshchenko's earlier database offered evidence for this. It found that two Bishops scored at least 62% against Bishop+Knight when each side has at least two Pawns each.
Trades of Pawns will, of course, change the value of Bishops and Knights. Timoshchenko came to the conclusion that Knights decrease by three to five percent after each pair of Pawns are swapped. Kaufman went further and concluded that the two Bishops were worth less than half a Pawn when less than half of the 16 Pawns had been traded -- but more than half a Pawn when most of the Pawns have been traded.
Spielmann felt that two Bishops+Rook+Pawn for two Rooks+Knight is even. Kaufman's database confirmed that but found that if a pair of Rooks is traded, the second player is slightly better. (His redundancy has been reduced.)
Nevertheless, the superiority of Queen+Knight is vastly overstated. When the Pawns are flexible and the Bishop is not bad, the edge is slight. Kaufman said the difference was "trivial" in his database survey.
p.191 - The January 2003 Chess Life is a resource I had overlooked until now.
Kaufman tested Lasker's chart in another way, as he reported in the January 2003 Chess Life. Kaufman used 25 different versions or settings of strong computer programs to play matches of 50 games to see what would happen if a game began with missing Pawns. He concluded that Lasker vastly overrated the value of center Pawns and "somewhat" underestimated the value of Rook Pawns but was otherwise fairly accurate.
Kaufman found that a player with an extra piece -vs.- three Pawns will have a winning advantage if he also holds the two Bishops. It would only be a slight disadvantage if the difference is four Pawns, he added.
This led some players to think that three pieces naturally dominate the Rooks regardless of extra Pawns. Yet Steinitz said two Bishops+Knight is only slightly better than two Rooks. Kaufman's database survey found evidence to support that but he said the superiority of the pieces was solely due to the two Bishops. If the imbalance is two Knights+Bishop -vs.- two Rooks there is no appreciable edge, he said.
The many references to Spielmann need to be pursued, as does the Kaufman article in the January 2003 Chess Life.