New chess players often make the mistake of advancing1I first wrote developing, but I’m not sure bringing the queen into the middle of the board at the beginning would be regarded as developing it with the queen in the early game, only to find it harried and pushed about by the opponent’s less valuable pawns, knights, and bishops. The queen, being worth 9 points, shouldn’t be traded with a knight worth only 3 points, so it is better left in the back ranks out of harms way.
But what authority does tradition have to say that a queen is worth so much more than a knight – after all, if the queen will just be left passive in the back ranks, is it really worth 3 knights? Or, a more careful challenge of chess piece values would be to ask, could a player with a radically different piece valuation system beat one using the standard values?
The answer, as suggested by the teaser quote, is to be found in the endgame. In the endgame2The best characterization of the endgame is when the board is sufficiently cleared that the king becomes a useful attacking piece; in the midgame it must avoid danger. many of the pieces are gone, and those that remain are largely able to use their full range of motion. The knight’s ability to jump over pieces is no longer so useful – it gains no new squares as pieces are removed, whereas the rook and queen gain greatly.
In the endgame the approximate relative values of the pieces becomes self evident: from a position with better quality pieces, fairly methodical play can take all the opponent’s remaining pieces with few or no losses. Here any imbalance in the pieces becomes most important: a queen using its diagonal attack can easily fork an enemy rook and king to gain the rook, but for a rook to similarly take an enemy rook while remaining out of danger requires skewering the enemy king to the enemy rook. A queen is therefore clearly stronger than a rook, and roughly evenly matched with two rooks, which by protecting each other are immune to such a naive attack. The standard piece values hold up surprisingly well even in extreme imbalances such as a queen vs three minor pieces, which can compensate for their poor mobility by the wide variety of ways they can protect each other as they move.
In the endgame of the longest ever chess game to be played in a world championship match, game 6 of the 2021 Magnus Carlsen vs Ian Nepomniachtchi match, Carlsen had a rook, knight, and two pawns to Nepo’s lone queen. Nominally this is a material advantage of 1 point, suggesting Carlsen is winning. The position reached after the final pawn exchange on move 115 is a theoretical draw but neither Carlsen nor Nepo have memorized a whole tablebase3The biggest complete chess endgame tablebase contains the optimal move for all positions with 7 or fewer pieces, counting the kings, and weighs in at 140 TB.: instead, Nepo is attempting to defend a draw while Carlsen pushes to secure a win by promoting a pawn. Carlsen demonstrates his reputation for squeezing water from a stone in the endgame in slowly, methodically, progressing his pieces up the board while giving Nepo’s queen no avenue to attack; after 136 moves and 8 hours victory is imminent and Nepo resigns.
As an aside, this illustrates a distinction between pawnless endgames and those with pawns remaining: the standard piece valuations make most sense in the presence of pawns, as promoting a pawn to a queen (usually) creates an unstoppable advantage. Typically the stronger side is able to use their advantage to advance a pawn, or force the opponent to sacrifice a piece for a pawn under threat of promotion, and in either case secure a decisive result. However in the absence of pawns the particular mating patterns that certain combinations of pieces can or cannot achieve, as well as which player is on move, matters more than the standard piece values; also draws are more likely, even with a material disparity.
So: if the standard piece values are sensible in the endgame, but the pieces change in how usable they are over the course of the game, why do we persist in using the endgame piece valuations through the whole game? Why not say a rook is worth more than a knight in the endgame but less in the midgame?
The reason is that either player is usually able to force the game to progress towards the endgame through equal trades while making only positional concessions, but in the endgame position matters much less than material strength. Suppose in the midgame game one side is up “the exchange”, say having a rook to the opponent’s knight. In the short term this seems like a bad deal, with the rook too obstructed to be as useful as a knight, but after most of the other material is traded a rook with a few outside4i.e., on the wings, rather than in the center pawns easily beats a knight with a few pawns.
As the movements of the pieces are symmetric, it is impossible for a player to offensively use a piece without making it vulnerable to attack from its opposite. Thus a player seeking equal trades can meet each offensively placed piece with its pair, forcing their opponent to either retain their advanced position and accept such trades, or concede the position and withdraw to a passive stance.
Substantial material sacrifices are therefore usually only used if they enable an immediate conclusion to the game, without reaching the endgame at all. Paul Morphy’s games are an excellent illustration: his contemporaries favored a more materialistic style, accepting sacrificed pieces even when it leaves them too bottled up to defend mate. E.g. in Morphy v Maurian 1866, Morphy opens by gambiting a pawn and knight, leading into a crushing attack that wins on move 16.
Some further examples will be illustrative.
In Leela Chess Zero v Stockfish, Leela makes an unusual long-term positional sacrifice of a bishop in the midgame. While not immediately leading to a decisive result, the move both neutralizes Stockfish’s aggressive pawn attack and enables Leela’s own attack. Ideally Stockfish would be able to trade down the pieces to enter an endgame with a winning bishop advantage, but its pieces are bottled up and unable to usefully defend.
One of my favorite games shows the limitations of using the endgame chess piece values. Rybka v Nakamura 2008 was a 3 minute game lasting 271 moves5Yes, that is both sides playing an average of 1.5 moves per second! in which Nakamura, a leading blitz player, took advantage of several weaknesses in the Rybka chess engine to secure victory. First, Nakamura brought the game into a completely closed position in which most pawns were locked against their opposition, and then sacrificed the exchange twice, until he had a bishop, knight, and 8 pawns against the engine’s 2 rooks and 8 pawns. Because of the closed nature of the board, the knight and bishop are more useful than the rooks, which only realize their full potential in an open position. The proper evaluation of the position was a draw, as neither side had any way to advance successfully, however Rybka believed it had a small advantage due its “better” material.
After much shuffling of pieces, Rybka sacrificed one of its pawns, fatally weakening its pawn structure: if it didn’t sacrifice the pawn, the game would end in a 50-move draw, but it falsely believed that it held the superior position with the two rooks and sacrificed pawn. Nakamura used his superior pieces to then capture several more pawns, as the rooks were poorly suited to defend them. As the board opened up, the rooks became stronger, but Nakamura’s strong pawn attack was more than compensation for not having rooks on an open board. He overran the computer’s position and ended with a rather rare 5 bishop checkmate.
AlphaZero v Stockfish illustrates a rare late game sacrifice of the exchange, in which AlphaZero trades a rook for Stockfish’s knight. Stockfish’s rooks and queen should be able to make good use of the wide open board, but instead are immobile behind a few pawns in the corner. Stockfish is unable to trade queens or rooks and eventually succumbs to the pressure from AlphaZero.
I first wrote developing, but I’m not sure bringing the queen into the middle of the board at the beginning would be regarded as developing it↩︎
The best characterization of the endgame is when the board is sufficiently cleared that the king becomes a useful attacking piece; in the midgame it must avoid danger.↩︎
The biggest complete chess endgame tablebase contains the optimal move for all positions with 7 or fewer pieces, counting the kings, and weighs in at 140 TB.↩︎
i.e., on the wings, rather than in the center↩︎
Yes, that is both sides playing an average of 1.5 moves per second!↩︎
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