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Almost everyone knows the standard rules of thumb: a minor piece is worth three pawns, a rook is worth five pawns, and a queen is worth nine. (I’m aware that this is debated; in particular many theorists think the exchange is worth less than two pawns, but at any rate this is the most famous and well known set of values).
When were these consensus material values first stated? Who came up with them, and how?
In Claude Shannon's paper of 1949, he quotes those values as part of his evaluation function:
Most of the maxims and principles of correct play are really assertions about evaluating positions, for example: -
(1) The relative values of queen, rook, bishop, knight and pawn are about 9, 5, 3, 3, 1, respectively. Thus other things being equal (!) if we add the numbers of pieces for the two sides with these coefficients, the side with the largest total has the better position.
(2) Rooks should be placed on open files. This is part of a more general principle that the side with the greater mobility, other things equal, has the better game.
(3) Backward, isolated and doubled pawns are weak.
(4) An exposed king is a weakness (until the end game).
These and similar principles are only generalizations from empirical evidence of numerous games, and only have a kind of statistical validity. Probably any chess principle can be contradicted by particular counter examples. However, form these principles one can construct a crude evaluation function. The following is an example: -
f(P) = 200(K-K') + 9(Q-Q') + 5(R-R') + 3(B-B'+N-N') + (P-P') - 0.5(D-D'+S-S'+I-I') + 0.1(M-M') + ...
He doesn't cite an explicit reference for these values, but appears to treat them as well-known. He does cite three obviously chess-related books published from 1937 onwards.
However, Nimzowitsch's My System was first published in 1925, and it is not immediately obvious that specific relative values are assigned to pieces; a text search for "piece value" yields only oblique references to the idea that a rook is so much more valuable than a pawn that the former should not be tied down to defend the latter. With that said, My System is a textbook about positional play, so could be said to have moved beyond simple material analysis.
Also first published in 1925 was Lasker's Manual of Chess, which starts from the very basics - the form of the board and the rules of moving pieces. Here, we do find a numerical description of piece value, near the end of the "first book":
We rivet our attention on the games of the experienced […] and among them certain regularities show very plainly. […] Hence, we know that ceteris paribus (all else being equal) knight and bishop are even, either is ceteris paribus worth three pawns, rook ceteris paribus as strong as knight or bishop and two pawns, queen very nearly as strong as two rooks or three minor pieces.
From this prose, we can extract B=N=3, R=5, Q is a little less than 10 (2xR) or 9 (3xB/N).
He then goes on to point out some situations where the qualification ceteris paribus is most definitely not true. But again, it is not immediately clear from the text whether Lasker was the first to explicitly write these values down, or whether he himself learned them from elsewhere.
A subsequent answer notes that Staunton published a similar set of values in 1847, but essentially quotes Q=10 instead of Shannon's value of 9; these values in turn appear to have been established even earlier. So we can see that Lasker may have obtained these piece values from Staunton (a very influential figure in chess, so Lasker would certainly have read him) and, before writing his own chess manual three-quarters of a century later, revised them based on his own experience.
It appears that Lasker revised his own values yet again for a later 1947 work, to values somewhat different from Shannon's: B=N=3.5, R=5, Q=8.5.
It is also worth noting that modern chess engines sometimes choose a different set of values entirely, especially when they are self-optimised. Stockfish uses N=4.16, B=4.41, R=6.625, Q=12.92, which roughly corresponds to devaluing an individual pawn more than anything else. Nevertheless, the "standard" values appear to have remained reasonably stable through the late 19th century and most of the 20th.
Before 1847
The value of the chess pieces was established in the XIXth century. The simple scheme 1-3-3-5-10 (rather than 9 which is common nowadays) was already accepted in the first half of the century (and maybe introduced even earlier, we gotta check Philidor, Stamma or even the Italian school).
I was surprised I couldn't find any reference to it in Bilguier & Von der Lasa's Schach Handbuch, published in 1853 and the main reference for decades - but maybe I haven't checked the right pages. Edit: Indeed, @A.Thulin found the value of the pieces in the Handbuch and mentioned it in a comment.
Anyway, Howard Staunton gave values to each chess piece in his own Chessplayer's handbook published in 1847. As you can check on page 34, the short 4th chapter is entitled On the relative value of the chess pieces and offers these very sophisticated numbers:
Pawn 1.00
Knight 3.05
Bishop 3.50
Rook 5.48
Queen 9.94
While Staunton admits that the precision of these values should not be exaggerated, it is obvious from his comments that many writers had already offered their own estimates beforehand. Most of the chess knowledge before Staunton spread through newspapers, so that might be where the first quantitative estimation of the value of the chess pieces originated.
