Is the Queen Worth 9.94 Pawns?

Chess players are obsessive creatures.  One player may make it his life’s ambition to become an expert on some arcane opening line, say the Soltis Line of the Yugoslav Attack of the Dragon Variation of the Sicilian Defense.  There is actually a page in Modern Chess Openings for that very thing.

Chess, like baseball, has a safe haven for every passionate interest, regardless of how odd it may be.  If you can conceive it, dream it, feel it, or just plain make it up, there will be games for you to study and others who will even share your compulsive disorder without thinking it in the least bit unusual.

Some people are attracted to chess through the light of mathematics, as a bug to a flame.  I myself have succumbed on occasion to this weakness, as some of my past blogs can bear witness.  One of the simplest concepts to understand in the intersection of chess and mathematics concerns the value of pawns and pieces. 

All newcomers to chess today are exposed to the ‘standard’ pawn values of the pieces.  Knights and Bishops are worth 3 pawns each, while a rook is worth 5 pawns and the Queen 9.  His Majesty the King, like all good despots, is regarded as invaluable and generally spared from such mundane comparisons.

After the newcomer has learned enough chess to lose with some semblance of honor, he or she discovers that Bishops are typically valued slightly more than Knights and, with even more experience, that a well-placed Knight on a protected outpost may be worth a Rook.  It all depends on the position.

But we all want to know what value the pieces really have.  Is the Queen worth 9 pawns, or is she really worth 9.94?  Better minds than mine have studied this question, and arrived at interesting answers. 

Let us begin with W. S. Kenny, who translated Philidor’s Analyse in an 1826 book to which he appended a Practical Description of Chess that says, “The arrangement of the pieces, according to their real powers, is as follows:  Pawn, Knight, King, Bishop, Castle, Queen.  The power each may be said to possess is about Pawn 2 1/5, Knight 6 ¼, King 7 ½, Bishop 9 ¾, Castle 15, Queen 23 ¾. … If the pawn’s chance of promotion be taken into the question, his value will be 4 7/8.”

Kenney’s values are distinct from, but nonetheless similar to, those given in Peter Pratt’s book Studies of Chess:  Pawn: 2, Knight: 9 ¼, Bishop: 9 ¾, Rook: 15, Queen: 23 ¾.   Pratt also gave a boost to pawns due to their power of promotion, but he gave them an ultimate value of 3 ¾ vs. Kenney’s 4 7/8.

In 1808 a book was published in London entitled Mr. Hoyle’s Game of Chess including His Chess Lectures with Selections from Other Amateurs.   This is the very same Hoyle who became famous for his book on games.  In this book Edmond Hoyle cited the exact same values as those that appear in Pratt above, including the boosted value of the pawn.  However, given that Hoyle died in 1769 he presumably wrote his book before Pratt wrote his in 1803.  Thus, it would appear that Pratt was citing his numbers quite literally “according to Hoyle”, as the cliché goes.

A 6^th English edition of Philidor’s Studies of Chess published in 1825 includes supplementary material by an editor whose name does not appear in the book.  This editor devoted nearly 100 pages to a mathematical analysis of the piece values, which he computed to be:  Pawn: 1.01, Knight: 3.23, Bishop: 3.62, Rook: 5.55,  Queen: 10.1  He stated that, “The pawn has, in its capacity of promotion, a dormant value, which is included in the estimate.”

My blog title refers to the queen’s value as 9.94 pawns, which is usually attributed to the famous Howard Staunton who gave the following values in The Chess-Player’s Handbook in 1847 and The Blue Book of Chess in 1870:  Pawn: 1.00, Knight: 3.05,  Bishop: 3.50,  Rook: 5.48,  Queen: 9.94.

Staunton, however, at least had the good sense to write that "the student...will find in practice the relative worth of his soldiers is modified by so many circumstances of time, opportunity, and position, that nothing but experience can ever teach him to determine accurately in every case which to give up and which to keep."

Wilhelm Steinitz adopted the same values as Staunton in his 1889 Modern Chess Instructor, Part I where he wrote that the values are an “approximate valuation”, which seems ironic to me given that they are cited to two decimal places.

Interestingly, it is likely that Staunton did not compute these values himself, but was simply adopting the values that were published earlier by Charles Tomlinson in his 1845 book Amusements in Chess.  Tomlinson used those very numbers in a ten-page chapter devoted to the values of the pawns and pieces.

To me one of the more fascinating valuations was devised by Hermann Vogler, who explained in the January 1916 edition of the Schweitzerische Shachzeitung that, “Nous trouvons la valeur absolue en cherchant le nombre des coups possibles d'une figure à partir de chaque case de l'échiquier et en additionnant les nombres des 64 cases.” (We find the absolute value [of a pawn or piece] by determining the number of possible moves of a figure from each square of the chess board and adding the numbers of the 64 squares.) 

Thus, from a1 a knight can move to 2 squares so we begin with 2.  To that we add the number of knight moves from a2, which is 3, from a3, which is 4, and so on for all of the squares on the board.  The final sum for knight is 336.  Doing this for the pawn and for all other pieces results in values of Pawn: 140, Knight: 336, Bishop: 560, Rook: 896, Queen: 1456, and King: 420.

So how do all these systems compare with the ‘standard’ values as given today?  It is easy to normalize all of these systems by dividing the piece values by the pawn values, in this manner converting all systems to the same scale where the pawn is worth 1.  The result is shown in the following table, using the unboosted pawn values for both Kenney and Hoyle:

 It would appear that Tomlinson’s system most closely approximates the modern values in common use today.  Interestingly all of these alternate systems value the Queen beyond her customary nine pawns.  There are numerous other systems out there, as well.

 Can you imagine trying to determine which side will obtain the advantage in a complicated position after four or five exchanges if you have to sum up fractional values or values to two decimal places with the clock ticking in your ear?  I, like most others, I suspect, will continue to use the modern values with a small mental boost for the bishop.  In any case, when you are thinking of these values, remember to choose your move carefully, in chess as in life.