The joys of chess – and the value of the pieces

12/21/2011 – Christian Hesse is a PhD from Harvard and Professor of Mathematics at the University of Stuttgart. He is also an avid chess fan and has written a number of articles for our news page. As a book author he has been successful with instructive and entertaining works on chess. Now one of his best is available in English. We bring you an interesting chapter to study and enjoy.

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The Joys of Chess is an unforgettable intellectual expedition to the remotest corners of the Royal Game. En route, intriguing thought experiments, strange insights and hilarious jokes will offer vistas you have never seen before.

The beauty, the struggle, the culture, the fun, the art and the heroism of chess – you will find them all in this sparkling book that will give you many hours of intense joy.

Christian Hesse is a Harvard-trained professor of Mathematics who has taught at the University of California, Berkeley (USA), and since 1991 at the University of Stuttgart. He has written a textbook called 'Angewandte Wahrscheinlichkeitstheorie'.

Chess and literature are his main hobbies, and he also likes fitness and boxing. His heroes are the ones who fall to the bottom and rise again, fall and rise again…

From the foreword by World Champion Vishy Anand: "A rich compendium of spectacular highlights and defining moments from chess history: fantastic moves, beautiful combinations, historical blunders, captivating stories, and all this embedded into a plentitude of quick-witted ideas and contemplations as food for thought."


The value of the pieces

Thus it appears that there is a reciprocal determination of value by the objects. By being exchanged, each object acquires a practical realization and measure of its value through the other object.
G. Simmel : Die Philosophie des Geldes (tr. by Tom Bottomore and David Frisby)

Even beginners know that different chess pieces have different values because of their differing strength. As they learn the game they are usually taught that compared to a value of one point for a pawn, the value of the knight, bishop, rook and queen are defined as 3, 3½, 5 and 9. These are of course approximations of the average values of pieces, since the actual value of a piece is of course dependent on the position.

More recent data-based investigations have established slightly different average values. The final row of Table 2 below documents the results of a detailed study by International Grandmaster Larry Kaufman based on some 300,000 games of chess between players of at least master level, that is with Elo-ratings of at least 2300 points. The value of the rook was set at 5 and the results have been rounded to the nearest ¼ point, as is generally the case in Table 2.

Kaufman’s conclusions are even more instructive and definitive than is obvious from the way they are summarised in the table. In the most cleverly thought-out way he has allowed the data to speak for itself and derived highly differentiated empirical knowledge from the immense treasury of master games. In the case of each of the distributions of material which he considered and based on all the games played with a particular distribution, he worked out the difference between the games’ Elo performance (performance rating) and the average Elo rating of the players (player rating). In order to avoid any distortion from side issues, some control factors were included. For example the ratings comparison was calculated separately for White and Black and then an average was arrived at. This allowed him to eliminate any skewing on account of the advantage of the first move. In addition, he only included those positions in which the specific piece distribution lasted for at least 6 half moves, in order to exclude cases in which it was present only as the temporary product of a tactical series of exchanges.

In this way, for every material constellation he arrived at a performance rating which was then converted into a number of pawns. Let’s now make a list of some of Kaufman’s interesting and extremely useful results:

  • The value of a rook pawn is on average some 15% less than the average value of the other pawns. The reason for this is that it can only capture in one direction rather than in two. This diminution of value is in general enough for it to be an advantage for a rook pawn to capture and thus promote itself to a knight pawn. This is also the case if the result of the capture is a doubling of pawns and even if there is no rook left to occupy the rook file which has just been opened.

  • Compared to two individual pawns, the value of doubled pawns is on average approximately 1/8 of a pawn less. But there are numerous extenuating circumstances. When doubled pawns arise, there is also at least a semi-open file which is created at the same time, changing the value of the major pieces. The evaluation of doubled pawns depends upon the presence or absence of the said major pieces. Empirical research demonstrates that when all the rooks are present the average diminution in value of the doubled pawns drops from approximately 1/8 to 1/16 of a pawn. When each side has a single rook, that change in value goes up to ¼ of a pawn and where no rooks are present to 3/8 of a pawn. The presence or absence of queens on the board leads to further, though admittedly tiny, modifications in the direction one would expect.

Isolated doubled pawns on semi-open files are not worth appreciably more than a single sound pawn. When they are on a closed file, the damage caused by the doubled pawns is only approximately ½ a pawn.

Other aspects which are relevant in the creation of doubled pawns and which play a part in the profit and loss account are as follows: doubled pawns arise as a result of a capture. Since most captures are made where possible towards the centre of the board, the creation of doubled pawns generally increases control of the centre, which can be evaluated as positive. On the other hand, what is negative about the creation of doubled pawns is the increase in the number of pawn islands. Another negative aspect is that doubled pawns cannot produce a passed pawn even when they form part of a pawn majority.

Furthermore, the downside to doubled pawns depends on how many pawns remain on each side of the board. If each side has 8, 7, 6, 5, 4, 3, 2 pawns on the board with an otherwise symmetrical distribution of pieces, the disadvantage of the doubled pawns lies in the range between 1/5 of a pawn (when there are 8 pawns) and ½ (when there are 2).

  • The average value of a bishop is greater than that of a knight, but the difference in value is totally determined by the additional value conferred on the bishop pair. On average, single bishops and knights have the same value, that of 3¼ pawns. This is a wonderful balance, when one considers that the same value can be attributed to two pieces which are so different in their way of moving and in their effect.

Of course, when compared to the knight the bishop has more mobility, but if the second bishop is not there alongside it (i.e. one has the bishop pair) then the side with the knight can play in such a way as to exploit the weakness that the bishop can only command the squares of one single colour.

The bishop is somewhat stronger in the struggle against a rook or – in the endgame – against pawns than is the knight. When the struggle is between a single bishop and a knight with other pieces and pawns on the board, then the side with the knight has a slight advantage when there are 6 or more pawns per side, the situation is level when there are 5 and when there are less than 5 pawns per side the bishop has an advantage of roughly 1/8 of a pawn. When there is the bishop pair, that side has an extra value of ½ a pawn. This bonus for the bishop pair is even more marked if one’s opponent does not have other minor pieces so that he can exchange off one of the bishops. On the other hand, the extra value of the bishop pair is less than ½ a pawn if more than half of the pawns are still on the board. If one has the bishop pair and the opposing bishop is a bad bishop, then one has an advantage of more or less a whole pawn. If almost all the opposing pawns are fixed on squares of a single colour and both sides have a pair of bishops, it is worth sacrificing a pawn to exchange off one’s opponent’s good bishop for a knight.


The following over-the-board situation during Kasparov’s WCh match against Short speaks volumes about his evaluation of the value of the bishop pair and is quite in line with Kaufman’s data analysis:

Let us continue with IM Kaufman’s empirical results.

  • The average value of the exchange (rook against a knight or a single bishop) is about 13/8 of a pawn. The advantage for the side with the rook is only 1 and 3/20 of a pawn if the opposing side has the bishop pair. If all the other minor pieces are still on the board, the value of the exchange drops by ¼ of a pawn. If, on the other hand, the queens and a pair of rooks have been exchanged off, it goes up by somewhat more than ¼ of a pawn. The following can serve as a rule: if one side has a rook against a knight and two pawns, then materially it is ¼ of a pawn behind. But if there is a possibility of exchanging major pieces, then it can gain a tiny material advantage. After even more intricate evaluation, it can be added that in the struggle of a rook against a knight the value of the knight increases by 1/16 and that of the rook drops by 1/8 for each pawn over and above the number of 5 on its own side. The modifications operate in reverse for numbers of pawns which are less than 5.

  • In the struggle between a rook and two minor pieces, there is generally equality if the side with the rook has 1 or 2 pawns more. Somewhat fewer pawns are required if both minor pieces are knights, and on the other hand 2 pawns are necessary if we are talking about the bishop pair.

  • The average value of the queen (if the opposing side does not have the bishop pair) is that of a rook, a minor piece and 1½ pawns. The knight is fractionally stronger than the bishop when supporting the rook in its struggle against a queen. The value of a queen and pawn is the same as that of two rooks, when no minor pieces are present. When both sides have 2 or more minor pieces, the queen does not need a pawn to equal the two rooks in value. In the situation of queen against 2 rooks with 5-8 pawns on each side, the advantage of the rooks is a tiny one; when there are at the most 4 pawns per side, the rook has an advantage of approximately ½ a pawn. A queen and half a pawn equals 3 minor pieces.

  • With international masters and those of higher playing strength, it is suggested that the advantage of having White equals approximately 40 Elo points. The value of an extra pawn with no corresponding compensation for the other side is reckoned to be roughly 200 Elo points.

The results of this empirical study can be applied in important ways. In the process of playing a game of chess, values of the pieces are continually being negotiated. Whenever there is the possibility of an exchange which will result in an asymmetrical material situation, Kaufman’s results can be of help in the evaluation and measuring of how useful it is to one side or the other. In addition, when there is already an asymmetrical distribution of material, every subsequent symmetrical exchange also leads to an advantage for one side or the other since the value of every constellation of material also depends on which pieces both parties still retain on the board. If one proceeds in this fashion, one gets a subtle feeling for the average strength of the present constellation of material. Thereafter one must turn to the question whether the actual constellations on the board for each player are better or worse than the average, i.e. whether they are faced with aspects of the position such as good or bad bishops, open lines, weak points, etc. which will lead to a change in the evaluation. In other words, the truth is on the board.

The following example demonstrates how to undertake the evaluation of a position by using the empirical results we have seen.

Kaufman,Larry C (2400) - DeFotis,Gregory (2425) [B53]
USA-ch New York (12), 1972

White to move – Position after 11...e6

Kaufman, who in 1972 was not yet equipped with his detailed empirical results, now exchanged his pair of knights for a rook and 2 pawns. According to conventional calculation, this is materially speaking a bargain since it wins a pawn. However to go into the finer details of the structure, the following must be factored in: Black’s bishop pair (a plus for DeFotis), White’s bishop (a plus for Kaufman), the presence on the board of the other pair of rooks (a plus for DeFotis), the exchange of queens (a plus for Kaufman), the pawn situation of 7 pawns against 5 (a plus for DeFotis), the number of rook pawns 2 to 1 (a plus for DeFotis).

The calculations portrayed above (without including the pawn situation) result in a slight advantage for White. But the latter has one rook pawn more than Black and all but one of the white pawns are still on the board, meaning that there are no open files for the white rooks, which reduces their value while at the same time increasing the value of the knight.

On the whole, after these corrections the material balance is more against than for White, which was proven in the remainder of the game. DeFotis went on to win the game. At first things went according to plan: 12.Nxa7 Rxa7 13.Nxe6+ fxe6 14.Bxa7


Black to move – position after 14.Bxa7

This is the position where Kaufman thought he had an advantage. Next came 14...d5 15.exd5 exd5 16.Bd4 Bd6 17.0-0 Ke7 18.b3 Bf5 19.Rfe1+ Kf7 20.Re2 Rc8 21.c3 Ne8 22.g3 Nc7, and Black won on move 42. The computer program Shredder did not see Black’s advantage either. After some four hours of thought it evaluated the above position as 0.46 pawns in White’s favour and suggested as a move for Black 14...Kc7. You too, machine?

This concern with attributing a numerical strength to the pieces is as old as the game itself. The Arab chess masters of the 9th and 10th centuries had their word to say about it. As-Suli made use of a scale of values based on the coinage in use at the time, the dirham. He standardized the value of the strongest piece in shatranj, the rook, as being one dirham, and on this scale he calibrated the values (based on his experience) as follows:

rook
1
dirham
knight
2/3
dirham
fers
1/3–3/8
dirham
alfil
1/4
dirham
central pawn
1/4
dirham
knight pawn
1/6–1/5
dirham
bishop pawn
1/6–1/5
dirham
rook pawn
1/8
dirham

As-Suli also believed that the king’s alfil was stronger than the queen’s alfil and that the f-pawn was superior to the c-pawn since it limits more strongly the opponent’s more important alfil.

His evaluation of the strength of the knight (recalculating the value of the rook to 5) lies according to his list in the vicinity of 3?. This value is comparable to the value worked out by von Bilguer and later empirically derived value of 3¼, and is far more accurate than Staunton’s suggested 2.75.

Staunton’s values (in The Chess-Player's Handbook) and those of Von Bilguer (in Handbuch des Schachspiels) are listed in the first two lines of Table 2 below.

There have also been purely mathematical attempts based on the characteristics of their mobility to define the relative strength of the pieces. One simple formula would define the relative value of a piece as the average number of squares which it threatens on the board relative to the total number of 64 squares. Alternatively, but mathematically equivalent, it is possible to estimate the relative value of a piece as the probability with which the king would be in check after a random placing on the board of a king and the piece in question. The commensurate figures (once again calibrated based on the value of 5 for the rook) can be found in the row marked Simple mobility in Table 2.

It has also been proposed that a better idea of the relative strengths based on these probabilities would be given if, after a random placing of the king and piece on the board, the piece would be giving check without the king being in a position to take it. These results can be found in the row marked Restricted mobility in the following table.

Source
Knight
Bishop
Rook
Queen
Staunton
2.75
3.25
5.00
9.00
v. Bilguer
3.25
3.50
5.00
9.25
Simple mobility
2.00
3.25
5.00
8.25
Restricted mobility
2.50
2.75
5.00
7.75
Kaufman
3.25
3.50
5.00
9.75

Table 2: Relative strength of the pieces (rook = 5), rounded to the nearest ¼ pawn.

And finally let us quote the views of Steinitz (taken from The Modern Chess Instructor), Lasker (from Lasker's Manual of Chess) and Bronstein (from The Sorcerer's Apprentice). Steinitz wrote: ‘The mathematical values of the pieces have been calculated as follows: pawn = 1, knight = 3.05, bishop = 3.50, rook = 5.48, queen = 9.94.’ Lasker’s table of values is far and away the most detailed. His unit of reference is the value of having the first move:

First move =  
1.0
Rook pawn =
0.5
Knight pawn =
1.25
Bishop pawn =
1.5
d-pawn, e-pawn   =
2.0
Knight =
4.5
Queen’s bishop =
4.5
King’s bishop =
5.0
Queen’s rook =
6.0
King’s rook =
7.0
Queen =
11.0

In contrast to this, Bronstein’s looks really rough and ready and unscientific: pawn = 1, knight = 3, bishop = 4, rook = 5, queen = 9.

In an article he wrote on this subject, GM Ian Rogers essentially confirmed the values found in the empirical database study by Kaufman, with the slight modification that the opinion of modern grandmasters tends to value the queen at 9½.

Larry Kaufman’s analysis, since it was both objective and based on the analysis of data, seems to me to be the most reliable. His statistical methods and explorative analysis allow Kaufman to distil from large quantities of data from master games the average value of the pieces. The result of each game is a single data point which makes a statement about the value of the pieces taking part. The totality of all these individual statements allows the calculation of the average piece values via the law of large numbers. In this way the strength of the pieces is extracted directly from the results of games played by competent players.

No matter which of the scales one consults, the order of the pieces according to their strength is always the same, as one might expect. Following on from there, let us finish this chapter with this beautiful five-part problem, which takes this ranking list of pieces and in exemplary fashion turns it on its head.

Krätschmer, 2001

White to move

(a) Position in diagram: Mate in 2
(b) (wNg7 instead of Pg7): Mate in 3
(c) (wBg7 instead of Pg7): Mate in 4
(d) (wRg7 instead of Pg7): Mate in 5
(e) (wQg7 instead of Pg7): Mate in 6

The solutions are:

(a) 1.g8N d2 2.Nh6#.
(b) 1.Ne6 d2 2.Nd4 Kxf4 3.Nf3#.
(c) 1.f7 d2 2. Bd4 Kxf4 3.Bf6+ Nd4 4.Rxd4#.
(d) 1.Re7 d2 2.Re2 Nd4 3.Rexg2+ Kxf4 4.Rxd4+ Ke5 5.Re2#.
(e) 1.Qe7 d2 2.Re4 fxe4 3.Qxe4 Nf5 4.Qxf3+ Kh4 5.Qf2+ Ng3 6.Qxg3#.

Five independent problems with five different key moves and the nub of it all: the stronger the piece which is on g7, the longer and the more complicated the route to mate.


Christian Hesse holds a Ph.D. from Harvard University and was on the faculty of the University of California at Berkeley until 1991. Since then he is Professor of Mathematics at the University of Stuttgart (Germany). Subsequently he has been a visiting researcher and invited lecturer at universities around the world, ranging from the Australian National University, Canberra, to the University of Concepcion, Chile. Recently he authored “Expeditionen in die Schachwelt” (Expeditions into the world of chess, ISBN 3-935748-14-0), a collection of about 100 essays that the Viennese newspaper Der Standard called “one of the most intellectually scintillating and recommendable books on chess ever written.”


Article on Christian Hesse's "magnificent trip through the world of chess" in an Austrian newspaper

Christian Hesse is married, has a ten-year-old daughter and a seven-year-old son. He lives in Mannheim and likes Voltaire's reply to the complaint: ”Life is hard” – “Compared to what?”.


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