Why Knight and (lone) Bishop are so nearly equal in value

[hgm]

Is it blind coincidence that B and N are almost indistinguishable in value? When empirically determining the value of unorthodox pieces, I also often find it very hard to decide which one is stronger. So B and N are not alone in this.

My theory of piece values predicts that the effective value of a piece in a certain material composition is suppressed by the presence of lower-valued pieces of the opponent. Why? Because the 'safe mobility' of a piece is directly affected by this. You can never stray on a square under attack by an inferior piece, no matter how well you control that square, because it will always at least give the opponent a favorable trade. This limitation makes the piece less useful. How much less will no doubt depend on the base value of the piece (Powerful pieces would have more effect on the squares where they now cannot go than weaker ones), as well as on the value of the lower piece (which correlates with the number of squares in makes inaccessible to the stronger piece).

I have not determined a quantitative formula for this yet, but the effect explains why a Q vs R+B trade is bad (for the Q side) when it was your only Q, but favorable when both sides have two other Queens (or Chancellors and Archbishops, as in Capablanca Chess, all far stronger than Rook). By removing the opponents R+B, even at the expense of an intrinsically more valuable Q, you increase the safe mobility of your remaining two Queens so much that it more than compensates. It also explains why 7 Knights beat 3 Queens. (The value of the Queens is suppressed very much (nearly a full piece) by so many opposing Knights, which interdict almost the entire board to them, once deployed.) And why with QR vs RBNN the Q side gains by trading Rooks (its own Rook is devaluated by the opponent's minors, and its Q by the opponent's R, the opponent suffering from neither of those predicaments, and both problems disappear with the Rook trade).

This effect has very interesting consequences, however, when applied to two pieces nearly equal in value. The stronger piece, which has to avoid squares covered by the weaker piece, loses some value because of this, while the weaker piece loses none. But if the 'bare' value difference is only very small, the value loss due to the need to avoid exchange with the weaker one could suppress its value to below that of the weaker one. So the stronger piece is actually worth less than the weaker one!

This of course is a paradox. The solution is that avoiding trading, causing you to to benefit less from the strong piece than the opponent benefits from his weaker piece, is bad strategy. When you don't shy away from the trading, the stronger piece can reach its full potential, forcing the opponent sooner or later to trade it (which was counter-productive to avoid). So with this strategy it will be worth the same as the weaker piece, as it will almost inevitably be traded for it (and if not, so much the better), and in the mean time it exercises its unrestricted power. So the value of a marginally stronger piece is dragged down to the value of the (opponnt's) piece just below it, but not further. As long as you treat the pieces as equal, and don't shy away from trading it.

Does this now mean that small value differences are undetectable? Fortunately this isn't the case. If you compare the pieces indirectly, by playing them not against each other, but against other material, with which it cannot be easily traded (so that they either both suffer a value loss, or neither of them does), you could see that the one with the larger 'bare' value indeed does better. E.g. when in doubt whether a Rook or a piece that moves as Bishop or King would be more valuable, play each of them against a pair of Knights (adding a pawn in the balance to make it more equal). Then both would now suffer a suppression by the presence of the Knights, so this would not distort the relative values in any way. But when they directly face each other, it might be masked which of them is stronger, because both would be ill adviced to avoid trading, so that trading would almost always occur.

This raises some questions about the B-N difference (for Larry!). This has been shown to corrlate with the number of Pawns present. But, as we all know, correlation does not necessarily mean a causal relationship. The number of Pawns correlates also quite well with total material. If we suppose the bare value of the (lone) Bishop is higher than that of Knight, the presence of opponent Knight(s) pulls its value down to Knight level, as it is doomed to be traded. But the later the game stage, the more likely it is that the number of Knights it faces is reduced, making it suffer less danger it will have to be traded for one. So that it is more likely to rise to its bare value.

So my question is: does the value of a B-N imbalance correlate in any way with the presence of the second Knight?

[Rein Halbersma]

+1 Post of the month!

[smrf]

Remember the elephantiasis effect. It is caused by the missing or decreasing of big pieces' suicide ability.

[hgm]

Yes, I remember it, and part of the theory I presented was inspired by you. And especially the 3Q vs 7N was based on your Elephantiasis effect. But the way I understood the Elephantiasis effect is that the super-piece devaluation is only part of it, and that another essential ingredient is the extreme difficulty to simplify by trading without incurring unacceptable losses. (E.g. Q vs 3N trades are impossible to force.)

[Don]

Is it blind coincidence that B and N are almost indistinguishable in value? When empirically determining the value of unorthodox pieces, I also often find it very hard to decide which one is stronger. So B and N are not alone in this.

My theory of piece values predicts that the effective value of a piece in a certain material composition is suppressed by the presence of lower-valued pieces of the opponent. Why? Because the 'safe mobility' of a piece is directly affected by this. You can never stray on a square under attack by an inferior piece, no matter how well you control that square, because it will always at least give the opponent a favorable trade.

I think that is a good theory and probably true. In Komodo the value of the bishop increases as the stage of the game is reduced - but probably more to the point is that Komodo considers safe squares in its mobility calculation. So a bishop is not "mobile" to any squares where it is up-attacked by a pawn. This takes care of pawns and the game stage takes care of non-up-attack mobility issues.

Of course all programs that do mobility will have the principle covered a little bit - because as pieces come off in general the value of the piece on average will rise.

Our major pieces also have the same consideration, a rook is not "mobile" to empty squares attacked by knights and neither is a queen. The only up-attacks not covered is queens to rook attacked squares.

Have you looked at the game, Arimaa? It's difficult to score that game because the pieces all have the same power, the difference is that a greater valued piece can "pull" or "push" a piece of lesser value. So the actual values of the pieces vary in a major way based on what is on the board, much more so that is the case with chess. A high valued piece can become the lowest valued piece on the board if the lower valued pieces are removed and it's difficult to write a good evaluation function.

This limitation makes the piece less useful. How much less will no doubt depend on the base value of the piece (Powerful pieces would have more effect on the squares where they now cannot go than weaker ones), as well as on the value of the lower piece (which correlates with the number of squares in makes inaccessible to the stronger piece).

I have not determined a quantitative formula for this yet, but the effect explains why a Q vs R+B trade is bad (for the Q side) when it was your only Q, but favorable when both sides have two other Queens (or Chancellors and Archbishops, as in Capablanca Chess, all far stronger than Rook). By removing the opponents R+B, even at the expense of an intrinsically more valuable Q, you increase the safe mobility of your remaining two Queens so much that it more than compensates. It also explains why 7 Knights beat 3 Queens. (The value of the Queens is suppressed very much (nearly a full piece) by so many opposing Knights, which interdict almost the entire board to them, once deployed.) And why with QR vs RBNN the Q side gains by trading Rooks (its own Rook is devaluated by the opponent's minors, and its Q by the opponent's R, the opponent suffering from neither of those predicaments, and both problems disappear with the Rook trade).

This effect has very interesting consequences, however, when applied to two pieces nearly equal in value. The stronger piece, which has to avoid squares covered by the weaker piece, loses some value because of this, while the weaker piece loses none. But if the 'bare' value difference is only very small, the value loss due to the need to avoid exchange with the weaker one could suppress its value to below that of the weaker one. So the stronger piece is actually worth less than the weaker one!

This of course is a paradox. The solution is that avoiding trading, causing you to to benefit less from the strong piece than the opponent benefits from his weaker piece, is bad strategy. When you don't shy away from the trading, the stronger piece can reach its full potential, forcing the opponent sooner or later to trade it (which was counter-productive to avoid). So with this strategy it will be worth the same as the weaker piece, as it will almost inevitably be traded for it (and if not, so much the better), and in the mean time it exercises its unrestricted power. So the value of a marginally stronger piece is dragged down to the value of the (opponnt's) piece just below it, but not further. As long as you treat the pieces as equal, and don't shy away from trading it.

Does this now mean that small value differences are undetectable? Fortunately this isn't the case. If you compare the pieces indirectly, by playing them not against each other, but against other material, with which it cannot be easily traded (so that they either both suffer a value loss, or neither of them does), you could see that the one with the larger 'bare' value indeed does better. E.g. when in doubt whether a Rook or a piece that moves as Bishop or King would be more valuable, play each of them against a pair of Knights (adding a pawn in the balance to make it more equal). Then both would now suffer a suppression by the presence of the Knights, so this would not distort the relative values in any way. But when they directly face each other, it might be masked which of them is stronger, because both would be ill adviced to avoid trading, so that trading would almost always occur.

This raises some questions about the B-N difference (for Larry!). This has been shown to corrlate with the number of Pawns present. But, as we all know, correlation does not necessarily mean a causal relationship. The number of Pawns correlates also quite well with total material. If we suppose the bare value of the (lone) Bishop is higher than that of Knight, the presence of opponent Knight(s) pulls its value down to Knight level, as it is doomed to be traded. But the later the game stage, the more likely it is that the number of Knights it faces is reduced, making it suffer less danger it will have to be traded for one. So that it is more likely to rise to its bare value.

So my question is: does the value of a B-N imbalance correlate in any way with the presence of the second Knight?

[Adam Hair]

Here is some data from high Elo, longer time control engine matches, using similar criteria Larry used in his material imbalance study:

Code: Select all

qr>=0p7 = 1 queen, 0 up to 2 rooks, 7 pawns 

B2 vs N2           Total Games  Centered Score     Centipawns
total                 9084           58.15           57.1
qr>=0p7               2842           55.15           35.9
r>=0p7                 902           58.5            59.6
qr>=0p6               3220           56.2            43.3
r>=0p6                1191           59.5            66.8
qr>=0p5               1568           60.35           73.0
r>=0p5                1125           63.75           98.1
qr>=0p4                559           61.65           82.5
r>=0p4                 738           64.3           102.2
qr>=0p3                                 
r>=0p3                 403           65.5           111.4
qr>=0p2                                 
r>=0p2 

B2N vs BN2          Total Games  Centered Score    Centipawns
total                76553           52.7            18.8
qr>=0p7              46066           52.05           14.3
r>=0p7                6465           55.4            37.7
qr>=0p6              24444           51.95           13.6
r>=0p6                4872           54.75           33.1
qr>=0p5               6157           54.2            29.3
r>=0p5                2407           57.6            53.2
qr>=0p4               1091           55.45           38.0
r>=0p4                 927           60.65           75.2
qr>=0p3                                 
r>=0p3                                 
qr>=0p2                                 
r>=0p2                                                               
                                 
B2 vs BN           Total Games  Centered Score     Centipawns
total                46787           54.1            28.6
qr>=0p7              14073           51.6            11.1
r>=0p7                3474           52.4            16.7
qr>=0p6              16737           53.75           26.1
r>=0p6                6622           54.85           33.8
qr>=0p5               8476           55.7            39.8
r>=0p5                5999           58.4            58.9
qr>=0p4               2813           57.95           55.7
r>=0p4                3954           60.7            75.5
qr>=0p3                603           59.45           66.5
r>=0p3                2128           60.9            77.0
qr>=0p2                                 
r>=0p2                1032           59.4            66.1

BN vs N2           Total Games   Centered Score    Centipawns
total                20479           52.4            16.7
qr>=0p7               6953           52.9            20.2
r>=0p7                1464           52.2            15.3
qr>=0p6               7571           52.35           16.3
r>=0p6                2630           51.65           11.5
qr>=0p5               3707           53.45           24.0
r>=0p5                2487           52.25           15.6
qr>=0p4               1215           52.6            18.1
r>=0p4                1717           52.8            19.5
qr>=0p3                                 
r>=0p3                 844           52.95           20.5
qr>=0p2                                 
r>=0p2                                 
                                 
B vs N              Total Games  Centered Score    Centipawns
total                64957           51.45           10.1
qr>=0p7               8201           50.45            3.1
r>=0p7                3384           47.45          -17.7
qr>=0p6              15498           51.15            8.0
r>=0p6                8739           48.85           -8.0
qr>=0p5              13530           52.35           16.3
r>=0p5               12857           50.8             5.6
qr>=0p4               7699           52.35           16.3
r>=0p4               13758           51.75           12.2
qr>=0p3               3266           51.95           13.6
r>=0p3               11546           52.6            18.1
qr>=0p2               1146           51.3             9.0
r>=0p2                8052           52.45           17.0

[stegemma]

I remember that in real games (human vs human) mobility and maybe future mobility is what player think about before exchange a knight with a bishop. Nobody think about the presence of other lower pieces but often he looks at the pawn structure. Most important is that nobody think about the fact that the needing to exchange lower pieces with our greater one is something that drop down the value of our greater piece. Maybe your idea is very new and could be somehow true but i think that only the whole position can say what piece is more important. In Satana i'm trying to adapt piece values to the position during the game but nothing works right, for now.


So, your idea could be a good heuristic... in fact, even the concept of the piece value itself is nothing more than an heuristic... so this is an heuristic raised at two power

[Vinvin]

Nice subject, but the brainstorming should be larger and concerns all pieces (rather than only N and B).
In my sens the value of piece is multi-dimensional, the best evidence is the king, he has a very big "value" (as he can't be lost) and he have a small "value" (as he moves slowly).

I made quickly draft list of concepts of piece values :
1) movement, action (be in few moves on a wishing square)
2) "price" (= classical values about 200,9,5,3,3,1)
3) "control" (control squares without moving)
4) fragility (wishing not to be attacked)
5) power (not clear in my head right now )
6) mutual protection, easiness to protect and to be protect (=coordination?)

examples :
a position seen some days ago here.
[d]kb4B1/p7/P6p/1K3P1P/3p3P/3P4/1p1N4/qB6 b - - 0 1
The trapped queen on a1 control squares, cannot moves (loses his power) but hold his value (=potential).

Another one to show mutual protection
[d]N3k2N/8/4q3/8/8/8/8/N2K3N b - - 0 1
The knights have a very low value for mutual protection...
(4 B or 4 N , far for each others vs 1 centralized queen) (or 7 N vs Q)

May be with growing power of computers, the new concepts will be more valuable because actually simple concepts are enough to define value of pieces ...

My best

[hgm]

Indeed. I did not touch on that, to not make it too lengthy a read, but of course piece values consist of multiple components. I definitely want to distinguish:

*) current tactical ability
*) non-tactical current or future advantages

The non-tactical advantage could be not immediately lose (King), promote in some distant future, convey mating potential against bare King, have the power to null-move (tactical, but not worth anything before deep in the end-game), etc.

Obviously only the tactical ability of a lower opponent piece determines how much your higher piece suffers from it, if the sum of the two terms dictates that you should avoid exchanging them for each other. This makes it possible to do interesting thought experiments. E.g. if we accept that in orthodox Chess Knight is worth slightly less than lone Bishop, and thus suppresses the value of opponent Bishops (but never to below the suppressor's own value) by its presence, we could wonder what happens if we start increasing the value of the Knight, by endowing it with non-tactical advantages. We could, for instance, equip it with the ability to null move. Or let it promote to a Knight that has an extra 1-step orthogonal move, when it reaches last rank. This would not affect its tactical ability in the middle-game, and thus presumably also no how much a trade-avoiding strategy would suppress the Bishop value. But it does affect whether would need a trade-avoiding strategy when handling the Bishop. Against an augmented Knight (N+) that is intrinsically more valuable than B, the Bishop would reach its true potential, because now it is the N+ that would have to worry about avoiding an exchange for B. This would suppress its value, so that its practical value would be dragged down to that of the Bishop, and tiny increases in its non-tactical value would be masked. Until the difference gets so large that it 'breaks loose', and even the full penalty of a trade-avoiding strategy puts it above Bishop. It is like the practical values 'snap' together, if they get too close in the process of continuously tuning the intrinsic values past each other.

The other factors you distinguish seem to go beyond piece values. In my terminology, the latter do not constitute just any eval term that is affected by the piece, but only those eval terms that are independent of the location of the piece and other pieces, so they depend on mere presence only. This is not to say that such terms would be unimportant, of course.

But distinguishing control and agility could be very useful for divergent pieces (which move differently from how they capture, e.g. Pawns). There agility would be determined by the non-captures, and control by the capture moves. Which could be quite different. (E.g. a piece that captures as a Queen and moves as a Knight is worth about 7, one that moves as Q and captures as N only ~5.) I expect the amount by which the presence of such a piece would suppress the value of higher pieces (such as Q) would depend on a weighting of these two properties (agility and control) that is about the same as the weighting determining the tactical piece value, so that you can compute the suppression from the latter, and don't need to distinguish the two aspects.

[Vinvin]

...
Have you looked at the game, Arimaa? It's difficult to score that game because the pieces all have the same power, the difference is that a greater valued piece can "pull" or "push" a piece of lesser value. So the actual values of the pieces vary in a major way based on what is on the board, much more so that is the case with chess. A high valued piece can become the lowest valued piece on the board if the lower valued pieces are removed and it's difficult to write a good evaluation function...

Amazingly, when I read the post for the first time, I thought about Arimaa game too !
I played Arimaa for some months, but I still too much chess addict