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Ideal Values and Practical Values (part 3). More on the value of Chess pieces.[All Comments] [Add Comment or Rating]
Joe Joyce wrote on 2011-04-25 UTC
To continue messing with piece values, let's look at different boards. The piece values given are based on a more or less standard 2D chessboard, rectangular in shape. Change that in any significant way, and you've changed the piece values. Take a 1D board. There are 2 ways you can 'cut' a 1D board from a standard 2D board, orthogonally or diagonally. Consider the value of a rook and a bishop on each board for the slider rook and the slider bishop - not the jumping bishops of One Ring Chess [LLSmith] for example. One one board, the bishop can't move, and the rook has unlimited [except for blocking pieces] movement. So on average in these 2 systems, the rook and bishop are exactly equal. This is carried over into some games played with standard pieces on a diamond-shaped board, where the rooks and bishops trade the number of squares they can move to without trading any other aspect of their respective moves. Clearly this increases the value of the bishop and decreases the value of the rook. 

Now consider the knight's move in 2, 3, and 4 dimensions. That move, the only 2D move in standard chess, explodes in higher dimensions - attacking 8 squares in 2D, 24 in 3D, and 48 in 4D, if I did my numbers right. Beyond that, it takes a Dan Troyka, a Larry Smith, or a Vernon Parton to go. But it's easy to see the value of the knight rises considerably in comparison to, say, the rook. 

So, to sum this up, we seem to have established that the value of a piece depends on both the kinds of other pieces on the board with it, not just the number, and it also depends on the shape of the board it's on. 

Does any piece have an intrinsic value? :)

George Duke wrote on 2011-04-25 UTC
Joyce's ''chessness'' of a cv was a comment once by Gifford:

Joe Joyce wrote on 2011-04-24 UTC
Happy Easter, all!

This is a totally fascinating topic I can't stay away from, even though I am terrible at it. I seem to be much better at asking questions and confusing the issue than I am at answering questions and casting light. Well, everyone has a role in life. 

David Paulowich recently commented that I believe the modern elephant, FA, is worth about half a pawn less than a bishop, and I fully agree, on any board they are liable to play on together. But if we change the rules a bit, maybe that answer changes. 

Mike Nelson made a comment [in 2003?] about pieces having a value that is relative to which other pieces are on the board, and gnohmon picked up on it a little. I'd like to take that idea, maybe add a little to it, and run with it, full-tilt, right over a cliff, or two or three. 

Let's start by asking what is the value of the queen in a multi-move game? There are various types of multi-movers, each of which may have a different influence on the piece values. A Marseilles variant has to play differently than a progressive variant. Are the piece values in all 3 games the same? Would the rooks get out faster in progressive, or not at all, because the game is over on a 5 piece attack? 

The game I recently posted can be considered a large Marseilles variant, with batteries. The batteries need to be charged for the piece to move. A king charges the battery of 1 piece that starts the move within 2 squares of that king. What is a queen worth under these conditions? It has unlimited movement, once. If it is then stranded, what happens to its value? Clearly, it becomes seriously reduced, as do all the other long range pieces. They act as slightly variable short range pieces, or as a one-shot missile. 

Here, let me suggest we have another potential measurement for the 'chessness' of a variant. On that scale, which can in theory be computer-evaluated numerically by someone like HG Muller, both Warlord and Chieftain show some distance. But I submit that it is likely Marseilles will show a little distance, and Progressive a fair bit more. While I haven't played either variant, one thing seems apparent, and that is when you expose a major piece, you are very likely to lose it before you can move it again. I'd think especially in progressive, all the pieces become 'one-shot', and in that sense, the values of the pieces contract toward each other along their range of values, or alternatively, and maybe more likely, all [non-royal] piece values fall toward 1, and the fall is Aristotelian - the more valuable pieces fall faster.

Jörg Knappen wrote on 2011-04-19 UTC
In fact, if there is any difference in the values between Grand Rook (RAF) and Chancellor/Marshal (RN), then I expect the Grand Rook to be stronger, because it cannot be approached by the opponent's King.

David Paulowich wrote on 2011-04-18 UTC

Ralph Betza wrote: 'The Chancellor is roughly equivalent to the Queen even though the ideal value of N is presumably less than Bishop: the Bishop is colorbound and its practical value is ever so slightly more than a Knight, combining it with R removes the colorboundness, and therefore is a classical case of 'combining pieces to mask their weaknesses and thus allow their practical values to be fully expressed'; and therefore one might expect the Q to be worth notably more than the Chancellor.'

'One hypothesis about why the Chancellor does so well is that the R has a weakness that is masked when N is added to form Chancellor. This weakness would be its relative slowness and difficulty of development, and perhaps its lack of forwardness (it has only one forwards direction).'

P=100, N=300, B=300, R=500, C=850, Q=900 are my preferred values on 8x8 boards. Sometimes I like to say that the value of a Bishop is 5/8 times that of a Rook on any square board, which would bump the Bishop up to 312.5 points here (an insignificant change, which does not affect my strategy when playing FIDE chess). Back in the 1990s I used to debate the relative values of Queen and Chancellor with Betza. Years later I came up with a way to compare these two pieces indirectly, by introducing some new pieces. The 'Elephant' moves like a Ferz or an Alfil, and is worth 50 points less than a Bishop (I believe Joe Joyce agrees with me on that). The 'Grand Rook' moves like a Rook or an Elephant, and is worth 50 points less than a Queen (I am casting my solitary vote for that value). The Grand Rook and the Chancellor are similar enough in design that I would expect them to have the same value on any square board.

Jeremy Lennert wrote on 2011-04-18 UTC
As I said, it's a wild guess, but the basic reasoning is that each of the
following seems it should be significantly stronger than the next:

Wazir > adding null-move to King (endgame value) > adding null-move to King
(average value) > triangulation on every piece in your army > triangulation
on a single piece

Empirical testing of the switching weakness is a nice idea, except that I
suspect that its effect is much smaller than all sorts of other effects
that we don't know how to control for.  For example, if you compare NW to
NF or ND (as Robert Shimmin did earlier in this thread), then NW is
switching while the others can triangulate, but that's certainly not the
only difference (and in fact, Shimmin's test indicated that NF > NW > ND,
which proves that *something* has a larger effect on his test values than
switching does).  Betza suggested that the NW's aptitude for perpetual
check may be relevant, and also the NF's ability to escort a Pawn unaided;
I think someone else suggested the NF gains because a Pawn cannot make a
stealthy attack against it.  Forwardness and mobility also differ between
the listed pieces.  Speed may also be an issue, since it takes a N three
moves to simulate a W move (in an open position), but only two to simulate
F or D.  And there's probably another dozen factors that each MIGHT be
more significant than switching.

So I can't imagine how you'd construct a *controlled* test of
triangulation/switching without a much more comprehensive theory of piece
values than we currently have.

Investigating the value of a null move should be easier, and it obviously
must be at least as valuable as losing a tempo to triangulation, so that
might give an upper bound (though perhaps a very loose one).  It wouldn't
tell us if switching is a disadvantage for some reason we haven't thought
of, though.

H. G. Muller wrote on 2011-04-18 UTC
I am not sure I entirely follow your line of reasoning leading to the very small value of the 'triangulation' ability. And even if I would, it might not be valid, because piece values need not be strictly additive. (E.g. A Queen usually beats 2 Knights easily, but 3 Queens badly lose against 7 Knights, all the in presence of Pawns.)

The Wazir is indeed worth 130-140 centi-Pawn.

I should perhaps point out that I don't believe anything Betza claims here, unless empirical testing happens to show the same thing as he claims. For the simple reason that his statements are purely based on theoretical considerations that do not surpass the level of educated guessing, and his empirical testing, done in Human play, is not statistically significant. (As this would require tens of thousands of games.)

So it would not come as a big surprise to me if testing would show that color-switching has no significant impact on the value of a piece at all. An interesting test for this could for instance be to play augmented Knights against each other, which get a single extra King move, either orthogonal (preserving the color switching) or diagonal (breaking it), and then see which performs better, and by how much. (Or play Knights where one move is replaced by an Alfil or Dababba move against normal ones.)

Jeremy Lennert wrote on 2011-04-18 UTC
Adding the null move to your King is certainly very helpful in a number of endgames...but so is having virtually any extra piece. For example, I suspect a Wazir added to either the K+R vs. K0 or K+P vs. K0 endgames would easily turn them back into wins (probably even if you crippled it by, say, removing its sideways capture; though I have not worked it out). Based on Betza's theory, a Wazir should be worth around 1.5 Pawns at most (probably less).

And those endgames show the null move at its strongest; averaged over the course of the whole game, is it worth even a quarter that much? A tenth?

On the reverse side, adding a null move to your King is obviously more than enough compensation for having your ENTIRE ARMY saddled with the 'cannot lose a tempo' weakness due to switching. That's regardless of the size of your army, which shows that giving the weakness to multiple pieces can't possibly be linear, but still, on a single piece it has to be worth only a tiny fraction of the null move.

So at a wild guess, we're talking about maybe a one centipawn penalty for the switching weakness, or even less? That's noise.

H. G. Muller wrote on 2011-04-17 UTC
I largely agree with everything you say there. It would indeed be very interesting to see if there is an advantage on equiping a color-switching piece with an extra null-move. Unfortunately Fairy-Max cannot be configured to include null moves on pieces; I would have to program the capability to null-move separately.

If the right to null-move would be granted to a side irrespetive of the pieces it still has (equivalent to giving the King an extra null move), I am convinced it would be a very significant advantage, as Rooks lose their mating potential against such a King. And Rook endings are the most common endings in Chess. So it would bring in many draws in otherwise lost positions. This despite the fact that the King in itself is not even color-switching. (And in KPK it would also help a lot, both for the attacking and defending side!)

Jeremy Lennert wrote on 2011-04-16 UTC
Interesting...I suppose I can believe that's a weakness, though I'd imagine it's quite a small one. The difference of giving this disadvantage to every piece in your army must be worth substantially less than the privilege for a player to make null moves--I wonder if anyone has made a reasonable estimate of that value?

Is that a disadvantage specifically of switching pieces, though, or are they just the most extreme case on a sliding scale based on the length of the shortest odd cycle the piece can make? It seems intuitive that a hypothetical piece that takes, say, 9 moves to return to its starting square is much less likely to be able to lose a tempo in practice than a piece that can do it in 3 moves (though I could be wrong).

That would also be interesting because it suggests pieces that capture and move in different patterns should be affected if and only if their non-capturing move is switching, regardless of their capture pattern.

H. G. Muller wrote on 2011-04-16 UTC
Well, how about this then:

with color-switching pieces it is not possible to lose a tempo, to bring the opponent in zugzwang in an end-game. Of course there will also be color-bound pieces that suffer from the same problem, because the are meta-color switching. But compared to other non-color-bound pieces it is a weakness.

Jeremy Lennert wrote on 2011-04-15 UTC
There are many non-switching pieces that also can't mate, though--and by your suggested definition of 'switching', there are many switching pieces (though not *color*-switching pieces) that can.

And Betza lists both can't-mate and color-switching as *separate* weaknesses in his list (IV&PV part 4).

So I still can't figure out why anyone believes that color-switching is a weakness, as opposed to merely an interesting trait.

H. G. Muller wrote on 2011-04-15 UTC
I think that color-switching in the sense of true colors is a disadvantage, because it guarantees the piece has no mating potential with and against an orthodox King.

Your first definition seems the correct generalization: if there are no loops of odd length, the accessible squares should break up into two disjunct sets, one set being reachable only after an odd number of moves, the other after an even number. Of course there could be higher-order color switching schemes, like for a piece that only moves N, W and SE. Note that a Ferz is both color-bound and meta-color alternating within the set of accessible squares.

I agree about the FAND.

Jeremy Lennert wrote on 2011-04-14 UTC
'FAND is a special case. This piece not only 'can mate', it can mate all by itself by force in an open position.'

This doesn't seem to be true. The only mating position for FAND unaided is K in a corner and FAND one A move away. But in order to force a K on b1 into a1, the FAND would need to threaten all of a2, b2, c2, and c1 (plus b1, if K's owner has other pieces), which it can only do from a0, which first of all is off the board, second of all is close enough for the K to capture it, and third of all is more than one move away from c3, where it needs to be to complete the mate.

WFND can mate unaided by threatening K with a D move and then duplicating K's moves until it is trapped on an edge. WFAN cannot, because K can move orthogonally to the threat.

Charles Gilman wrote on 2010-11-14 UTC
My first paragraph was a quote from the page itself that I was questioning. I was highlighting what the Rook gains from adding either Bishop or Knight move, and what both the Bishop and the Knight gain from adding the Rook move.
Could I also point out that '-boundedness' is not the right term here? Bounded is the past tense of the verb to bound, meaning to jump or leap or hop (in a general sense rather than the specific Chess ones), and as an adjective it means having a boundary so '-boundness' would be more correct - although so is the even briefer '-binding'. There are analogies with other verbs - you can ground an aeroplane and get a grounded aeroplane, but if you grind coffee you get ground coffee - not that I'm offering any.

H. G. Muller wrote on 2010-11-11 UTC
I don't see what is being explained here. The Kaufman values for solitary B and N are exactly equal with 2x5 Pawns on the board; with fewer Pawns the Bishop has a small edge, with more Pawns the Knight. As 5 Pawns is a quite typical middle-game case, that is about as equal as it can get. Only the _second_ Bishop (if it is on the opposite color, which of course it always is) is worth a lot more than a Knight, the so-called pair bonus, which amounts to half a Pawn.

Now the Chancellor (RN) is about half a Pawn weaker than a Queen (RB). So how come 'the Chancellor is doing so well'? Seems to me it is not doing well at all. Before adding R they (i.e. N and B) were equal, after adding it the B has gained appreciably more than the N. In fact about as much as the pair bonus, which could be interpreted as due to lifting the color-boundedness.

Such interpretations are a bit dangerous, though, at least when used quantitatively. The 'lifting of color-boundedness' argument could also be used when adding N to B or R. But there it would have to explain away nearly a full 2-Pawn difference between R and B, as RN is only marginally stronger than BN in practice. And it is a bit strange that lifting the color-boundedness in one case would buy you 0.5 Pawn, and in another case, combining even less valuable pieces, 1.75 Pawns.

Anyway, it seems to me that attempts are made to 'explain' something that is the reverse from what is true.

Charles Gilman wrote on 2010-11-11 UTC
'One hypothesis about why the Chancellor does so well is that the R has a weakness that is masked when N is added to form Chancellor. This weakness would be its relative slowness and difficulty of development, and perhaps its lack of forwardness (it has only one forwards direction).'

The fact that the Rook has just the one forward direction does not explain the lack of difference between Rook+Bishop and Rook+Knight as both wopuld gain from the non-Rook move. The more likely factors are (a) that adding the Knight move to a Rook with eight adjoining allies allows it to leap out of that space in a way that a Bishop move does not and (b) just adding the Rook move to a Bishop removes its colourbinding, adding it to a Knight rewmoves colourswitching, which is the Knight's property of always moving to the (not just an) other colour. Both compounds - and for that matter Bishop+Knight - can move to squares both of the same colour and of the opposite one. There are other kind of switching - the Silvergeneral is rankswitching (always moves an odd number of ranks), the Fibnif and Mushroom fileswitching (always move an odd number of files), and the Ferz and Camel both. Can everyone see why pieces that are both rankswitching and fileswitching are colourbound?

George Duke wrote on 2009-12-07 UTC
Betza never answered Aronson's last question: ''Ralph, didn't you also test the NH (2,1 leaper + 3,0 leaper for those unfamiliar with funny notation)?'' Ralph never explained H, it's just a useful letter for 0,3, that Gilman later revives instead as Trebouchet from George Jeliss and the problemists. /a

Peter Aronson wrote on 2003-07-24 UTC
Ralph, didn't you also test the NH ([2,1] leaper + [3,0] leaper for those unfamiliar with funny notation)? How did it fit in the rankings? (While it doesn't particularly apply to the current discussion, I always thought that the NH would be a good extended Knight for a 10x10 board.)

gnohmon wrote on 2003-07-24 UTC
'Part of the advantage of the augmented Knights over the Rook may be a
Zillions artifact--Knights are strongest in the opening, Rooks in the
endgame. Zillions sometimes has trouble getting to an endgame, where
masters would'

I seem to recall having written a short piece on the relationship between
opening values, endgame values, and the strength of the players, that is,
why the
Blackmar_Diemer Gambit is winning between 1800 players and losing between

Opening advantages favor normal humans, endgame advantages favor GMs, in
both cases because of the precision of play required to overcome an
opening advantage and then profit from the endgame advantage. Exceptions:
Tal, RJF.

gnohmon wrote on 2003-07-24 UTC
'ND, NA = 27
NW = 38
NF = 82 (!)'

NF is better, but not by that much. NF might be notably better when
Grandmasters play; but for normal masters, NF is barely better, hardly
notice it at all.

Your extreme results should be taken as a sign that you should distrust
the tool you used to take this measurement. Of course, lacking other tools
you will continue to use it! However, your faith in the value of its
measurements will be diminished.

The relative order is correct: NF is best, NW is second-best, NA and ND
are a bit behind. The quantification is way off. The 'quantum' of
advantage would be about 30 in your scale, and so NW is worth ND, close
enough. Eleven centipoints is nothing.

Therefore your tool has some value. If several different unreliable
measurements give the same result, one may have some degree of confidence
in the combined result.

gnohmon wrote on 2003-07-24 UTC
My experience playing the game of Different Augmented Knights against a
strong chessplayer convinced me that the differences in value could safely
be ignored -- and it was this that gave me the confidence to proceed with
the next step, the Colorbound Clobberers.

My own computer-versus-computer simulations showed a slight advantage to
NF over the others. In addition to its ability to move to either color
square, the NF can escort a Pawn to promotion against a King, unaided by
its own King, and also very important in the endgame a NF can mark time:
NFe2-f3 keeps d4 defended! (Of course, ND can also sometimes do some of

NW has exceptional ability to draw by perpetual check (saving a lost

When playing games using the NF, the most noticeable thing is that it
seems to have extreme flexibility. NFe2-g3 attacking Qh4 and defending the
K-side is great, but so is NFe2-f3 doing the same things! These choices
are powerful weapons in the hands of a strong player.

Numbers always underestimate the Rook. The new thought of 'King
interdiction' may play a large part in this. NF and Fibnif have some
interdiction ability, but not as much as WF....

Calculation would be so much easier if there were more known data points.
Instead, we begin with values known only for R, N, B, Q, and P; plus
as-Suli's estimates for A and F; plus recent chess variant experience
that NN is equal to Rook; plus old CV experience that on the cylindrical
board B == R. Developing a comprehensive theory of piece values based on
so few known data points is not easy.

Archangel worth only a Q? I'll buy that, based on the idea that the
displaced R move is worth a bit less, and the F move is duplicated. I fell
for a moment into the intuitive trap of envisioning the empty-board
mobility!, and of course in the late late endgame the Archangel must be
superior to Queen. 

I have the advantage of being a strong chessplayer, which means that not
only can I attempt to establish new known data points by playing both
sides of a game, but also once in a while I can talk some other strong
player into playing some wierd game against me. As a professional computer
programmer, I also had the chance to run comp-v-comp test series years
before anybody else. Over the course of 25 years, I have added a few known
and fairly trustworthy piece value data points: 

1. Ferz beats Wazir. I can't say by how much.

2. Augmented Knight equals Rook, or pretty close to equal. Of course, as a
competitive player, I'll choose NF every time, and try to win based on
its advantages! 

3. Commoner beats every other 2-atom piece because of its severe endgame
advantages. It has the largest absolute mobility, but its advantage is
much greater than the simple mobility numbers would indicate. A calc that
could 'predict' this would be wonderful!

4. What else? Is that all, in such a long time? Either I should have
worked harder or it is not so easy. 

Somebody reading this might have more money than math; if so, your
contribution to this developing science would be to pay grandmasters to
play with different armies (sponsor a tournament). This would create new
known data.

As you can see, developing known values by playtesting is extremely
A good theory for calculating values would be such a help...

I have estimated the value of a Reaper and a Harvester and a Combine, but
I have fairly little faith in the correctness or exactness of these
estimates. These are simple and logical pieces, easy to estimate with the
current methods (easy to estimate though the estimate may be wrong!).

King Interdiction is a very promising new idea. Gryphon has double
interdiction! Is its practical value much greater than my estimate? Note
that until we can calculate the value of interdict, a NN ought to calc
higher than a Rook.

Another possible avenue of exploration is the interaction with Pawns.
There are many Pawns, and promotion is usually decisive. A Rook behind a
passed Pawn at a2 has value all the way down to a8 although current
calculations do not give it any credit for that. A Bishop supporting a
Berolina Pawn? Nobody knows..

In these few lines, I have pointed out what I think are promising
questions to explore. This is all pure speculation. Feel free to take a
different approach.

Michael Nelson wrote on 2003-07-22 UTC
More thoughts on augmented Knights:

Part of the advantage of the augmented Knights over the Rook may be a
Zillions artifact--Knights are strongest in the opening, Rooks in the
endgame. Zillions sometimes has trouble getting to an endgame, where human
masters would.  If my conjecture is correct, setting Zillions to deeper
plies would show the gap reducing or increasing much more slowly than
normal for repeating a Zillions calc at higher plies.

I suspect your results are not anomalous among the augmented Knights. The
NF has yet a third advantage--it cannot be driven from an outpost square
by an undefended pawn! All other augmented Knights can (as can the Rook,
but outposts are more important for short range pieces). This factor is
also almost certainly a part of why the Ferz is stonger than the Wazir.  

I would be curious to see what the numbers are for the various augmented
Knights vs Rook and each other if Berolina Pawn are used. I predict NW the
strongest but with a smaller gap, and Rook significantly better vs
augmented Knights (easier development as well as can't be attacked by an
undefended pawn).

Robert Shimmin wrote on 2003-07-21 UTC
Michael --

Here's the data on the augmented knights, obtained from Zillions vs.
Zillions using 5-ply fixed-depth searches.  The augmented knights are
placed in the rook positions and played against the orthodox army.

All the augmented knights give an advantage vs. rooks, probably in part
due to their ease of development and the rook's lack thereof.  The
following values are the various pieces' advantages over the rook, in

ND, NA = 27
NW = 38
NF = 82 (!)

The standard deviation for these measurements is about 10 cP.

Michael Nelson wrote on 2003-07-21 UTC

That is puzzling. Are there value gaps between the other augmented Knights
or do they test out fairly equal? Value of NF vs. R I could argue either
way as their moves are so unrelated.

I would think that the NF would be the strongest augmented Knight (even
though less mobile than NW) as it masks two Knight weaknesses:
colorswithching and inability to move a single square. 

NW masks one step inablility but isn't as forward as NF.

NA and ND mask colorswitching and give a a lot of coverage to the 2-square
distance.  These are very likely quite well mathced: NA more forward, ND
more mobile.

I really never had though of colorswitching as a major disadvantage, I
have even doubted it is worth considering. On the other hand one of the
nice things about Rooks is that they are neither colorbound nor

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