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Ideal Values and Practical Values (part 3). More on the value of Chess pieces.[All Comments] [Add Comment or Rating]
Robert Shimmin wrote on Thu, Jul 10, 2003 02:46 PM UTC:At the end of 'About the values,' Ralph mused on whether the anomalous
excess value of the queen was due to excess forking power or nonlinear
mobility; also how to account for pinning power.
I think I can account for all this in a rough way. Forking and pinning
are sort of the same thing if you think of a pin as a fork with both tines
pointing in the same direction. So let's calculate a number that's very
like crowded-board mobility, but instead of finding the average number of
squares a piece can attack, let's find the average number of two-square
combinations that a piece can simultaneously attack.
Now let's consider the practical value of a piece as a weighted sum of
mobility and this forking power. Because it gives nice results, I like
the sum PV = M + 0.043 FP. The results for a few common pieces are below.
The magic number is 0.67.
Piece Mobility Forking Practical % from
Power Value Forking
Knight 5.25 13.06 5.81 9.6
Bishop 5.72 16.38 6.42 11.0
Rook 7.72 29.23 8.98 14.0
Cardinal 10.97 62.77 13.67 19.7
Marshall 12.97 84.53 16.61 21.9
Queen 13.44 91.32 17.37 22.6
Amazon 18.69 179.95 26.43 29.3
The playtestable result from this is an amazon is worth about a queen and
a rook. Does anyone have the playtesting experience to say whether this
is too high, too low, or about right?Robert, I think you are on the right track. I think the Bishop needs a reduction due to colorboundness, and 10% would make it equal to the Knight. The Amazon seems a little high. Perhaps this is because the Amazon's awesome forking power is a bit harder to use--for example, forking the enemy King and defended Queen is terrific if you fork with a Knight, but useless if you fork with an Amazon. I think that it is neccessary to take the forwardness of mobility and forking power into account--indisputably, a piece that moves forward as a Bishop and backwards as a Rook (fBbR) is stronger than the opposite case (fRbB). Nevertheless, your numbers aren't bad at all as is. They seem to have decent predictive value for 'normal' pieces ( a 'normal' piece moves the same way as it captures, and its move pattern is unchanged by a rotation of 90 degrees of any multiple). Various types of divergent pieces will need corrections--I would assume that a WcR (moves as Wazir, captures as Rook) is stonger than a WmR (capatures as Wazir, moves as Rook) and that both are a bit weaker than the average of the Wazir value and the Rook value.
Without doing lots of arithmetic, I'll just comment that enormously powerful pieces like the Amazon are actually less valuable than their overall mobility would indicate due to the levelling effect. I quote Ralph from Part 4: '...what's more, if one minor piece is a bit more valuable than another, some of the surplus value is taken away by the 'levelling effect' -- if the weaker piece attacks the stronger one, even if it is defended the target feels uncomfortable and wishes to flee; but if the stronger piece attacks the defended weaker piece, the target simply sneers.' While Ralph refers here to minor pieces, it seems to me to be a generally applicable concept. Isn't that why we don't develop a Queen too quickly, so it's not chased all over the board by less valuable pieces?
Robert Shimmin wrote on Thu, Jul 10, 2003 09:23 PM UTC:I once tried to take the levelling effect into account via the following scheme: a piece can neither occupy nor attack a square where it is either left en prise or attacked by a weaker piece. The result is that the minor pieces can more easily occupy the center, where they are more easily defended, and the major pieces must occupy the edge, where they most easily avoid attack. The numbers I got for levelled crowded board mobility were (I forget the magic number, but it was somewhere between 0.6 and 0.7): Knight: 3.71 Bishop: 4.31 Rook: 5.56 Queen: 8.98 Aside from giving a slightly overstrength bishop and a decidedly understrength queen, the calculation was a great deal of hassle. In short, it was rather disappointing because the results were no better than a straight-out mobility calculation, even though they took into account something the mobility calculation neglects. Which may mean the mobility calculation works as well as it does because a lot of its errors very nearly cancel out. I would love to think of a better way to include a levelling effect, but haven't come up with one yet. One note though: the levelling effect is not inherent in a piece's strength, but in the strength of pieces that are less valuable than it. So if the amazon is the strongest piece on the board, then all other things remaining equal, it suffers from levelling no worse than the queen would if it were the strongest piece, because the ability of the other pieces to harass it remains the same.
I wonder what thoughts Robert and others have about multi-move mobility and its influence on value. For simplicity of figures, let's calculate empty-board mobility starting on a center square. In one or two moves, a Rook can reach all 64 squares, while a bishop reach 32. On the other hand, a Wazir can reach 13 and a Ferz can also reach 13. Are crowded-board, averaged over all starting square numbers for two-move mobility of use for piece values? Would it be necessary to also calculate three-move, etc mobility? Another question from the numbers above--does this indicate that the Bishop is affected more detrimentally by colorboundness than the Ferz is?
I've had this thought (2nd-move mobility etc.) before, and I think the correct way to express it is this:
<p>
Averaged over the possible locations on the board, let M1 be the average number of squares that can be attacked in one move (crowded-board mobility), M2 the average number of squares that require two moves to attack, etc. Then the practical value might be some weighted sum of these quantities:
<pre>
PV = k1 M1 + k2 M2 + k3 M3 + ...
</pre>
Of course we don't know these weighting values. But it is reasonable to believe the value of being able to attack a square diminishes by the same factor for each tempo required to do so, and if so, there's really only one adjustable parameter:
<pre>
PV = M1 + k M2 + k^2 M3 + k^3 M4 + ...
</pre>
This is at first sight a very promising approach, since it lets us lump a number of 'weakening' factors such as colorblindness, short range, etc. into one root cause: not being able to get there from here. Also, it provides an alternative explanation for the anomalous extra strength of queen-caliber pieces. Moreover, it would for the first time give a basis for calculating the practical values of pieces that move and capture differently.
<p>
However, there's one problem I've run into when I've pursued thoughts along these lines. The probability of being able to rest on a square is different from the probability of being able to pass through a square, so we need a second 'magic number' to calcuate the various M-values. Also, because the number of squares strong pieces can safely stop on is smaller, it may be necessary to make this value smaller from strong pieces than for weak pieces to account for the levelling effect. (Although I've <i>almost</i> convinced myself the levelling effect may cancel itself out for M1, I'm far less certain that it does for M2, etc.) Anyway, I've rambled about this enough. I think it's a very promising path to go down, but there are at least two arbitrary constants we need to know to go down it.
Excellent the ideas pointed out by Robert Shimmin. I have used informally something like that once, evaluating piece values for a game, but not with rigurosity, it was only a flash idea that I have not analized well. Parameters perhaps can be calibrated with the use of simulation of standar games, I´m going to think a little more about it.
For anyone who was curious about my previous prediction that an amazon may be a full rook more powerful than the queen, I ran the following experiment. Whether it means anything is up to you to decide. I ran scripted Zillions to play against itself for 500 games where black's queen was promoted to amazon, but black was missing its queenside rook. At strength 4, results were 249-62-189, or 85 ratings points in white's favor. At strength 5, results were 265-57-178, or about 110 ratings points. For comparison, samples of 1000 games each found pawn-and-move to be a 135-point advantage at strength 4 and a 260-point advantage at strength 5, while giving white two opening tempi instead of one is a 50 point advantage at strength 4 and a 140-point advantage a strength 5. Based on this, I would guess that the amazon falls short of being a full rook stronger than the queen by perhaps half a pawn, but that still leaves the amazon a pawn stronger than a queen and a knight.
Robert With regard to the multi-move mobiltiy calculation, I think we can ignore levelling effects at the M2 etc level as well--levelling effect can't be calculated on a per piece basis at all. For example, in FIDE Chess, the levelling effect brings the queen's value down--but add a Queen to Betza's Tripunch Chess and the levelling effect brings its value up! I think the correct way to allow for the levelling effect is to calculate all piece values ignoring it, then correct each piece value by an equation which compares the uncorrected value to the per piece average (or perhaps weighed average) value of the opponent's army. So the practical value of a piece depends on what game it is in.
gnohmon wrote on Thu, Jul 17, 2003 05:26 AM UTC:This discussion is wonderful, about 3 levels up from excellent. I'll try to reply to everything at once... Michael Nelson 'inverse relationship between the geometric move length and the ratio of the mobility of a rider', but isn't that ratio already accounted for by the probability that the destination square is on the board? 'Clearly this suggests that the Rook has an advantage over short Rooks', why didn't I think of that? I may be wrong, but at first sight this looks like a brilliant thought! Maybe it is K interdiction; I wonder how you'd quantify that? 'This suggests that the Wazir loses more value from its poor forwardness', continues and concludes a compelling and powerful sequence of logic. Then there follows a plaintive plea for some mathematical type to get interested and find a way to quantify it. Where have I heard that plea before?, I ask myself with a wry grin, and mentally give myself 3 points for the rare use of the word 'wry'. Robert Shimmin 'PV = M + 0.043 FP'. This also looks like something brilliant. You urgently need to run your numbers for the Knightrider! I was surprised that the Bishop had such a high '% from forking'; never thought of it as a great forker because when Bf1-c4, the square a6 is not newly attacked; but perhaps I forgot that Bc4xf7+ also attacking g8 is a kind of fork that I have played a million times -- the B forks 2 forward when it captures forward! Nelson 'WmR ... WcR' my feeling is that when a piece captures as A but moves as B, if A and B have nearly equal values then the composite piece is roughly equal to the average, but when A and B are vastly different, the composite is notably weaker than the average. Does it matter whether capture or move is stronger? I think not much difference if any, because mobility lets the piece with weak attack get more easily into position to use its weak attack; but this opinion is largely untested. Lawson (Hello!) mentions the levelling effect; Shimmin talks about having tried to calculate it! Wow! I made a great many calculations that did not work out, and the failures contributed to learning. I disagree that a top Amazon suffers no worse than a top Q from levelling; say it suffers a bit more, because sometimes Q can get out of trouble by sacrificing self for R+N+positional advantage, but Amazon needs more and thus is more difficult for that kind of sacrifice. 'Which may mean the mobility calculation works as well as it does because a lot of its errors very nearly cancel out.' Yes, it may mean that. The mobility calc seems to work but there's an arbitrary magic number in there, the results are approximate, how can you have full faith in this methodology? Someday there will be something better, but until then my flawed mobility calc is the best we have. Bummer. '135-point advantage at strength 4 and a 260-point advantage at strength 5' -- makes me feel good, worth of advantage varies by strength of player, as predicted. Several '[multi-move calculation]' I think the idea is very interesting that the mmove cal might intrinsically compensate for many of the value adjustments that we struggle with.
It's wonderful to hear from the Master on this topic. I really mentioned the geometric move length becuse you mentioned it in the article--the key point was the comparison of mobility ratios to value ratios and the Rook discrepancy. We need about 10 orders of magitude above excellent for Ralph's work on the value of Chess pieces--I would nominate it as the greatest contribution to Chess Variants by a single person. I am convinced that the capture power and the move power are not equal, but that the difference will only be discenable when extreme. An example--compare the Black Ghost (can move to any empty square, can't capture) to a piece that cannot move except to capture, but can capture anywhere on the board (except the King, for playability)--clearly the Ghost is weaker, though its average mobility is higher. I feel that WcR will be perceptibly stronger than WmR but I could be wrong. I suspect the effect is non-linear with a cutoff point where we don't need to worry about this factor. I also think that the disrepancy will be less than the discrepancy between the actual value of the WcR and the average of the Wazir and Rook values. This discrepancy may be non-linear as well.
I would not call the magic number arbitrary--it is empirical, it cannot be deduced from the theory, but I think the concept has an excellent logical basis. For piece values we want to have sometihing that allows for the fact that the board is never empty, that takes endgame values into account, but is weighted towards opening and middlegame values. So let's take a weighted average of the board emptiness at the opening (32/64) and the board emptiness at its most extreme in the endgame (62/64). Let's weight them in a 3:2 ratio to bias the average toward the opening. This gives a value of .6875 -- right in the middle of the range of magic number values that Ralph uses! The 'correct' value can only be determined by extensive testing and it might well be .67 or .70 -- but I am quite certain it is not .59 or .75! A way to verify this would be to do some value calculations for a board with a different piece density that FIDE chess, then see if the calculated magic number for that game yields relative mobility that make sense (as verified by playtesting). Sticking to a 64 square board, imagine a game with 12 pieces per side. This game has a magic number of .7625 -- I predict that the Bishop will be worth substantially more than the Knight in this game. Now a game on 64 squares with 20 pieces per side. This game's magic number is .6125 -- I predict the Knight is stronger than the Bishop in this game.
<blockquote><i>
Sticking to a 64 square board, imagine a game with 12 pieces per side.
This game has a magic number of .7625 -- I predict that the Bishop will be
worth substantially more than the Knight in this game.
</i></blockquote>
<p>
Take FIDE Chess, and remove the Rooks and their Pawns. Is the Bishop
really worth substantially more then the Knight in that case? I find
myself with unconvinced.
Mike Nelson wrote: 'I would not call the magic number arbitrary--it is empirical, it cannot be deduced from the theory, but I think the concept has an excellent logical basis.' May I add, an empirically determined constant is no less scientific. For those who remember high school physics, it is rather like the gravitational constant, which has been measured very precisely to make the equations fit the evidence. This is all OK, because results that depend on it can be applied to accurately predict events in the real world. Of course, it is even better if we find a way to calculate the 'magic number'.
Mike Nelson wrote, 'I feel that WcR will be perceptibly stronger than WmR but I could be wrong.' I think there is more going on here than just mobility when we compare a WcR and a WmR. My opinion is that tempo matters significantly. A WcR cannot move quickly, but its long-range threats are immediate, for it captures at distance. A WmR threatens only at short range, and must take the time to move to make an immediate threat. Furthermore, in the endgame, a WcR can interdict the King across the board, a WmR cannot. Therefore, if given the choice between the two, I will choose a WcR. I would happily trade a WmR for a minor piece, but I would think long and hard about losing a WcR for a minor piece. Although I have only discussed the specifics of these two pieces, the concepts (king interdiction, threats without loss of tempo) are general considerations, that, like leveling, affect the values of pieces in ways that would be difficult to calculate. Some pieces have abilities that are more useful than their calculated value would imply. In Omega chess, the Wizard moves as a Ferz or Camel (WL in Betza notation). Although they are colorbound, I prefer them to Bishops and Knights because they can make threats beyond a pawn chain.
The is an ideal test bed for the WcR vs WmR question and also the question of asymmetric move and capture vs symmetric move and capture. Run three sets of CWDA games: 1. Remarkable Rookies vs. Remarkable Rookies with WcR in the corner 2. Remarkable Rookies vs. Remarkable Rookies with WmR in the corner 3. Remarkable Rookies with WcR vs Remarkable Rookies with WmR If I can find the time, I will run some Zillions games over the weekend. In thoery, the short Rook used in the standard Rookies is equal to the WcR and the WmR. I predict that testing will show WmR the weakest and the other two quite close, but the only result that would really surprise me is for the WmR to beat the WcR consistently.
gnohmon wrote on Fri, Jul 18, 2003 04:03 AM UTC:I think of the magic number as arbitrary because I was there... In the early 1980s, I wrote a computer program to do the value calc for a large range of magic numbers (0.50, 0.51, and so on). Then I printed out the results and picked the value that I liked best. This seemed very arbitrary to me; yes, given the idea of average crowded-board mobility, some magic constant is needed; and yes, the idea of crowded-board mobility has a strong feel of Truth to it, which somewhat justifies picking it in such a crude and self-predictive way. But because I was there I never have felt strong faith in the magic number!
gnohmon wrote on Fri, Jul 18, 2003 04:12 AM UTC:When you consider 'can mate' (although K+WcR vs K appeared to be a draw after 30 seconds of blindfold analysis, another 30 seconds shows me a way that might work. Danger from the '50 move' rule!), When you consider 'can mate' and the new idea of King interdiction, the WcR becomes hugely stronger than the WmR. However, White Ke4 WcR at h8, Black Pa5 Kc4, White loses but a WmR at h8 should draw; a demonstration of how sometimes mobility can be better.
gnohmon wrote on Fri, Jul 18, 2003 04:16 AM UTC:A Bishop would be delightful in Xiang Qi, wouldn't it? However, if each side has 20 pieces, the B merely has to wait a bit longer for the board to empty out and make it strong. The Knight's advantage in the opening would last a bit longer, making it overall a bit stronger, but still worrisome to give up B for N... Consider the game of 'Weak!' in this context.
Peter Hatch wrote on Fri, Jul 18, 2003 06:24 PM UTC:Excellent ★★★★★I've noticed that for the R1 through R7, the practical values seems to be proportional to empty board mobility. So if a Rook is worth 4.5 pawns, here are the calculated values and Betza's comments on their actual value from the short rook and Wazir pages: R6 is 4.339 (worth a rook, most of the time) R5 is 4.018 (a weak rook) R4 is 3.536 (more than a bishop, but only slightly) R3 is 2.893 (a bit weaker than a bishop, but close) R2 is 2.089 (clearly less than a knight) R1 is 1.125 (little more than a pawn) My guess is that this is because a combination of practical concerns make the endgame the prime determinant of a rook's value. Only one forward direction, king interdiction, being stuck in a corner at the start, and the bishop and knight not gaining power in the endgame as fast may all contribute. Or it could be something else entirely.
Peter brings up an interseting observation about Rook values approximating empty board mobility. Yet the short rooks seem a little weak by this standard, just as the usual crowded board mobility makes long Rooks too weak. The Rook's special advantages over the Bishop and Knight (interdiction, can-mate) are endgame advantages--so empty board mobility or at least a higher than normal magic number might be the way to quantify the value of different length Rooks among themselves. An R7 is much superior to an R3 in both can-mate and interdiction. And Rook disadvantages (lack of forwardness, hard to develop) apply regardless of length so they would cancel out in this comparison.
With regard to the WcR vs the WmR, I wonder if the tendency at least in the endgame is for the capture power to be more important offensively and the non-capturing movement to be more important defensively. I also wonder if unbalnced pieces in general tend to belong to the category of 'it's worth x, but you really should trade it before the endgame.' In the late endgame, an R4 might be superior to both WcR and WmR by a perceptible margin.
In response to Ralph's comment, I've done the forking power calculation for a few more pieces. The magic number is 0.67 Piece Mobility Forking Total % Fork ------------------------------------------------------- Nightrider 7.82 29.53 9.09 14.0 Rook 7.72 29.23 8.97 14.0 One thing I've noticed (and should have expected) is that the 'forking power' value is very close to being proportional to mobility squared. These pieces illustrate about the most variation I can create in FP for 'normal' pieces of about the same mobility. Archangel is gryphon + bishop. Piece Mobility Forking Total % Fork -------------------------------------------------------- Archangel 13.10 98.07 17.32 24.4 Queen 13.44 91.32 17.37 22.6 FAND 13.56 95.38 17.66 23.2 Clearly, these differences are too small to test. So while we know there is some superlinear dependence of value on mobility, we can't yet say whether that is most related to forking power, multi-move mobility, or what.
gnohmon wrote on Mon, Jul 21, 2003 07:48 AM UTC:Excellent ★★★★★Excellent work, but I am amazed. The specific endgame that convinced me a NN is worth a R is (NN + Pawns versus R plus Pawns) and in this endgame it's all about the amazing forking power of the NN. Your calc doesn't show what I saw in this endgame. This might be worth thinking about. My anecdotal evidence is not the same as your numbers.
gnohmon wrote on Mon, Jul 21, 2003 08:02 AM UTC:Excellent ★★★★★'Archangel is Gryphon plus Bishop'. If your numbers do not show it as supeirior to Q, mustn't that be an eror in the numbers? FAND is a special case. This piece not only 'can mate', it can mate all by itself by force in an open position. Its shortness is compensated by its excssive ability to mate. Today's primitive science of value calculation inevitably underestimates this piece. I developed the concept of theoretical values so that I could simply describe this piece as being worth 5 atoms, same as the Queen. Now you've found a calculation that gives this same result. This is a huge accomplishment!, if the calc holds up for other pieces. (Looks like it does, so far, I think?) A reliable calculation for the theoretical value would make it possible to spend much more effort on attempts to calculate practical values.
Perhaps Ralph's conjecture that mobility has a non-linear (yet fairly close to linear) relationship to value is the real starting place for these calculations, rather than forking per se. What kind of non linear equation would we be looking at if we assume without proof that that the Spielmann values (N=B=3.0 pawns, R=4.5 pawns, Q=8.5 pawns) are correct?
gnohmon wrote on Mon, Jul 21, 2003 08:45 AM UTC:Excellent ★★★★★'very close to being proportional to mobility squared' I always thought that forking power should depend on number of directions. A recent comment made me wonder if I had, in effect, been underestimating the forking power of moves such as Bc4xf7+ (attacking both Ke8 and Ng8, for example 1 e4 e5 2 Nf3 Nc6 3 Bc4 Bc5 4 b4 Bb4 5 Bb4 c3 6 Qb3 Na5 (in 1985, in the Harding-Botterill book 'The Italian Game', this was ignored as a simple error). After 8 Bxf7+ Kf8 9 Qa4 c6 , White needs to play Bxg8 to avoid losing a piece. In 2003, opponents on FICS will play 6...Na5. This discussion of theoretical piece values ties directly into actual practical everyday playing of FIDE Chess! If there were a mathematical calculation for the Max Lange Attack or for the Evans Gambit, it would make me unhappy; but if this calculation opened the way to inventing a Max Lange equivalent in the Rookies versus Colbberers game, I'd be happy overall. In this discussion, we are asking questions that go far beyond the norm, and if our findings ever allow one to answer 'what's the best move in *this* position according to *these* rules, I think that none of us will be happy with the result. Basta Philosophy! 'Mobility squared'. 'Mobility squared' was always the sort of thing I felt iffy about. A simple math, seems so attractive, as a chess master I doubted that things were so clean. In my early calcs, I know I tried to use something squared, maybe geom dist, and later I shied away from simple squared. Maybe something squared is correct! If you prove I was wrong you may win the Nobel Prize for piece values research. (This is no joke, Do a web search, find how many professional mathematicians link to my values pages, and how few try to contribute.) I always feared handwaving. 'How to Lie with Statistics' is a very good book, and it is very applicable to our field of endeavour. I would always rather miss a discovery rather than present a flawed arg for it. Thus I am prejudiced against anything squared. It seems too simple. However, I will listen; and my own personal judgment is far from final, as I may be superceded. What I am trying to say is that a good result may turn out to be a false lead. I mean, today you get numbers that look good, tomorrow raises doubts. In order to feel this sort of pessimism, you need to be old enough to have gone through a few cycles of Eureka! and Oops!. Maybe you have something golden. I hope so. It is late. I was thinking of deleting this whole comment and remaining silent, Instead, I will trust your judgment to take it for what it is worth.
gnohmon wrote on Mon, Jul 21, 2003 08:54 AM UTC:Excellent ★★★★★'Or it could be something else entirely.' Hooray! That is the sort of doubt that I feel! I am so uncomfortable about having everybody take my primitive efforts as golden. It could be otherwise entirely.
<i>'Archangel is Gryphon plus Bishop'. If your numbers do not show it as
supeirior to Q, mustn't that be an eror in the numbers?</i>
<p>
The crowded-board mobility calculation for the Gryphon predicts that is very similar in value to a Cardinal. Because the Gryphon can move like F anyway, adding Bishop to its move is really only adding the longer Bishop-moves, and the Archangel is only one half-knight stronger than the Gryphon, and a piece one half-knight stronger than a Cardinal-class piece is a Queen-class piece, or so the numbers say.
<p>
Still, my gut instinct would think, like you, that an archangel would be noticeably superior to the queen. This is something that can only be resolved through playtesting. If the numbers are right, well, this somewhat bolsters our faith in using the mobility calculation to say two pieces are roughly the same in value. If the numbers are wrong, it is very interesting, because both the queen and the archangel are well-balanced pieces, and the first step to improving a theory is trying to identify those cases where it fails.
<p>
After calculating the 'forking power' for a number of these pieces, these are my second thoughts...
<ul>
<li>What I call 'forking power' is really only the average number of two-square combinations attacked by a piece, and therefore treats pins and forks as the same phenomenon. Perhaps this is wrong.
<li>If two pieces have similar mobilities, they will almost certainly have similar FP's. Therefore, the theory can't make any predictions that couldn't also be made by invoking a multi-move mobility component to value, or simlper yet, a power-law dependence of value on mobility. I originally thought up the archangel to think of a piece with similar mobility as the queen, but vastly greater forking power. But this 'vastly greater' wasn't nearly as great as I hoped. :(
<li>Not all two-square combinations of attack are created equal. Maybe two squares in the same direction are more or less valuable than two squares in different directions.
<li>In short, I originally invoked this definition of forking power because it was something that would be calculated without any undue difficulty or assumption, and would be much larger for strong pieces than weak pieces, so with the right scaling factor, could be made to give the queen the right value. It may be an improvement to the theory, but I can't think of any test for it that would distinguish it from a few alternate improvements.
</ul>
On another note, can anyone think of any reason why NF ought to be noticeably stronger than the rook, or any of the other knight+(one atom) pieces? Zillions wins rather more often with it than the others, and I'm trying to puzzle out if this is a quirk of Zillions', or if it could be duplicated in theory or human playtesting.
Robert, 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 colorswitching.
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 centipawns. ND, NA = 27 NW = 38 NF = 82 (!) The standard deviation for these measurements is about 10 cP.
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).
gnohmon wrote on Thu, Jul 24, 2003 04:32 AM 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 this.) NW has exceptional ability to draw by perpetual check (saving a lost game). 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 expensive. 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.
gnohmon wrote on Thu, Jul 24, 2003 04:43 AM 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 Thu, Jul 24, 2003 04:51 AM 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 human 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 GMs. 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.
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.)
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
'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?
H. G. Muller wrote on Thu, Nov 11, 2010 09:49 AM 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.
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.
'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.
H. G. Muller wrote on Fri, Apr 15, 2011 10:13 AM 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.
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 Sat, Apr 16, 2011 08:47 AM 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.
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 Sun, Apr 17, 2011 09:30 AM 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!)
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 Mon, Apr 18, 2011 08:57 PM 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.)
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.
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.
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.
Joe Joyce wrote on Sun, Apr 24, 2011 04:12 PM 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.
Joyce's ''chessness'' of a cv was a comment once by Gifford: http://www.chessvariants.org/index/displaycomment.php?commentid=19665.
Joe Joyce wrote on Mon, Apr 25, 2011 10:19 PM 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? :)
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