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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.
<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>
'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.
'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.
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?
'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.
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.
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.
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.
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.
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.
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.
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.
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!
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.
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.
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'.
<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.
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.
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.
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.
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.
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.
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.
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.
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