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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.
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