Comments/Ratings for a Single Item
With all paws being on their original positions, all players having 31 points to spend (and having to spend exactly 31 points), players being able to choose at max 3 rooks (to not mess with castling rules) and being able to choose at max 7 pieces (the back rank): Each player would be able to choose 7 different ways of starting the game. The math and the positions on wolfram alpha: http://tinyurl.com/bzodn2c Of course this assumes that choosing the same pieces, will produce the same starting position. With players that brought the pieces being able to choose where each piece will start this number will be higher.
To make absolutely sure I have a good calibration of the Pawn-odds score, which is the yard stick I measure all piece-value differences against, I did a 2N vs R+P match. (I had never done that before with the mating-potential-aware Pair-o-Max.) This ended in favor of the Knights with 64% score, which is an advantage of 0.93 times the Pawn-odds score (which is ~15%). The difference R-N has been implicitly measured in 2 steps, the N+N+P vs R4+R4 difference and the R4+P vs R difference. I quoted 20cP and 0cP for these earlier, but I should really have said these were 0.2 and 0 times Pawn-odds. From the latter it follows R4 = N + 0.6, and R = R4 + 1, so that R = N + 1.6. While now we have 2N = R + 1.93. Some algebra then shows 2N = N + 1.6 + 1.93, or N = 3.53 times Pawn odds. With the Kaufman value N = 325 cP, this gives Pawn odds = 92 cP (= 15% score advantage) This agrees excellently with the conclusions drawn from 2W vs N, but are a bit more reliable. R = N + 1.6 = 3.52 + 1.6 = 5.12 x Pawn odds, or 472 cP. That is 28 cP below the classical value, but it is for a Rook trapped behind Pawns. Recalibrated, R4 becomes R - 92cP = 380 cP. The measurements of R2, R3 and R4 have become a bit ambiguous, as these pieces should also suffer from the quarter-Pawn penalty behind Pawns. But in some of the starting positions they were combined with Pawn odds, and although they never started on the open file caused by the Pawn removal, one of them was nevertheless very close to it. So in those cases one of the two Rn might have suffered appreciably less penalty. So I really should redo these measurements under better controlled conditions. I am a bit dissatisfied with how long they take, though. I am currently playing 40 moves/min games. Perhaps I should switch to 40 moves/10 sec games. I am not sure that Fairy-Max is strong enough to deliver sufficient game quality at those speeds, though. Perhaps I should make its search a bit less minimalstic, using real move sorting, so that it can use killer heuristic and all these other nice techniques to reduce branching factor...
Unfortunately I never made webpages for this. I did post occasionally results here. I have now tried to measure a Wazir directly against a Pawn. This was kind of interesting. An obvious way of doing it would be to replace the f- or g-Pawn (the one in front of the Bishop in the shuffled setup) by a Wazir. My Chess intuition immediately revolted against this, however: such a position has none of the characteristics of traditional Pawn odds, which I used to compare other material imbalances against. The Wazir nicely fills the hole that otherwise would be such a liability. When I was doing Wazirs vs Knight one Wazir was replacing the Q-side Knight, and the other put just in front of it, moving up the b- or c-Pawn one step. To better compare with that, I took a position that did put a Wazir in front of the Q-side Knight, as an extra piece, and deleted the f- or g-Pawn in compensation as Pawn odds. In fact, the two positions are somewhat reminiscent to comparing the value of an extra Pawn to that of Pawn odds. If I 'replace' the f-Pawn by a Pawn, I of course get an exact wash, as I did not change anything. But if I do it by deleting the f-Pawn, and adding an extra Pawn on c3, I do faithfully create the Pawn-odds motif. Of course any Chess player would know this position is much worse than the usual initial position. Doubled Pawns are worth less than two normal Pawns. I tried both W vs P positions, and indeed the measurements give significantly different values. A Q-side Wazir and K-side a Pawn hole only show a 50.9% score in favor of the Wazir, hardly significant (1-SD error = 1%), and only 6 cP when taken at face value. But when a K-side Pawn is replaced by a Wazir, the score increases to 55.3% percent in favor of the Wazir (33 cP). That is significantly more (4 SD)! So a Wazir behaves a bit like a Pawn. Put on an open file to replace a Pawn it is worth about 27 cP more than when trapped behind another Pawn. Much like a doubled Pawn. Unlike a Pawn a Wazir can move sideways, but it takes so many moves to get from the Q-side to the K-side that this will be too expensive in itself. Unlike with doubled Pawns, there now the situation Wazir behind Pawn or Pawn behind Wazir are no longer equivalent. I expect Wazir in front of Pawns to be better, but did not measure from such positions. This is related to the earlier remark that Rook-like pieces are difficult to develop when they start behind Pawns. Remarkable enough the 27cP difference I measure between open-file Wazir and behind-Pawn-wall Wazir is almost exactly the deficit I usually measure in the Rook value with my method (475cP in stead of 500cP). Anyway,when I use W = 1.06 x Pawn odds, the 2W vs Knight with Pawn-odds result (40cP in favor of the Knight) would say 3.12 x Pawn odds + 40cP = 325cP (=N), from which follows that Pawn odds = 91cP. So this particular Pawn in the early opening seems worth somewhat less than the abstraction of the 'average Pawn' (which probably corresponds to a well-centered Pawn in the mid-game). That puts the Wazir (behind Pawns) at 1.06 x 91cP = 97cP (where N=325cP). A developed Wazir, (on open file) would be 1.33 x 91cP = 121cP. Compare that to an old result where I played 2 Ferzes against Knight in the same setup (replacing Q-side N by F, and putting the second right in front of it, moving up the Pawn). There the pair proved worth ~300cP, i.e. 150 each average. (Undoubtedly a lone Ferz will be worth slightly less, and the second slightly more, due to its color binding.) Of course the Ferz does not suffer from being behind a Pawn wall; it sneaks out between them.
Is there a page somewhere that summarizes all of your experimental results for piece values?
Wazirs really seem to be awful pieces. Replacing a Knight by two Wazirs (on the Queen-side, moving the Pawn in front of them up one square) make the Wazirs lose even with the help of additional Pawn odds (on K-side) by about 40 cP (WWP vs N: 44%, WW vs N: 31%) This would bring the Wazir even slightly below a Pawn, if N=325 and P=100: (325 - 100 - 40)/2 = 94cP. Could be that classical Pawn odds is worth slightly less than the average Pawn, however. E.g. when the Pawn-odds score of 15% would represent only 90cP, the 6% score deficit in WWP-N would be 36cP, and we would get (325 - 90 - 36)/2 = 100cP, slightly more than the initial f/g Pawn. I guess I should really measure W vs P directly, replacing the f- or g-Pawn by a Wazir, to see who does better.
(In reference to Jeremy's note about comment display weirdness: It occurred when the associated page was deleted; the hanging comment is handled strangely. If an editor who knows more about the php code can fix this for the future that would be nice; for now I've just moved the comments for the deleted sovereign chess page into a non-page comment thread.)
Fairy-Max would have no problem with some of these. Asymmetry is no problem per se, as the move in every direction has to be defined separately.Long-range leapers (bison) is also no problem; any board step is possible on 8x8 (a large lateral step could limit board width, to prevent the piece from jumping over the guard band, and wrapping around the side edges of the board, but 16-stepSize is still possible). Nightriders could lead to some instability, if the search encounters positions where mutual perpetual check is possible. (This means the program would occasionally hang, and lose on time.) I am not sure this is even possible in theory when there is only a single Nightrider present. As to promotion: Fairy-Max is programmed to promote pieces number 1 and 2 when they reach last rank. It does not matter how these move. If that is not enough, a simple change in the code could make a different number of pieces promote. It would be a bit tricky to make them all promote into different types (but doable). It should hardly matter for the value of a piece to what exactly it promotes, however. Promoting to a piece of approximately the right value should be enough. Bent riders is currently not possible, as are pieces with a minimum range. They are basically the same thing. It should be possible to patch the code to do those, though. The current system is based on toggling a given set of bits in the initial move step & rights to get secondary step & rights. For non-hoppers this happens after every step, but I just made a change to allow suppression of that for the first 1, 2 or 3 steps, to do the limited-range sliders. After that (as the range counter gets negative) it would toggle every step again. But the code can be trivially changed to only toggle when the counter is zero. This one-time toggle could then switch a primary step with neither capture nor non-capture rights after a certain number of steps to a secondary step with rights for both capture and non-capture. Or it could change the direction of the step. This patch would lose the possibility to do Crooked Bishop or Mao-Rider and such. (But who cares?) Piece drops are out of the question. For those it would be easier to modify a Shogi program. I would think that also holds for 'swap pieces'. They would need move generation that does not fit at all in Fairy-Max' table-driven system of looping over directions and distance for every piece. You would have to loop over the board in stead, looking for friendly pieces, and to perform the move the occupant of the target square, which normally would be captured, would have to be moved back to the from square. (OK, perhaps this part can be made standard for 'capturing' own pieces.) 'Rifle Chess' pieces would also require a different MakeMove and UndoMove procedure, which now consists of clearing away the captured piece (which for e.p. capture could be somewhere else), clearing the from-square, and putting the moved piece on the target square (possibly promote it there). And for UndoMove clear the target square, put the original piece on the from-square, and put back the captured piece. I guess that when 'shooting' a piece, you would have to skip the second and third part of the MakeMove, based on some new (or redefined) move-rights flag. Or fool it by forcing the target square used during MakeMove to become the from-Square, so that it becomes sort of an e.p. null-move. This might be doable, after all. The challenge is how to preserve the info that this has to be done if the move is not generated from scratch, but the best move of a previous iteration (possibly passed through the hash table). Capturing mid-move also seems hard. Perhaps this can be done by changing the hopper code (removing the platform, and only allow hops if it is an enemy. If you can do double captures this way, it requires even more code changes (to remember the secondary victim, but not impossible). So I guess most of this can be done by making dedicated versions by modest patching. But it can't be done all at the same time. (But this is probably not really needed; as long as you don't want to implement complete games.)
There seems to be a bug on the recent comments page: Mark Bates' comment, which is on another game, immediately folllows H.G. Muller's comment (below) with no intermediate heading row, as if they were on the same game, and yet the two comments after that, which are both on Buypoint Chess, each have their own separate heading. Not sure where to report that.
On the topic of Rook development, it seems possible to me that an R4 or R5 might have its development hindered less than an R7 would, since by advancing past the pawn wall the piece will also become somewhat more centralized, which seems of some benefit to a short rook but probably not to a full one. But I certainly could be wrong.
On the topic of software, For the Crown includes some sufficiently exotic pieces that it seems unlikely I will find any software that can represent all of them without modification (I'm a programmer and could potentially perform some modifications myself)--but testing only a subset of them would still have some value. You can download the rulebook from here (near the bottom of the page) if you're interested in the details, but some highlights include:
- Asymmetrical pieces
- Long-range leapers (bison)
- Bent riders
- Pieces that can promote as if they were pawns
- A long-range rider with a minimum move distance
- A piece that can exchange places with another friendly piece as a move
- A piece that stays in its original square when making a capture
- A piece that can capture mid-move and continue moving
- A piece that, when captured, goes into it's owner's hand and can be dropped back into play
The ideal software would also handle multiple royal pieces on a side (win by capturing all of them, no restriction on moving into check; corollary: no stalemate) and the option to start with pieces in hand that can be dropped into an empty square on the first rank in place of a move (some pieces can be dropped on the first or second rank).
There's an expansion coming out in a month with yet more weird pieces (example: a piece that moves once "for free" each turn, before you move a different piece).
Maybe you could have the cost increase once it is used
The difficulty of developing is inherently included in the piece values of the limited-range Rooks. I do replace Knights and Bishops in the FIDE setup with those. In fact I think the Rook itself suffers from that too. It is not replacing a Bishop, but I don't think it really matters much whether it starts on a1 or c1. With FIDE-like Pawns, Rook-like pieces suffer. (This makes it interesting to do piece-value measurements in Berolina, to see if the Rook-like pieces go up in value compared to the FIDE context.) I guess that what disturbed Betza was that the Cannon is a piece of a value that ought to be developed early. So that would mean that especially R2 and R3 suffer a lot, but that R4, R5, ... R suffer less. After all, Q is easy to develop, but it is a classic mistake to do so. You want the heavy pieces to remain in the back until the light ones have cleared the field for them. Btw, the Cannon should suffer a lot less than R2/R3. Behind its own Pawns the capture move of the Cannon is developed from the very beginning. In the early days of Fairy-Max I did test the Cannon, btw. From what I recall it was indeed about the value of a Knight. The Cannon is a tricky piece, however. The fact that it has a move that is explicitly dependent on the presence of other pieces makes it a non-constant, and in the late end-game it loses its value completely. One should be skeptical whether a program not taking this into account, but handling it as if it had a constant value, could give you an accurate value determination. In my dedicated Xiangqi engine I noticed that with Cannons it is extremely important that you know when to get rid of them. Wait too long, and you are stuck with worthless Cannons. As to the software: The latest WinBoard release ( http://hgm.nubati.net/WinBoard-4.7.0.exe ) contains a pre-installed Fairy-Max in the package. This is not the 'Pair-o-Max' version I am using now, which has been patched for taking account of mating potential of the pieces through some simple heuristics (where the most important recipe is that any score is divided by 2 if the leading side has no Pawns, and even by 8 if he is less than 350 cP ahead in piece material (and pawnless), unless he has a light piece that is marked as having mating potential in its piece definition). The source of that version is on my website, though, and I can easily make it available as a Windows binary that could be used to replace the Fairy-Max in the WinBoard install. The way you would have to define pieces for Fairy-Max is not very user-friendly, though. I recall having mentioned Spartacus, but I am not sure anymore why at that time I thought that Fairy-Max would be to simple for what you needed. Please refresh my memory. Some limitations are very deeply ingrained in Fairy-Max' design. Some other things can be added in a relatively simple way, and sometimes I make the effort to do that. Lately I added the capability to do limited-range sliders of ranges 3, 4, 5, and this heuristic for judging mating potential. Perhaps this already solves the problems I perceived at that time. If the problem was board-depth, it would be pretty hopeless with Fairy-Max. Problem is that the Spartacus project is in a sort of sleeper state; I have not worked on it for more than a year, and it has dropped on my list of priorities since I know this year's Computer Olympiad will be in Japan. So I am mostly working on improving my Shogi engine now, and the chances I will get to do any work on Spartacus before August are very slim.
IIRC, when Betza was considering cannons, he concluded that a standard cannon (move along rook lines, capture by hopping along rook lines) was likely stronger than a bishop, but thought that replacing bishops with cannons in the FIDE opening might not reveal this advantage because it would make development too difficult. He commented that "even replacing bishops with rooks is not that easy", or something like that. I'm curious, have you tried a test where one side replaces its bishops with additional rooks? Also, is your testing software available for others to use? I try to assign values for importing the For the Crown pieces into something like buypoint chess, and it would be nice to check my guesses against a computer. I think I asked this once before and you suggested I wait for Sparticus, but it's been some time...
I am close to completing play-testing of the R2, R3, R4 limited-range Rooks. My methodology has been to create 4 different opening positions, the standard FIDE position and the 3 derived from it by swapping Knights and Bishops for white, black or both. Each position then gets some of the FIDE pieces replaced by the piece-under-test, for white or for black (so that in the end I have 8 starting positions). The strong side can get the f- or g-Pawn deleted (the one in front of its King-side Bishop) to make the chances more even. These 8 positions are then each played 200 times with a pair-bonus- and mating-potential-aware version of Fairy-Max ("Pair-o-Max"). This gives 1600 games for each material combination. The statistical error of the score percentage in that many games is 1%-point (1 standard deviation). The results show that deleting an extra Pawn causes a score difference of 15-16%-point, so that the statistical error is about 7 centi-Pawn on the oeverall strength difference of the armies. As I usually take imbalances that involve a pair of the pieces, this would result in a standard error of 3cP in the individual piece value. The combinations I tested, and the result converted to centi-Pawn were: WD+WD vs N+N equal R2+R2 vs N+N -110 cP R2+R2+P vs N+N -10 cP R3+R3 vs N+N +40 cP R3+R3 vs N+N+P -60 cP R3+R3 vs B+B -25 cP R4+R4 vs N+N+P +20 cP R4+R4 vs B+B +70 cP R4+R4 vs B+B+P -15 cP* R4+P vs R ~equal* *) not fully converged yet; games are still running This suggest the following piece values on the Kaufman scale (N=B=325cP): WD: 325 R2: 270 R3: 345 R4: 385 R: 485 This produces a value of the ordinary Rook that is a bit low (the Kaufman value is 500), but this is rather typical for these empirical measurements. Note that Rooks are known to be worth a lot more on open files then when boxed in by Pawns, and the Kaufman values are probably based mostly on positions where the Rooks have come into play through open files. (It would be hard to create an imbalance involving Rooks otherwise in a normal FIDE game.) Note that the scale above is rather sensitive to the assumption that the Pawn value is 100cP, which for this particular Pawn (a 50-50 mixture of an initial f- or g-Pawn) is of course debatable. The only way to measure that is to play Rook (or R+P) vs two minors. My guestimate for Wazir is 130cP; it would be good to measure this as well (e.g. play W+W vs N and W+W+P vs N).
Final result for the R2: Replacing Knights by R2, and favoring the latter with Pawn odds (deleting the Pawn in front of the K-side Bishop, i.e. f2/f7 or g2/g7, as I shuffled the minors in the initial setup to force more variation in the games) still left a small advantage for the Knights: 2N vs 2 R2 + P: 51.75% (592+ 536- 472=) Without the Pawn odds 2N vs 2 R2: 67% (865+ 319- 416=) The statistical error in 1600 games is 40%/sqrt(1600) = 1% (1 SD). The difference between the two results gives the effect of a Pawn (~15%). So the advantage 2N vs 2 R2 + P translates to 11.7 cP +/- 6.7 cP. With N = 325 and P = 100 this gives R2 = 269 +/- 3.3 cP. I never did a direct measurement of Wazir. I did play 2 Ferz vs Knight once, and the result (a quarter Pawn) suggested F = 150 (or actually 2F = 300, as there will be a pair bonus involved). With the rule of thumb that forward moves contribute twice as much value as sideway or backward moves (determined by taking out the various moves of the FWADN piece), that would make the Wazir 2.5/3 times the Ferz, i.e. W = 125 as best guestimate for the Wazir. SO we have W = R1 = 125 R2 = 269 R = 500 This result is a bit sensitive to the value used for the Pawn, however. There are many differennt Pawns (center, edge, backward, isolated, passers, protected passers, connected passers), and most strong Chess engines do use an opening value 80-90 for the Pawn. In the standard setup the f-Pawn is the most valuable, (deleting it has no compensation in terms of development, and leaves a bad unsafe spot for the King), though, and deleting the g-Pawn in the shuffled setups produces an isolated h-Pawn. Too bad Fairy-Max does not have limited-range sliders in its repertoire. I can do the R2 only by virtue of defining the W moves and lame-D moves as separate 'directions'. It can handle at most two-step lameness, however, so I cannot use a similar trick for R3. Perhaps I should change the code to keep a range counter, set to a value extracted from the move-rights flags (I think I still have 3 unused bits in there), and let it count down with the slider steps, and abort when it reaches 0. (So that starting value 0 gives unlimited range.) Then I could also do CWDA's Rookies army, which features an R3. [Edit] OK, I managed to do this. So R3 measurements are under way! [Edit2] A pair of R3 seems to fall almost exactly between 2 Knights or two Knights + Pawn. That would make R3 ~ 375, about as much as your first Bishop of a pair.
OK, I see. Yes, it makes sense that color binding should somehow incur a penalty, and that combining would take this penalty away. OTOH I remember doing some measurements with a 'switching Bishop', which had a single-step backward non-capture on it. This was hardly an improvement over the normal Bishop (when both played in pairs), perhaps something like 25 cP. And there was a similar, perhaps slightly lower increase (~15cP) of the Knight value when I equipped it with such an extra non-capture, so it seems that a large part of that bonus is tactical. This was all measured before I had Pair-o-Max, but that should not matter much in this case, as none of the involved pieces has mating potential. The Bishop pair bonus could have an effect, though. I am just not sure what the effect would be. So perhaps I should just redo the measurement with Pair-o-Max. I am currently running 1600-game matches with Pair-o-Max of Commoners vs Knights and Woody Rooks (WD) vs Knights. The Commoners already lost 819-781, or about 51%-49% (1-sigma error 1%), i.e. hardly significant. The other match is still running. After that I will do R2. Adjutants etc. would be a very interesting case as well. I will keep it in mind.  The WD pair versus Knights ended with almost exact equality (810-790 in favor of WD), well within the 1-sigma error. I am running the R2 matches now, replacing the pair of N by a pair of R2, and handicapping the Knights with additional Pawn odds or not. Early results suggest the R2 pair (very) lightly loses to the Knights even with the help of Pawn odds. That suggests Knight minus 60cP. On the Kaufman scale that would make R2 = 265.
"Unchained bishop" is a rather vague concept at the moment. It is my model to explain the excess value of Queen and Archbishop (Janus/Paladin) compared to their raw components. The bishop itself is hindered by board geometry and pawn structures (there is always a so-called bad bishop in the team) to move from one good position to another good position. Combining it with some other piece lifts this restriction and some of the value of a queen (specially the queen-chancellor difference, maybe more) comes from the "unchaining" of the bishop. Your measurement of the Archbishop's value suggests that adding a knight is sufficient to "unchain" the bishop. I don't think that colourboundness is a big issue for the bishop. It may be testable by comparing BDD (Duchess or Adjutant) to the Bishop-Panda compound; the latter is not colourbound, the former is, while the pieces are very similar to each other in other respects. At last, I am interested in the outcome of the R2 tests, since I made a prediction of its value. Depending on the knight's value (300 or 325 cP) it should be one or two quanta of advantage (30 to 60 cP) less than a knight.
Averaging mobility over the board is not a very accurate method, because in a real game pieces tend to seek out squares where their mobility is high. What is an "unchained Bishop", btw? It should be easy to measure the value of R2. It should be lower than WD, which I found to be only slightly less valuable than Knight. But I should redo that measurement with "Pair-o-Max", as the original one was done with the regular version of Fairy-Max, which was not aware of mating potential. As to non-linearity: for short-range leapers with N moves I found the approximate empirical formula N*(30+5/8*N) centi-Pawn. So the non-linearity does not seem extremely large, and a piece with 24 targets is only ~1.5 times more valuable as 6 times a piece with 4 targets. I never succeeded in measuring the value of Bison, because indeed the initial positions I tried were not tactically quiet. The mildly blockable version of Bison (the multi-path Falcon) was very close to Rook in value. Perhaps I should try Shogi/Makruk-like starting positions (all Pawns on 3rd / 6th rank); later I found this to work great for measuring Grasshoppers.
Well, you somehow decided that the values would follow the formulas:
R2 = R1 * (1 + p)
R3 = R1 * (1 + p + p^2)
R4 = R1 * (1 + p + p^2 + p^3)
and so forth.
That looks like a mobility calculation to me, but whether you choose to call it that or not, the fact remains that your "interpolation" is following a curve that you derived based on ONE of Ralph Betza's ideas regarding piece value while ignoring many other important ideas that he also had about piece value.
I'm saying that I don't think the intermediate piece values are actually going to fall along that particular curve.
Jeremy, I don't do a mobility calculation. I just steal Ralph Betza's idea for mobility calculation to do an interpolation of piece values. Both endpoints (the values of wazir and rook, e.g.) are empirical piece values coming from playtesting; therefore the interpolated values are also piece values including all the factors affecting the piece value. And yes, it is only an interpolation, not a calculation from first principles. I think there is still some point in it, seeing the different values of "magic" for Rook, Bishop, and Queen. And seeing that an "unchained" bishop is worth almost a rook maybe explains the surprising fact that the Janus/Paladin/Archbishop is worth almost a Chancellor/Marshall. At least, this is my current interpretation of the data. The next riddle to solve is What constitutes the pair bonus?
JÃ¶rg, I'm not sure you've given due consideration to board geometry. Betza's mobility calculations attempt to account for both the probability that a move will be blocked by another piece AND the probability that a move will be blocked by the edge of the board. If you assume that pieces are distributed randomly, then the odds of being blocked by a piece are p^(distance-1), as your formula suggests, but the odds of being blocked by the edge of the board follow a completely different pattern (e.g. for a rook, it's 1 - (distance/8)). That means it can't be accurately represented simply by plugging a different value for p into the polynomial shown in your post.
You also seem to be assuming that piece value is directly proportional to mobility. Most people believe that value has a super-linear relationship to mobility; the evidence being that compound pieces tend to be worth more than their component parts. There are also many things other than mobility that might affect a piece's value; Betza attempted to compile a list here.
Finally, you might want to consider that Betza believed a full Rook had a value closer to 3/2 of a Knight than to 5/3 of a Knight. If that's the case, then it's unclear whether a piece priced at 4 points should aim to be worth 4/5 of a Rook or 4/3 of a Knight--a difference of perhaps 10%, or roughly the difference in your calculated mobility between an R4 and a R5.
Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.
This gives the following formula for the value of a full rook:
R = R1 * (1 + p + p2+ p3+ p3+ p4+ p5+ p6)
Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.
The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is â‰ˆ0.715.
The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.
The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.
Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.
If we use Betza's theory of ideal piece values, then the gnu, gazelle, and bison are each four 'atoms', and therefore should be worth about 7 pawns each (similar to his estimate for the bishop+knight compound, though Muller has some empirical evidence suggesting that piece may be stronger than its mobility suggests). The buffalo is 6 atoms, and so should be worth more than a queen (5 atoms). We can't simply multiply the number of atoms by some magic number, though, because value grows faster than linearly. Maybe 11-12 pawns? The wizard is 3 atoms and colorbound, so should be comparable to the FAD (4 pawns). However, Betza also comments that pieces with long leaping moves are dangerous in a game with an FIDE-ish starting position, because they may be able to make swift, unblockable attacks on the enemy back rank and win heavy material in the early game. That could possibly elevate the value of any of the above pieces. Assuming the Panda cannot be blocked on the squares that it doesn't reach, then it has about 75% of the average crowded-board mobility of a Rook (with magic number = .7). It might get a bonus for faster development. I would guess between 3 and 4 pawns.
This is an intriguing idea. I notice that some of my favorite fairy chess pieces are missing, and I'm not sure how they would rate in terms of value. Looking through some of Ralph's pages, I haven't seen these pieces featured at all. This is something I'm curious about anyway, whether or not I would apply it to some variation of Buypoint Chess. In particular, I'm wondering about: * The Gnu a.k.a. Wildebeest (Knight+Camel) * The Gazelle (Knight+Zebra) * The Bison (Camel+Zebra) * The Buffalo (Knight+Camel+Zebra) * The Omega Chess Wizard (Ferz+Camel, colorbound) * The Panda a.k.a. Slip Rook
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NLJ 12 (10 but has SERIOUS checkmate power)
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