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 Gothic piece values, An empirical computer study
H.G.Muller
Posted: Dec 30 2007, 04:28 PM


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The following unequal-material matches were played so far:
CODE

Q-BNN    (172+ 186- 75=) 48.4%
Q-BBN    (143+ 235- 54=) 39.4%
C-BNN    (130+ 231- 71=) 38.3%
C-BBN    ( 39+  86- 11=) 32.7%
A-BNN    (124+ 241- 67=) 36.5%
RR-Q     (174+ 194- 64=) 47.7%
RR-CP    (131+ 227- 74=) 38.9%
RR-AP    (166+ 199- 67=) 46.2%
RR-C     (188+ 170- 74=) 52.1%
RR-A     (197+ 162- 73=) 54.1%
QQ-CC    (131+ 55-  30=) 67.6%
QQ-AA    (117+ 60-  39=) 63.2%
QQ-CCP   (112+ 72-  32=) 59.3%
QQ-AAP   (112+ 78-  26=) 57.9%
CC-AA    (102+ 89-  25=) 53.0%
Q-CP     (164+ 191- 77=) 46.9%
Q-AP     (191+ 186- 55=) 50.6%
Q-C      (215+ 161- 56=) 56.3%
Q-A      (219+ 138- 75=) 59.4%
C-A      (187+ 182- 63=) 50.6%
A-RN     (261+ 122- 49=) 66.1%
C-RN     (273+ 101- 58=) 69.9%
A-RNP    (247+ 121- 64=) 64.6%
C-RNP    (242+ 144- 46=) 61.3%
NN-RP    (262+ 127- 43=) 65.6%
NN-RPP   (221+ 141- 70=) 59.3%
P-.      (233+ 132- 67=) 61.7% (g-pawn)
P-.      (218+ 160- 54=) 56.7% (c-pawn)
PP-.     (235+ 144- 53=) 60.5% (g+c)
PP-.     (253+ 129+ 50=) 64.4% (g+b)
R-BP     (187+ 187- 58=) 50.0%
R-BPP    (170+ 209- 53=) 45.5%
R-NP     (226+ 148- 58=) 59.0%
R-NPP    (218+ 153- 61=) 57.5% (g+c)
R-NPP    (195+ 172- 65=) 52.7% (g+b)
BB-BB'   (208+ 164- 60=) 55.1% (pair vs anti-pair)
B-N      (395+ 314- 154=) 54.6%
B-NP     (331+ 384- 149=) 46.9%
BB-BN    (459+ 278- 127=) 60.5%
BB-BNP   (371+ 346- 147=) 51.4%
BB-NN    (262+ 109- 61=) 67.7%
BB-NNP   (221+ 139- 72=) 59.5%
BB-NNPP  (197+ 168- 67=) 53.4%
RR-BBPP  (201+ 175- 56=) 53%



Remarks:
The Pawn-odds score came out at 61.7%. This is a bit lower (11.7%) than the 14% advantage a Pawn seemed to give in earlier test. The difference is not really statistically significant though. OTOH, shuffling the pieces might more often give compensation when deleting g2 (because of opening lines for Bishops or Queens) than it would when starting always from the Capablanca opening position (where deleting g2 gives no compensation). Deleting the c-Pawn (not opening a line on the King) seems to give on the average even more compensation.

Most remarkable is that a pair of Chancellors hardly succeeds in beating a pair of Archbishops. The standard error in these 216-game matches is about 3.3%, so there is a 68% probability that the value of CC is between that of AA and AA+0.4 Pawn. This would make C only 20 cP stronger than A. Against a pair of Queens A pair of Chancellors even scored less than a pair of Archbishops (but not significantly; the statistical error in this difference is 4.6%).

Also against two Rooks there is little difference in performance between C and A. In both cases the Rooks have the upper hand, but by much less than a Pawn. When given an extra Pawn (RR-CP) the Chancellor gets the upper hand much more than Q has against RR.

R seems suspiciously weak from N+N vs R+P results, but also in normal Chess this is known to be a badly losing trade if you do it early in the game. Mostly because you give the opponent a numeric Piece majority, which he will start using by attacking your Pawns more often than you can defend them.

R vs N+P gave an advantage of 9%, or about 3/4 Pawn odds. That does not seem crazy from the standard 8x8 value of these pieces. However, R vs N+P+P hardly gives a lower score (7.5%), which would suggest the second Pawn isn't worth anything at all. This is clearly non-sensical. I will have a look at the individual games to see if there was something fishy going on here. Also here I might have to retry with another Pawn deleted as second Pawn.

The Bishop pair gave a huge advantage over two Knights: significantly more than Pawn odds. Even when given a Pawn in compensation, the Knights lose badly. The results suggest that the Bishop pair is worth about 150 cP more than two Knights, which seems to much to me. But many tests of B vs N are to come:
I will have to determine which part of the advantage is due to the value of a single Bishop over a single Knight, and how much is cooperativity between the Bishops. A trick I used before in 8x8 Chess is to play both sides with 2 Bishops, but put the Bishops for one side on the same color. I might try that here too.

Note that A vs R+N+P gave a very anomalous result: hardly worse than A vs R+N. C beats R+N a lot more than it beats R+N+P, as you might expect.

To get an idea of the value of a Bishop pair, I played a pair against an 'anti-pair', i.e. two Bishops on the same square color. It is not sure that such an anti-pair is completely without compensation (i.e. exactly twice as valuable as individual Bishops), as they can now be used to form batteries. On the other hand, they might be in each others way. But in any case, a B pair seems worth significantly more than an anti-pair, (55.1% score, while the standard error is only 2.2%), so pairing or anti-pairing is definitely an important effect. Trading your Knight for the opponen's first Bishop also gives an appreciably better score (BB-BN, 60.5%) than trading it against his second Bishop (B-N, 54.6%).
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Cartaphilus
Posted: Dec 31 2007, 04:10 AM


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I don't think you get any information from running tests with a pair of Archbishops or a pair of Chancellors. It ain't gonna happen in real play, and even if it did, for superstrong pieces, the marginal utility of the next piece added goes way down.

A Chancellor is strong because it can reach so many squares on the board and only the Queen reaches more. Toss another Chancellor on the board and you have 1 square less for it to roam and many squares where it will interfere with the range of the other Chancellor. When like piece movements overlap there is no need for the 'double coverage'.

You want a real tough question? Tell me what is better to have, a Chancellor and 2 pawns and 2 Bishops, or a Queen and an Archbishop? And explain why.
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H.G.Muller
Posted: Dec 31 2007, 05:09 AM


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You make a wrong assumption. When I play with two Chancellors, there is not an extra super-piece on the board, because I replaced a Queen by a Chancellor. So the number of super-pieces is the same, and it is just their character that is different.

Of course there is the danger of a systematic error because there is an unusual cooperativity between two like pieces, which I already admitted. Therefore I will also do tests deleting wC and bQ, rather than replacing wC->wQ and bQ->bC, to get a position that could occur in a real game. But because this halves the difference you want to measure, it needs 4 times as many games to achieve the same statistical accuracy. And I can already predict you with 95% confidence: the resulting piece value will be exactly the same as derived from the comparison of pairs. There is no synergy between pairs of Archbishops, like there is between pairs of Bishops. The superpieces have so many moves that unlike super-pieces also do cooperate very well. (e.g. you can form batteries with C+C or A+A, but equally easy with C+Q or A+Q).

You argue for a kind of anti-cooperativity for like pieces, because exchange symmetry would reduce the number of different positions you could reach. I doubt how important that effect will be in practice, as it usually takes too many tempi to interchange two pieces to actually reach those positions. If such an effect does exist (which is certainly interesting for the development of ab-inito theories piece-evaluation, but of little practical importance, since, as you say, such positions will not occur in games), it should be cancelled to lowest order in, say, the QQ vs CC measurements: both sides do lose a factor 2 in the number of possible positions because of the exchange symmetry. So the bias you warn against would only occur during QC vs CC, (which I don't do) but not in Q vs C or QQ vs CC (which I do).

As to your question, I would be inclined to say (from the scarce data I have so far)that it hardly matters. It would be positional factors like mobility, King safety, Pawn structure and game-stage which decide the preference. Which the engine of course does take into consideration when the actual choice occurs. The good piece values are mainly needed to prevent the engine from picking the positionally best case with the highest win probabilities because it was 'vetoed' by an unreal large difference in piece values.
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Cartaphilus
Posted: Dec 31 2007, 05:13 PM


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Sorry to sound rude, but it sounds like you are just making stuff up. I'm just not convinced this is meaningful information. You're playing games at a very fast rate with an untested evaluation function as if it is "the truth", then lumping the games together and drawing conclusions from this. Suppose I was playing the side that I believe has the advantage and I scored 80% against your program? What conclusion could you make then?

Several programmers have had "good ideas" on paper that just didn't work out once play tested against a strong human player. The old GothicChessLive.com site was a great place to meet and play live timed games and test out new Gothic Chess theoretical novelties. From what I read in your other posts, you don't believe anything but massive amounts of automation produces worthwhile results.

Joker80 might be off to a good start against the weak opposition that it has seen so far. I think you will be surprised when players like M_TAL and me score 75% or better against it at normal time controls once it is released. It can have the best piece values in the world, if it plays a weak opening, it's got one foot in the grave already.
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GothicInventor
Posted: Jan 1 2008, 01:14 PM


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Carta,

Try to start the New Year off in a better manner! H.G. is a very experienced programmer. If he is taking a stochastic approach to determine his piece weights, and the experiment he described is just not your "cup of tea", there is no need to insult his work.

Let's face it: There is no "magic formula" that works every time for every situation, or software programs would win every game every time with 1 second of search!

If you have something to offer that is a constructive demonstration of a better approach, share it with us.

Happy New Year to all!
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GothicInventor
Posted: Jan 1 2008, 01:49 PM


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QUOTE (H.G.Muller @ Dec 30 2007, 05:14 AM)
Last night the QQ-AAP run finished with 57.9%, (112+, 78-, 26=) and QQ-CC with 67.6% (125+, 61-, 30=). This latter score seems (slightly) larger than the Pawn-odds score, which I am now verifying by playing QQ-CCP. When confirmed, this would mean that QQ >~ CC+P, or in other words, that Q >~ C + P/2. This is more or less expected.


Gtohic Vortex 2.2.5, which is in the current tournament, uses Q = C + 0.75

QUOTE (H.G.Muller @ Dec 30 2007, 05:14 AM)

Note that with two equal super-pieces there are only 216 different initial positions in stead of 432, so that the statistical error in the scores has gone up to 3.5%. But because the difference in piece value we wanted to measure was present twice, ithe resulting error has to be divided by 2 in determining the value of a single piece, it is actually more accurate than a 416-game measurement of a single Q vs a single C would be. But there might be a systematic error if there is a synergy effect between pieces of an equal pair (like is well esteblished for Bishops). There is no reason to expect that in the case of the super-pieces, though; it seems mainly a consequence of color-boundedness.

The QQ-AAP result might be more surprising, although it confirms earlier fMax results. This score is definitely less than Pawn odds would produce; it is slightly larger (but not significantly, considering the 3.5% statistical error) than half of it. So it seems that QQ >~ AA + 3/2 P, or Q >~ A + 3/4 P. That would still leave C > A, but not by much (about 25 cP). To confirm this I am now playing a direct match CC-AA.


The results of Chancellor vs. Archbishop will depend a great deal on which stage of the game it ended. The Archbishop is more dominant in the opening and middlegame. The Chancellor should be considered to be "just a Rook" until the endgame or until there are 12 pawns or fewer on the board. By this I mean, you don't play 1. a4 2. a5 3. Ra3 to "develope the Rook", nor should you engage the Chancellor so quickly.

If your games terminated before move 35-40, I would expect the Archbishop results to be higher than your average of the collected data. Games going beyond this, I would think the Chancellor results are better.

Vortex now uses C = A + 1.5

If yout think about the pieces this way: C = R + N, A = B + N, then C ~= A + k(R - B), where k is some constant denoting how well the Chancellor benefits from having additional movement imparted to its Rook (A piece that can reach every colored square) versus how much benefit the Archbishop gains from having its single-colored Bishop be able to reach all colors (a much better gain for it.)

For my purposes, I chose k = 3/4, reducing the gain by 25% when comparing standard 8x8 chess values of R = 5 and B = 3.
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H.G.Muller
Posted: Jan 1 2008, 02:02 PM


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QUOTE (Cartaphilus @ Dec 31 2007, 05:13 PM)
Sorry to sound rude, but it sounds like you are just making stuff up. I'm just not convinced this is meaningful information. You're playing games at a very fast rate with an untested evaluation function as if it is "the truth", then lumping the games together and drawing conclusions from this. Suppose I was playing the side that I believe has the advantage and I scored 80% against your program? What conclusion could you make then?

Several programmers have had "good ideas" on paper that just didn't work out once play tested against a strong human player. The old GothicChessLive.com site was a great place to meet and play live timed games and test out new Gothic Chess theoretical novelties. From what I read in your other posts, you don't believe anything but massive amounts of automation produces worthwhile results.

Joker80 might be off to a good start against the weak opposition that it has seen so far. I think you will be surprised when players like M_TAL and me score 75% or better against it at normal time controls once it is released. It can have the best piece values in the world, if it plays a weak opening, it's got one foot in the grave already.

I am not presenting anything as "truth", except that the test runs I describe had the results I report, and I am telling what conclusions I would draw from them and why. If you believe I am forging these results in stead of actually playing them: Just take your own Gothic Chess program (or one you downloaded from the internet), play the games yourself, and report the results here if they are very different. Anyone who doesn't believe the reported results can check them, as I have witten exactly how I conducted the tests.

If you don't believe the conclusions, well, tell me what your conclusions from the fact that C+C vs A+A ends in only a 53% score. And if you think no conclusions can be drawn from these experiments, tell us what experiments you would recommend (or better yet, do and share with us) to determine if an Archbishop or a Chancellor is the stronger piece.

If you played the side you think had the advantage against my program, and scored 80%, it would prove NOTHING about piece values. As the 80% might just as well mean that you are so much stronger than the program that you can even win 80% from an inferior position, and would perhaps win 95% from an equal position, and 99% with the position reversed. So you would have to play BOTH sides, and the results should be close enough to 50% so that averaging the results would be meaningful, to eliminate the difference in player strength (and play with both leading and trailing move to eliminate the advantage of having the first move). Only then we could start to draw conclusions. In my tests it is easier, as the two players are identical. So any bias in score must be due to an imbalance in the initial position.

If good players can win against Joker80 because it makes opening mistakes, it would not worry me at all. But this is so far a purely hypothetical situation, and I am certainly not going to worry about it now. In normal Chess, Joker was never handicapped very much by not having a book. I also don't see why it would surprise me if Joker80 can be beaten 75%. After all, it is just the 10x8 version of a mediocre Chess program. The 8x8 version would be beaten by 95% or more by top engines. It would just tell me there is room for improvement.

But is seems to me that this is a bit premature. You still have to prove that you can score 75% against Joker80. You will get the opportunity soon enough.
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Posted: Jan 1 2008, 04:30 PM


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QUOTE (H.G.Muller @ Jan 1 2008, 07:02 PM)
You still have to prove that you can score 75% against Joker80. You will get the opportunity soon enough.

Hello H.G.,

Just so you know, Cartaphilus is the resident "hot head" and he tends to spout off on occasion. He has the anger issues of Fischer and he's a rather good player. But when he loses, he acts like a 5-year-old who just had his lollipop stolen by his older brother.

Carta, I'll be nice and not issue a warning here. H.G. is doing a fantastic job with this 10x8 programming efforts and we welcome him with open arms.

Please show him more respect.
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H.G.Muller
Posted: Jan 1 2008, 04:50 PM


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Well, no offense taken! Criticism is the basis of scientific progress, and always welcome. It helps to keep me sharp. I certainly hope Cartaphilus would be intersted to play Joker80, and point out its weak spots.

My approach to measuring piece values seems new, (I could not find reports of similar attempts, not even in normal Chess), and should be scrutinized for possible artifacts and pitfalls. Some of the scores I get when comparing the familiar pieces are a bit suspicious, and might point to systematic errors.

But I think the basic approach is sound. The way I look at it is that I am measuring the degree of control a certain piece army will allow the players over the path the game takes through the game tree. The latter will always have a large number of leaves with scores that are widely distributed around the root score, as either side can win or lose a lot of material in the path from root to leaves. Each side will try to steer the actual game in the direction of the end-leaf nodes that are good for him, and the moves when it is his turns with the pieces he has are the tool for that. The side with the pieces that offer him the best control over the game tree will have the best chance for winning.
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GothicInventor
Posted: Jan 1 2008, 08:20 PM


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QUOTE (H.G.Muller @ Jan 1 2008, 04:50 PM)
But I think the basic approach is sound. The way I look at it is that I am measuring the degree of control a certain piece army will allow the players over the path the game takes through the game tree. The latter will always have a large number of leaves with scores that are widely distributed around the root score, as either side can win or lose a lot of material in the path from root to leaves. Each side will try to steer the actual game in the direction of the end-leaf nodes that are good for him, and the moves when it is his turns with the pieces he has are the tool for that. The side with the pieces that offer him the best control over the game tree will have the best chance for winning.

I think the idea is worth exploring. I can only see one potential defect.

The scores are the result of the search, which shares the same information and same evaluation function, despite different material distributions being present.

So, what you are uncovering are values that help the Alpha side of Joker80 best deal with the Beta side of Joker80, and/or vice versa.

A series of engine matches against other programs under the same constraint will tell you to what degree the approach is valid. The difference in the performances would be a meaningful "first derivative" and reveal whose engine has piece values that are skewed the most. The trend analysis of this data would help fine tune the values by means of a least-square-best-fit line.

Then, authors can adjust their weights, and do the experiment again. Eventually there will be a constant % which cannot be improved upon, and these would be the "real piece weights" to settle on for each program.
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H.G.Muller
Posted: Jan 2 2008, 09:09 AM


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Yes, you are perfectly right. In principle, one should iterate these values to self-consistency: if the score suggests that R+R is weaker than Q, despite the fact that you programmed both engines that R+R is better than Q, you should repeat the test with engines programmed to consider R+R worth less than Q (and preferably exactly as much less as the score would later suggest) to get a correct value for the difference. As the programmed value have a tendency to be self-fulfilling prophecies. If you tell the engines a Queen is worth less than a Pawn, it will almost immediately find a way to 'gain' a Pawn at the expense of its Queen, as this would be almost impossible for the opponent to prevent. And thus, playing one side with a Queen (and the other not) will not treally produce better results than playing it with an extra Pawn.

But tht is really a very extreme example. In practice the effect of having moderately wrong piece values is not very big. If I play normal Chess between engines that are misguided to believe that a Queen is worth only 7 Pawns, Q would still be worth more than R+P. So the wrong trades it would like to engage in are mostly Q vs R+N or R+B. It does not happen that often that trades like that are possible; in most games Q would still be traded against Q. Because if white can give its Q for a black R+N, the engine playing black will also think this is a good trade for white, and avoid it! So as long as both engines have the same erroneous piece values, the fact that they are erroneous only affects a minority of the games, where one of the engines 'blunders' and allow the trade.

The most risky case is actually where you say that two pieces (say C & A) are exactly equal in value. Then C vs A exchanges will be reasonably common. If either C>A or A>C there will always be one side trying to avoid trading them until it is to late and the game has simplified to a stage where it would be very difficult to force such a trade. And than the piece that is in practice the most powerful will start to gobble up pawns more efficiently and give better support to its own advancing passers, producing the score bias in favor of it.
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Cartaphilus
Posted: Jan 2 2008, 10:19 AM


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All I know is I can't beat Vortex 2.2.5 if I play it at 1 minute per move on my machine. It just seems to see everything I have in store for it. The scores are also strange. Like +1.75 in positions that I think I'm doing OK in, then, BLAM! I get knocked on my butt by some tactical shot I missed that is hard to see!

Before I would "get used to" Vortex's play and I could make headway against the blasted thing. After a couple of months I could beat it regularly. No more! It just is too strong and outplays me all the time.

I don't know if it's the new piece values or the new search or both.
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GeneM
Posted: Jan 3 2008, 07:20 PM


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QUOTE (H.G.Muller @ Dec 30 2007, 04:28 PM)
The following unequal-material matches were played so far:
[CODE]
Q-RR     (174+ 194- 64=) 52.3%
QQ-CC    (125+ 61-  30=) 67.6%

H.G.,

Could you please explain one concrete example is simplified language.

In the Q-RR case, do I understand correctly that White had a queen but lacked both rooks; and that Black lacked a queen but had both rooks?

And in this Q-RR case, White won 174 games but lost 194 games, right?

(174 + (64/2)) = 204
(194 + (64/2)) = 224
204 + 224 = 428
224/428 = 52.3% , right?


It seems odd that a value greater than 50% is used to describe results where White lost most games (than Black lost), yet greater than 50% is also used when White won more games (as in QQ-CC).

Thanks.
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H.G.Muller
Posted: Jan 4 2008, 09:58 AM


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This is probably a mistake of mine. The program I wrote to extract the score from the PGN file only prints percentage and number of draws. I calculated number of wins and losses from that by hand, and might have mixed them up while posting.

I will change my result-extraction program tonight, to print all data automatically, so that further mixups will be prevented. But one thing I am sure of: the Queen had the better deal in this setup. Much to my surprise, this is why I checked it extra carefully. I will add several new combinations tonight; they have been accumulating while I had no time to post.

[edit] OK, I figured it out. I did give white the Rooks, and black the Queen, because I hate to subtract, and thus assigned the pieces that I expected to be strongest to white. But the score was below 50%, so I renamed it to RR-Q, and posted 100% minus the measured percentage. But I forgot to exchange the + and - games.

I now renamed it back to RR-Q, for better comparison with the other combinations I now have done against a pair of Rooks.
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H.G.Muller
Posted: Jan 11 2008, 08:42 AM


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Note that I added a lot more combinations to the overview in the second post of this thread.

For the superpieces, the results seem to be quite consistent. Next to matching them in pairs, I now have also matched them individually, and individually against a pair of Rooks. The results of this are all consistent with A and C being both approximately 1 Pawn weaker than Q, with R+R somewhere in between.

The latter still seems quite suspect, though. With the amount of data collected so far is certainly not a statistial fluke, but I wonder if this might be a systematic error of the method. For comparison I should try to run R+R vs Q in normal 8x8 Chess, where correct piece values are almost certain, to see if Rooks suffer from underestimation there.
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