Help! Why don't fairy chess pieces have values?

H.G. most chess players I know of can watch a game replay with a brief time delay like that one. The entire game is online to replay one move at a time at the link https://triceschess.com/articles_04.shtml , if you look at November 20, 2021 and that wasn’t even a Letchworthshire game so he wasn’t bragging as you say. And about your piece weights: you claim to have computed good values yet you insist on this ‘non royal king’ which isn’t even a part of chess, nor any real game that I know of that has a following. To me it sounds like you’re avoiding the discussion about pieces we do use such as the Archbishop and Chancellor to try and make an irrelevant point with your non royal king. You’re deflecting more than the U.S. press secretary trying to defend the president. Just drop all the non royal talk it’s just nonsense. If you’re so sure the game was misplayed with those amazing moves, follow the replay and show us your better moves. Otherwise leave the expert opinions to those of us who are experts.

Well, like I said, I am currently not interested in any Trice's Chess games. The issue I have with Letchworthshire it about his posting that I quoted above. My only goal is really to debunk that statement, preventing people to attach any value to a theory about piece values that is obviously non-sensical.

The value it predicts for the Archbishop is not a very convincing proof of that, as to the uneducated any value between 5 and 9 would sound plausible, and most of the moves of the Archbishop do not go to adjacent squares. So for the super-pieces counting safe checks gives you results that are only marginally different from counting total mobility. So no matter how crappy the prescription through which you convert such mobilities to piece values, you will always get values in the plausible range. And if the difference is small it is all too easy to contest the correct value in a shouting match, using irrelevant arguments like "I would beat you at this game", and "15 years ago you wrote a computer program that is weak".

The non-royal King, on the other hand, is a much more convincing demonstration of how the safe-checking theory miserably fails to predict correct piece values, as it has only moves to adjacent squares, which that theory says should be discounted. So the safe-checking theory predicts a value of zero, which is so obviously non-sensical, that everyone immediately would recognize it as wrong without the possiblility to be led astray by false but devious arguments.

When those who want to obscure the truth cannot deny that the predicted value is utterly wrong, they will of course fall back on the argument that this is irrelevant. E.g. because "no one uses that piece, as we all know that Trice's Chess is the only chess variants that is ever played, so what is not in Trice's Chess doesn't really exist". But I don't think that would earn them a cigar from anyone. If a theory predicts the wrong value for one piece, you cannot trust the value it predicts for any piece. Surely the value it predicts for the Archbishop will not be 100% off, like it was for the non-royal King, but it might easily be, say, 20% off. Which makes its predictions as good as useless, as anyone with half a brain could have guessed such an inaccurate value without the aid of any theory. "Wow, how cool! It moves like Bishop and Knight, which they tell me are worth 3. So it must be worth at least 6!" No need even to count its moves.

As long as we do not agree that safe-checking theory is garbage, discussing anything else just is a distraction.

Not only is the safe-checking theory questionable, but also the claim that there was a mathematician named Taylor who came up with it in 1876.
The page from Ed Trice that Letchworthshire appears to talk about, is very sloppy in linking to its sources of information, but I still found this:

https://mathshistory.st-andrews.ac.uk/Biographies/Taylor_Henry/

This article does mention checks, but not specifically safe checks.

Since all orthodox Chess pieces check in the same way as they capture other pieces, the mention of 'checks' in this context is just a peculiar way to express that you should count the number of squares that would be under attack by the piece on an otherwise empty board. Obviously a piece that could only deliver check, but not capture any pieces, would be worth only a fraction of the normal piece. So it appears that this 1876 article describes the old-fashioned 'average mobility' method, that nowadays is re-invented nearly every other day by someone who thinks he has discovered the ultimate method for calculating piece values.

But at least this makes sense of the claim that the paper was receiving large appraisal; Taylor was the first to attempt finding an arithmetical basis for the empirical values of Chess pieces, and the idea alone to attempt this must have been considered a stroke of genius. And the average-mobility method is of course not so obviously flawed as this 'safe checking' idea; if he would have presented the latter in a paper he would have been the laughing stock rather than being considered a genius.

The method still has some obvious problems, though. It even lacks consistency in the case of pieces that are not able to access the entire board. The problem is that the averaging over all locations of the piece only makes sense if the piece is equally likely to be in any of those. But of course in real games the players would develop the piece to good squares, and avoid the very poor squares. No one would willingly move his Knight to a corner, (unless a capture to there results in a material advantage that would far outweigh the positional disaster), so who cares how few moves a Knight in a corner has?

As a numerical demonstration, we can count the total number of Knight moves that can be made in Chess, and if onne does this correctly it adds up to 336, which divided by 64 would give the Knight an average mobility of 5.25. But suppose that we play instead with a special rule to handicap the Knight, banning all moves into and out of the corners. For a Knight that isn't in a corner already, this would leave 320 moves between the 60 squares it can now access, for an average mobility of 5.33.

So by making the Knight obviously weaker by imposing a handicap on it that takes away some of the things you could do with it, the average-mobility theory predicts its value goes up! But a player of course doesn't need such a handicap rule to keep his Knights out of the corners; he can play as if this rule existed, and increase the value of his Knights, and thus his playing strength, by using this superior strategy.

A more-accurate theory of value guestimation would have to account for this 'strategy effect' on pieces that have a location-dependent mobility. And this is a subtle thing, as moves that go to a square where you'd rather not be are worth less than moves to a good square, starting from the same location. There always will be some base value of the moves, however: even for going to a square where you would have zero moves, so that you would be trapped for the remainder of the game, or even disappear as a 'kamikaze move', it might still be useful to go there. E.g. as part of a tactical exchange. When a piece is destined to be recaptured, you would not care how poor the square is where it goes; it would not survive there anyway.

Other aspects that a serious theory should address are that some moves cooperate better than others. E.g. a Bishop would gain more from an additional orthogonal 3-square jump as from a similar 2-square jump, even though 3-square jumps on average give it fewer extra mobility (as it strays off board more often). This because the former breaks the color binding. Moves to orthogonally adjacent squares cooperate better than moves scattered all over the place, because they equip a piece with mating potential. So quantity is not everything; quality should also be taken into account. Things like speed, forwardness, concentration are properties of the entire move pattern, and cannot be simply valued as a sum of their parts.

And then there is of course the well known effect of general synergy of moves, where extra moves make it possible to better aim the already existing moves to do damage to the opponent, apart from being able to do damage themselves. Which makes a Queen significantly more valuable than a Rook plus a Bishop, even though the latter two together have exactly the same mobility as the Queen. You would also have to account for the fact that during most of the game the board is not nearly empty.

This link will shed more light on Taylor, which cites the paper and an excerpt from it:

https://mathshistory.st-andrews.ac.uk/Biographies/Taylor_Henry/

Indeed, that is exactly the link Evert gave. And as you can see from the excerpt, there is no mention of 'safe check'; the prescription of placing the King is just a particular method for counting how many squares a piece would attack on an empty board.

No doubt ground-breaking for its time, but wrought with all the deficiencies I mentioned in my previous posting.

Excerpt means “part of” i.e. not the whole thing. Taylor defines the “safe check” on page 2 of his paper, had you even bothered to read it. Way to demonstrate not only your bias, but your ignorance also. Here is the proof that Taylor used the term “safe check.”

Indeed, if you quote something it is always better to quote the relevant part.

OK, so he found that his average-mobility did give a very wrong value for the B-N difference, and then started to make arbitrary modifications to it for tweeking the outcome. Apparently not realizing that these modifications did more damage in other cases (such as the non-royal King) than what it would repair in the B-N case.

When there are only 4 values you have to reproduce (the Pawn acting as standard, and the King being priceless), and do not care about any other, it is of course very easy to get things exactly right if you can draw on hundreds of (mostly nonsensical) ideas.

The largest contribution to the poor B-N ratio predicted by the average-mobility theory is of course the fact that during most of the game a chess board is far from empty. So that sliders (Q, R, B) usually would have far fewer moves than on the empty board where the counting was done. Most of their moves can be blocked in many places. Taking the board occupation into account thus also corrects the B-N difference in the desired direction, without completely wrecking the predicted value of the N-K difference.

This whole idea that undsafe checks would not contribute to the value was just a stupid idea; As measurement of the value of many different piece types has shown, these moves usually contribute more to the value than any other move.

We don't know what exactly Henry Taylor claimed to be the applicability or implications of his paper. If we wanted to verify that we would need to find and read his entire paper. Letchworthshire and mr. Trice do claim something wrt to this paper, but they have not shown any intention so far to simply provide a link. We'll have to search a bit harder ourselves, or rely on a little screenshot from Letchworthshire every now and then.

True. But this is of historical interest only, not of any practical interest. What Taylor might have claimed, and what not, and whether other people believed his claims, or not, doesn't later the values of chess pieces in any way.

Note that the entire concept of piece values is based on an idea that is just an approximation, namely that the strength of an army can be expressed as the sum of intrinsic values of the pieces it contains. That this assumption is flawed is most clearly demonstrated by the fact that 7 Knights easily beat 3 Queens (with equal Pawns on both sides), on an 8x8 board.

“Claimed” is too strong of a word to apply to Taylor’s work. He states his premise that the RELATIVE strength of a chess piece may be considered to be proportional to the number of times a piece can deliver a check to the enemy king without itself being trivially recaptured on the next turn. It’s not a bad conjecture. His paper does not offer a comprehensive evaluation function. It’s a starting point. Trice’s paper, titled “80-square chess,” extends Taylor’s work in two important ways. First it derives equations for rectangular boards not just one that’s perfectly square. Second, it includes the new compound pieces of Archbishop and Chancellor. He noted also that a baseline of pawn = 1.0 on an 8x8 board and a pawn = 1.0 on a 10x8 board would mean piece values on 10x8 do not necessarily compare directly to their 8x8 equivalents because the pawn weights at 1.0 are actually different metrics. One of the pawn weights is denser than the other.

It is actually a very bad conjecture, because it would produce a relative value zero for the non-royal King, which only can deliver unsafe checks. It only results in values that are still plausible for pieces that hardly have any such unsafe checking moves (relative to their total), where its effect is too small to make the error it introduces noticeable. The more unsafe checks a piece has, the more it spoils the value estimate, until it finally produces a 100% error for pieces that only have unsafe checks.

If a conjecture that only 'works' in cases where it has negligible effect isn't a bad conjecture, I wouldn't know what is...

And indeed, Pawns are poorly suitable for acting as a standard in any case, as even moderately strong players already distinguish many different kinds of Pawns, depending on their placement and that of other Pawns, even on the same board (backward, isolated, doubled...). A 7th-rank passer on a central rank is worth a great deal more than a 2nd-rank Rook Pawn (perhaps even 5 times as much). So when expressing the values of other pieces in 'Pawn units', you'd better define what kind of Pawn you are referring to.

Of course the value of a Pawn is not solely determined by its tactical strength. A large part of its value comes from its ability to promote to a piece that is so valuable it would immediately decide the game. Counting moves completely ignores that aspect. Theories based on that can only predict the value of a piece that moves as a Pawn, but cannot promote. Promotion on 10x8 isn't worth any less than promotion on 8x8, so the difference between the Pawn-based scales on both boards aren't as different as pure move counting would suggest.

Your non royal kings are irrelevant. And for the one millionth time, you’re talking about an evaluation function not piece values. Don’t you know the difference?

Sure. Your theory is perfect, and all the the million cases for which it is totally off are irrelevant.

The point is that for Pawns the value, which formally is the average of its contribution to the evaluation, is ill defined. The value is small to begin with, and the variation in evaluation is very large compared to that of other pieces. Because Pawns cannot easily improve their evaluation by moving elsewhere. For mobile pieces like a Queen it isn't very important where on the board they stand, as this can usually be corrected in a single move. So the deviation from the average is on the order of a tempo. The large variation in the Pawn evaluation makes it critically dependent on what set of positions you average over to get the value. Not a desirable trait for a standard unit.

Sum total of H.G. academic papers published by peer-reviewed journals: 0. Total papers published by Ed Trice regarding piece values: 1. Over a billion times as many as you. He beat your Gothic Chess program. And now, after a few months of trying, I finally beat him in the game he invented.

How funny that you bring this up at precisely this moment. I invite everyone to have a look at the announcement of this years physics Nobel prize, which was awarded two days ago. My name appears 3 times there, in the references to the academic papers that describe the research for which the prize was awarded (at the bottom; ref. 18, 23 and 36).

If you feel that blackmouthing other people's reputation is a good way for convincing people of your point of view, it would be wise to do some research first. Otherwise you run the risk people would take you for an utter fool.

A helpful overview of my publication record can be found here. So it appears I got 90. Beats me why you think that would change the value of the non-royal King in a chess game. Usually the number of citations is considered more relevant than the number of papers, b.t.w.; this to filter out papers that no one considers useful.

I meant 0 papers published about chess values.

Taylor has 1 paper published on piece values.

Trice has 1 paper published on piece values.

Yet you belittle both of them.

Why?

Letchworthshire schreef:

"I meant 0 papers published about chess values.

Taylor has 1 paper published on piece values.

Trice has 1 paper published on piece values.

Yet you belittle both of them.

Why?
 "

Why? Because the presented theory predicts an obviously wrong value for the non-royal King, (namely 0), of course. Not to mention all the Shogi Generals...

Nothing personal. But having a publication in a peer-reviewed journal is not the same as elevation to sainthood. Such publications merely serve to present your experimental results or theoretical ideas to the scientific community, so they get the opportunity to criticise or corroborate those, and in the end consensus will be reached on what is the truth. The peer review is supposed to weed out the biggest nonsense, but it is far from fail-proof. Especially if there only is one referee. (For prestigeous journals like Science there are three, which are not only supposed to assess the scientific soundness, but also the importance, and if only one is not happy the paper is rejected.) Experimental results cannot be verified, and implications of a theory other than what the authors mention might escape their attention. Referees are not expected to completely duplicate the research effort; they just glance at the paper, judge originality, and are prepared to accept claims that are not obviously contradicting known facts on good faith.

Some referees scrutinize submissions better than others; Trice was lucky they did not send the paper to me for refereeing. (Yes, I have refereed papers for the ICGA Journal!) It probably would have been rejected, if he had not been able to adapt it such as to get a plausible value for the non-royal King.

Of course 'luck' in this case is a dubious asset; if a referee allows you to present nonsense to the community, the community will act as thousands of additional referees, and some of those are bound to recognize the flaws, and expose them in public. Journals like Physical Review Letters even have a special 'Comments' section where they publish criticism on papers they published before (again refereed for validity, of course).

The save checking theory is rather primitive by modern standards (now that far more complex calculations can be done with the aid of computers in a matter of seconds), but for 1876 it was probably 'state of the art'. I already pointed out most of the effects it does not account for, and those for which it does account in a wrong way (i.e. the safe checks). That it was published doesn't make these flaws disappear, and does not make it immune for criticism.

I can very easily write a paper about piece values, e.g. about the algorithm of the Interactive Diagram guestimates them. This does take into account most of the effects I mentioned, so it is a far more refined algorithm than Taylor's or Trice's, and would thus be considered sufficiently novel to warrant publication. (It also gets too low an Archbishop value, b.t.w.) The referee would very likely not reject it, as it never predicts obviously wrong values like zero. Even if you were to be the referee, arguments for rejecting it as you presented here so far (such as "You should not publish this, because I beat a chess program this author once wrote") would fall on deaf ears with the editors, after they had a big laugh. But why should I? No one is interested in piece values anymore; all strong chess engines use trained neural networks for evaluation nowadays. How many citations did the Trice paper get?

They do have value, you just don't know what they are.