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SOHO Chess. Chess on a 10x10 board with Champions, FADs, Wizards & Cannons.[All Comments] [Add Comment or Rating]
Kevin Pacey wrote on 2018-12-31 UTC

Now that you mention it, in the Champagne Chess preset's index page thread, I quietly gave, with edits to previous Comment(s) of mine, a number of mutator variant ideas I've since came up with, which perhaps didn't deserve their own pages (who knows, they may get tried out in actual play via unofficial presets created later on). I did a similar thing with a number of other edits to index/rules pages of mine. So far I have somthing like 8 mutators altogether that may get tried out some day. The use of the Bishop-Pawn piece type could be one way to help spawn further variants and/or mutators of such, depending how much I or someone else feels up for it at some point. Hopefully any such games could make for an artistic use of that piece type.

The Bishop-Pawn compound piece is, unlike a Dragon (i.e. Knight-Pawn), currently not listed in the CVP Piececlopedia, at least with a dedicated hyperlink (if given at all). It'd be slightly less powerful than a Dragon because the pawn component's available capturing moves would already be part of the bishop component's available diagonal movements, all assuming that like for the Dragon, the piece would not be allowed to promote. Depending on a given variant's setup, I'd assume that like for a Dragon, if it starts on the second rank, it can make a pawn's double step. Like for a Dragon, it would also be able to make en passant captures when possible. The piece figurine is also available in the Alfaerie: Many piece set, as .bp :

John Davis wrote on 2018-12-30 UTC

I'm not sure this adds to the conversation, but in "A Guide to Fairy Chess" by Anthony Dickins. It lists the Dragon ( Pawn + Knight ) which you used in Champagne Chess. It also has Gryphon, Griffin as ( Pawn + Bishop ). I have thought the PB might be useful in some cases. 

Kevin Pacey wrote on 2018-12-29 UTC

I don't quite understand the bit about my apparently sometimes arbitrarily ignoring colourbinding, at least (the math parts of your last post are a little bit over my groggy head, tonight anyway, except I'd suggest my method for calculating at the least the first component of the compound left some margin for error, as in hindsight I clearly should have got 3.5 in a perfect world, rather than 3.625, on 8x8, and there was a similar sort of slight error for the 10x10 case - it too should have produced a final answer of 3.5, all in line with what you point out [else I'm unclear at the moment where "3.75" comes from]). First, note I never removed the implicit (i.e. built-in) colourbinding penalty (whatever it is) when considering 9/10 of a B as one component for the compound piece's value as I estimated it. For calculating chess values, this happens too, when one makes the compound piece Q=R+B+P and takes its value from the equation just given, without in any way discarding the colourbound penalty a B has built in (whatever it is).

A Q is as a result not a colourbound piece, similar to the compound piece that I estimated the value of is not a compound piece, in spite of having a colourbound piece as one of its components (i.e. the same story as for a Q). So, secondly, note that for either compound piece there is no now-non-colourbound-piece bonus explicitly used in the equations involved (for a Q, the lack of any binding it has is implicitly taken into account, along with any other factors created by the combining of its B and R components, by the Ps worth of cooperativity between the two components). Hopefully it will not confuse things to note also that a wazir and ferz are worth about the same (I treat them as =) in spite of a ferz being colourbound - in that case the pieces are very small in value, plus other factors are involved that help the ferz' value. Thus, I get a Waffle (WA) the same value as a ferfil (FA) in the case of those compound pieces, in spite of the fact one is colourbound and the other is not. This happens once again because an equation for compound pieces is being used, where I choose to use a P as the amount of cooperativity involved (I may get things significantly wrong on some occasions by normally using a pawn for cooperativity in the case of compounds, but at least it greatly simplifies my life, for the time being anyway).

There could perhaps be some piece type dreamt up that I could have big trouble handling as a compound piece, as a way of handling the masking of binding that occurs in a case like that of a Q. I thought such a type of piece improbable or uncommon, and it certainly might force me to see binding penalties in a different light. Otherwise, treating pieces as compounds whenever possible seemed attractive to me early on when estimating values, and I try to milk that cow for all it's worth. :)

If you ask me how I might assign a B a binding penalty etc. when evaluating it from scratch, I'd have huge trouble being sure I'd weighed all of the possible significant factors, but to try to meet you halfway, here's how I'd use my least well worked out method, a sort of crude weighing of pros and cons, between two piece types I'm considering, where the first one has a known value - in this case it'll be a knight (on 8x8), which I'll say is worth 3.5.

Characteristics of a N:

1) Leaper (thus x2 bonus. e.g. compared to a B, is built into its value, if what I've read on CVP is the common wisdom);

2) Average cells reached on empty (8x8) board = 5, which is half of a B's average of 10 (thus B deserves about a x2 bonus compared to a N, IMHO);

3) Short-range compared to a B, i.e. lacks speed in comparison (thus B deserves about a x2 bonus compared to a N, IMHO);

4) Can reach every cell on the board (i.e. B is colourbound, deserving a x0.5 penalty, as compared to a N, IMHO [thus before this final stage you might say I was tentatively thinking a B worth 2xN's value, further bonuses or penalties pending]).

At this point I've pretty well used up all the big pros and cons I can think of, and happily they balance (suggesting B roughly or exactly = N), which may make you somewhat happy given your computer study results, though I'd note there are many finer things I did not try to weigh (impossible as it is even to list them all), which might ever so slightly tilt the balance in favour of a B, such as that a B can at times trap a N on an edge of the board, while a N cannot do the same.

My problem is, without treating a 9/10 B combined with 1/4 wazir as a compound piece, this crude method would probably fail to work out so well for me when comparing the piece to a N, as colourbinding is no longer something that's clearly on the table (though one thing to note is that making a 1/4 wazir move in order to change the colour of cell the piece is on costs a tempo (as often/normally is the case) plus a lot of speed, but it's not so clear why such would carry a big penalty, and about how big it might be, trying to weigh things crudely). Fortunately for me (and my sleep at night) I can treat the aforementioned piece as a compound one.

H. G. Muller wrote on 2018-12-28 UTC

Well, to avoid divergence of the argument, I will limit my current reply to the following observation:

According to your method, you get a value for this '1W-replaces-1F Bishop' which is (somewhat) larger than what you assign to a regular Bishop. Which could actually be correct, because it has almost equal mobility, but is not color bound. But that then is a coincidence, because in no way did you invoke the fact that the W move lifted the color binding. This way you avoided (for totally unclear reasons) to involve the 50% penalty you use in other cases of color binding, and only by virtue of ignoring that penalty you could avoid to be off by a factor 2.

But now start your method from the '1W-replaces-1F Bishop', take away its W step, and give it back its F step. Exactly the same calculation would now apply: removing the W step also reduced the average number of moves to 90%, the F step that you add is also 1/8 of a Guard. But you won't get back the value of the Bishop. Instead the calculated value increases again, to about 3.75 (on 8x8). A correct method should have worked both ways, and predict both the correct value of the 1W-replaces-1F Bishop from the ordinary Bishop, as well as the other way around.

And if you would not have ignored the fact again that removing thw W step causes color binding, you would have ended up with a B value of ~1.9. What sense does it make to have a rule that says you should charge a 50% for color binding, except when you don't feel like it?

Kevin Pacey wrote on 2018-12-27 UTC

Once again I'm not sure how to argue with your most recent post, H.G. For that reason, and for the sake of not risking discussing too many points at once, which may in turn multiply (as it seems has been happening), I'll just mainly confine myself for now to answering your query of me, namely:

If I would make a new piece, by starting with a Bishop and replacing one of its Ferz moves by a Wazir move. Would you now argue that a normal Bishop is worth only half as much as this piece, because displacing that one move to a neigboring square made it color bound? Or would you argue that the piece is worth one Pawn more than a Bishop because it is the combination of 1/4 Wazir with a piece that was only handicapped so little compared to a normal Bishop by missing this Ferz move that it had no effect on the value?

The calculation I'd make for such a piece's value is a bit complex. First, I decide that such a piece is treatable as a compound piece of sorts, then I figure out the value of it's components in stages. However, first I need to assume a given board size, namely 8x8. That allows me to figure out what fraction of a bishop is left when a ferz move is taken away from it. A normal B on 8x8 has 10 moves on average on an empty 8x8 board (i.e. 7 minimum, 13 maximum), so taking away a ferz move gives 9 moves on average (just averaging the new minimum and maximum cases, maybe none too precise a thing to do). Thus I'd work out what 90% of a B is worth as the first component of the compound piece.

The second component of this compound piece would be 1/4 wazir (depending on if it was the forward step it would be 2/5 of a wazir, or if it was a sideways or backward step it would be 1/5 of a wazir, based on how I've implemented your previous discussions about the direction of a piece step, but you specified 1/4 wazir this time, perhaps to make things a little easier for me). In this case it does, because I can easily use the value of 1/8th of a guard instead of 1/4 of a wazir (more or less the same thing) which avoids what seems like a worse slight error I get if I used (wazir-P)/4 rather than using (guard-P)/8 - the latter produces the same value as wazir/4 (the only times fractions of pieces might IMHO clearly work out very 'nicely' for me as it were is when division of a piece by 2 is performed, such as a guard broken into its ferz and wazir components, each having values that I deem to be equal, with a Ps worth of co-operativity first subtracted).

Thus, First Component + Second Component + Pawn = value of the [compound] piece you enquired about, according to the way I'd do the calculation (for now, with my imperfect way of doing things). This becomes:

(B-P)x0.9 + (wazir-P)/4 + P = approx. value of compound piece, or (B-P)x0.9 + (guard-P)/8 + P, which becomes:

(3.5-1)x0.9 + (4-1)/8 + 1 = value of compound piece (note I rate B=3.5 and guard=4 on 8x8),

and thus value of [compound] piece that you asked about = 2.25 + 0.375 + 1 = 3.625 on 8x8, which I'd note is clearly far from twice the value of a B on an 8x8 board.

One of the other problems I have to cope with is that this sort of method wouldn't work (in any sort of fashion, at all) if e.g. a wazir's value was necessary to use in a calculation, and it was worth a pawn or less, since wazir-P would then be worth zero or a negative value. This is indeed the case for the value I give a wazir on 10x10, i.e. I put it at 0.75 on that size board (which you objected to, as well, and I'll more or less pass on that, except to note for now that a wazir crosses the board slower on 10x10 than 8x8, which I count as a tangible consideration all the same, though other considerations pro and con may be possible, especially depending on the armies deployed).

Anyway, for 10x10 figuring out the first component I'd do similarly as before, but for 1/4 of a wazir to be computed I'd now definitely first desire to compute the value of a guard, then hope to use the value of 1/8 of a guard (a similar thing as 1/4 of a wazir), i.e. hoping that a guard is worth more than a pawn on 10x10 (if not, I simply have to use wazir/4 rather than (wazir-P)x0.25, with any difference/error being rather small anyway). It just so happens my home formula for a guard's value puts it at approx. 2.5 on a 10x10 board, so I'd figure out 1/8 of a guard by using (guard-P)/8 = 0.1875 (happily the same as wazir/4, again, as it always would be), which in turn is what I'd use for the second component of the compound, if I were to compute its value for on 10x10. The first component I see as worth 12/13ths of a B, so now the compound's value that you asked about (if on 10x10) would be approx. (with having B=3.5 still, on 10x10):

(B-P)x12/13 + (wazir-P)x0.25 +P, or (3.5-1)x12/13 + (guard-P)/8 + P, or

value of [compound] piece you asked about (if on 10x10) = 2.308 approx. + 0.1875 + 1 = 3.496 approx., which I'd note is again nowhere near twice as much as a B's value on the given board size.

I'll have to admit again that my formulae and methods are not completely perfect, and seem unsound (in particular with fractions of pieces), but the values I get with them to date don't seem ever too far out of the ballpark, to me at least. One interesting thought experiment might be what to make of the value of some sort of 'half of an archbishop'. Doing things my way, (archbishop-p)/2 would be the answer, rather than archbishop/2 (or is half an A worth something different altogether?), and clearly A/2 would give a value greater than a B or N if one uses one, or even two, pawns worth of cooperativity between the bishop and knight components. For now, I still just assume one pawn's worth of cooperativity between those two components, so for me on 8x8 archbishop =N+B+P=3.5(approx.)+3.5+P=8 (noting Q=R+B+P=5.5+3.5+1=10), and thus (archbishop-p)/2=3.5 is the value I get for half an archbishop, i.e. about the value of a N or B. That's opposed to archbishop/2=4, or greater than the value of a minor piece. At any rate, there seems to be some sort of consistency with quite a few of the piece values I've come up with over time, in spite of the lack of perfection, at least it seems to me so far.

In a previous post in this thread I dealt at length with how I estimated the value of an alfil and a dabbabah, including how I factored in such things as my x0.5 binding penalties, plus counteracting bonuses for leaping ability and speediness, if you wish to see instances of how I've handled binding penalties in my personal calculations, when compound pieces are not deemed at issue. I've done calculations for the value of a knight in Alice Chess using a way to take into account the type of binding to it that happens on the two boards there, and I came up with a value for the N (and other pieces) close to what the rules page notes gave for piece values (maybe by ZoG?) in the case of that game, at least. The values are on my 4D Quasi-Alice Chess rules page, in the Notes section (note I used the initial chess base values N=B=3, R=5, Q=9 in that particular case, perhaps to try to match the chess base values I thought were probably initially used in the preliminary calculations made, for Alice Chess, by ZoG - I did all that work long ago).

H. G. Muller wrote on 2018-12-24 UTC

Well, that the Amazon value is just Queen + Knight is what I found when playing games where one player had one of its Knights removed, and an Amazon instead of a Queen. Neither player turned out to derive an advantage from this imbalance, in a match of a couple of hundred games. I was as surprised as you are. Perhaps at some point a piece is already so mobile that some kind of saturation sets in, and extra moves just don't provide that much extra. There also could be a risk penalty for 'putting so many eggs in one basket'.

My point was that your calculation is not self-consistent (and thus certainly wrong) if you use different methods for splitting pieces as for combining them, as split pieces can be recombined to give back the original piece, and that then should not suddenly have a different value.

But we were not really talking about splitting or combining here: you were comparing Knight and Camel, and calling the Camel the closest thing to a color-bound version of a Knight. (Indeed the Camel is the 'conjugated' piece of the Knight, i.e. it would be a normal Knight on the 45-degree rotated 'board' formed by the squares of one shade.) So we are talking of modifying moves to go one square instead of another. You applied a 50% penalty fot the resulting color binding, and that is totally off.

According to this estimation method, a Knight would initially not lose any value when I started to replace the (1,2) leaps one by one for the corresponding (1,3) leaps (as that would not cause color binding), until I replaced the very last move (after which it is color bound), after which it would suddenly halve. I don't think that would happen at all, but that the value decrease would be gradual. Yes, the Camel is significantly weaker thana Knight on 8x8. But IMO that is just because the (1,3) leaps are too large for the board. In games that pit Knight vs Camel I see that the Camels get usually lost in the end-game without compensation, because when they are chased away out of the center, a single move brings them so close to the edge that they hardly have any moves left (as their return to the center will remain barred). So that they are then trapped there. On a (much) larger board you would not have this problem at all, and a Camel might even be worth more than a Knight despite the color binding, because of its larger speed.

This is why I asked about the modified Bishop, (rather than an enhanced one), but I did not see an answer yet. So let me ask you again:

If I would make a new piece, by starting with a Bishop and replacing one of its Ferz moves by a Wazir move. Would you now argue that a normal Bishop is worth only half as much as this piece, because displacing that one move to a neigboring square made it color bound? Or would you argue that the piece is worth one Pawn more than a Bishop because it is the combination of 1/4 Wazir with a piece that was only handicapped so little compared to a normal Bishop by missing this Ferz move that it had no effect on the value?

P.S. It seems very wrong to rate a Wazir (4 captures, 4 non-captures, access to the full board) lower than a Pawn (2 captures, 1 non-capture, confined to the forward part of a single file until it captures, and even then confined to a triangle), on boards of any size. Promotion is surely worth something, but not that much, and it also gets more difficult on deeper boards. Even if you leave your Wazirs just sitting on the back rank as a sort of goal keeper, a Wazir must be able to trade itself for a passer that breaks through.


Kevin Pacey wrote on 2018-12-24 UTC

The one thing I've done so far is to treat a super-bishop (aka promoted bishop in shogi) as a compound piece where I add a B's value plus a wazir's value plus a pawn, on 10x10 (for my Sac Chess variant). On 10x10 I rate a B worth 3.5 (as opposed to a N being just set=3 there, unlike on 8x8), and I rate a wazir as worth 0.75 there (half of what I rate it as on 8x8). So Super-bishop=B+wazir+P=3.5+0.75+1=5.25 on 10x10, which feels about right to me, especially as most people seem to value it as about worth a rook on 8x8. Note I'd similarly rate a super-B as worth 6 on 8x8, slightly more than R (I set to 5.5), in line with Greg's post about the super-B compared to the R a while back in another thread, i.e. re: (8x8) Pocket Mutation Chess (both are put in the same piece type class in terms of value by that game's inventor).

Observe also that colour-binding is built into a B's known value, and having a B as a component of a compound piece still includes that built in binding penalty (whatever it is) for the B; the masking of the colour-binding by the addition of another component (in this case, a wazir) is taken into account by (in addition to adding a wazir's value) adding a pawn's value (only), much as Q=R+B+P in chess. So, there is no sudden doubling of the Bs value as it were, in the case of a super-B (or a Q) compound piece, the way I do this particular calculation (i.e. as a sum).

At the risk of repetition, that (not always perfectly applicable) Q=R+B+P analogy, when used for estimating the value of compound pieces, often seems (to me at least) to produce results that aren't too badly off, when I've run with it in order to make many of my estimates. The case of valuing an archbishop on 8x8 being one quite possible exception, however, though on 10x10 at least, I wonder if that piece might be nearly as potent as on 8x8 - I sense this when I play 10x10 Sac Chess, though I do sense a certain potency of an Archbishop when I play 10x8 Capablanca Chess. My guess is that the N component of the archbishop suffers from less influence on 10x10, the largest size of these three sizes. It's also quite possible a B enjoys a 10x8 board even more than a 10x10 one (indeed I rate a B as 3.75 on 10x8), so that may explain why an archbishop seems extra strong to me on 10x8, in spite of a slightly less influential N component (than if it were on 8x8).

One thing that still makes no sense to me, btw, is if Amazon at best =Q+N in value (as I recall the wiki for that piece implies), then why zero co-operativity between the Q and N components? That's why I feel still more comfortable with Amazon set=Q+N+P at the moment. There also may be a similar problem for Guard set=3.2 on 8x8, if ferz and wazir are each approx. 1.5 (as I vaguely recall the wiki for each more or less gave), as the co-operativity seems all but shockingly low between ferz and wazir, if so.

Note I still rate a rook as worth 5.5 on 10x10 (as I would for any number of board sizes), since for one thing I don't believe a B's value should ever be 4 or greater (since it can't often restrain 4 pawns in an endgame - though a problem for me may be that if I set Guard=4 [incidentally =ferz+wazir+P, perhaps] on 8x8, as some chess authorities have done similarly for a K's fighting value, the same reasoning, about not restraining 4 pawns in an endgame, might be argued), and a rook should pretty well be worth about a B and 2 pawns on any board size, at least for square or rectangular boards.

There are, I imagine, many things I have yet to try to take into account when tentatively evaluating piece values, such as what Betza has written about pieces with negative values.

Note a colourbound penalty of e.g. x0.5 can be just one part of an estimating process, possibly. There can be offsetting bonuses, such as a x2 bonus for a leaper. There can also be a x0.5 penalty for non-capturing movements that make up part of how a piece moves, too (then there's forward as opposed to sideways or backward movements by a piece, and how to reckon with the valuation of that). At the moment my repertoire of formulae and methods is limited, but, again, I try to keep my life simple when possible, and I hope to compare my estimates with existing ones, if any, to get a feeling for if I am in the right ballpark before giving an estimate of my own.

In the case of the (colourbound, but 8-target leaper) camel, its value of 2 is still close to N/2 on 8x8 (especially if N set to =3.5 is used, which is even close to Guard, if that's set to =4), coincidence or not, which gives me some encouragement. Incidently, I assume a leaper bonus is not used for pieces (or components of them) that take just a one cell step, which is why I see a N in a way more different from a ferfil than a N is from a camel, in spite of a ferfil being closer to a N in value (on 8x8, in all cases).

Aside from all that, I hope you (and everyone else) are enjoying the holiday season, H.G.

H. G. Muller wrote on 2018-12-23 UTC

Well, a 50% penalty for simple color binding (i.e. access to 50% of the board) is really a luducrous over-estimate. More reasonable would be 10%, and then only for the case that you do not have the pair. You cannot really believe that adding a single non-capture backward step to a Bishop (which would lift the color binding) would double its value?

What do you think a piece would be worth that can do all moves a Bishop could do to non-adjacent squares (i.e. the Tamerlane Picket), plus all Wazir moves (to make up for the lost Ferz moves)?

Kevin Pacey wrote on 2018-12-23 UTC

I am reminded now that I was going to mention that I once considered a camel to be be the closest thing to what could possibly ever be a colourbound knight, rather than e.g. a ferfil. Originally I considered a camel close enough to a N that I simply valued it as N/2 (usual colourbound penalty I apply being x0.5), but I saw that a camel was rated as 2 on 8x8 by a number of people and I also eventually accepted that the extra reach of a camel makes it worth a little more than N/2. You can certainly feel the effects of the reach, even on 10x10.

Kevin Pacey wrote on 2018-12-23 UTC

I happen to be up late, ready for bed soon, but I saw your post right after it was made, H.G.. Fwiw, for one older CV invention of mine (4 Kings Quasi-Shatranj), I had noted what I had estimated the value of an Alibaba would be on 10x10, based on my imperfect formulae and methods, and I had worked it out to be worth exactly 2, though I made no attempt to distinguish what it would be worth in different phases of a game, with the given armies involved.

H. G. Muller wrote on 2018-12-23 UTC

While Googling I got a 'surprise hit' on a posting of my own I had completely forgotten about! It seems I did study the value of the Alibaba once. In the opening, against a normal FIDE army on 8x8 a pair was worth 2 x 2.35 (where N = 3.25). I had no means to test how much of this was pair bonus. An individual Alibaba in the end-game seemed more like 2.00.


H. G. Muller wrote on 2018-12-20 UTC

Yes, that is the study in question. For your 'quibble' to carry any punch, you would at least have to show that it makes any difference which rating bin you take, i.e. whether there are any cases at all where only considering games of players rated 2300-2400 and only 2400-2500 Elo players or 2500+ Elo players would make any difference beyond statistical noise. (Which, for the 2500+ only bin would probably be intolerably large.)

As I explained, where this to be the case it would cast severe doubts on the usefulness of the concept 'piece value' in the first place. Which can only be partly cured, but by discarding the result of the 2500+ (and presumably also the 2400+) players, and including more games of 1900-2200 rated players. As those are the ratings relevant of players that would pay attention to piece values, while GMs and super-GMs tend to use a more 'holistic' evaluation of the board where positional factors usually dominate material concerns. (Weaker players cannot give those factors the weight they would deserve, because they are not able to identify them well enough. So that these terms for them would just represent random noise and are better ignored.)

Kevin Pacey wrote on 2018-12-20 UTC

Again I find myself unsure how to argue with all of your previous post in this thread, H.G.

Something that is a small quibble I have is that according to the following link, Kaufman's study of material imbalances involved the study of games of 2300+ vs. 2300+ FIDE rated players (assuming the link refers to the proper Kaufman study), i.e. implying that at least at times the games did not involve opponents who were each 2500+ (i.e. minimum grandmaster level):

H. G. Muller wrote on 2018-12-12 UTC

Of course I don't think that B=N=3 is just an approximation for beginners, other than that it ignores to mention that the Bishop pair is worth protecting. Even in the hands of human GMs the equality turns out to hold in practice. Now you have been complaining that the games that showed this were just by any GM, and not only by the top 200, but you haven't shown that this would actually make any difference. It is nothing but whimsical thinking and clutching at straws. One can always conjecture that things would be different if ... (and then some condition that is impossible to fulfill, like having God play a million games against himself). But there is zero indication that this might affect the statistics. I, on the other hand, have of course investigated if the results would be dependent on the level of play. Because raising the level of play by computers is costly in terms of time needed to generate the games, so that I prefer to not needlessly use a high level of play, and wanted to know what speed I could afford before heavily investing in measuring all kinds of imbalances. With the imbalances I tested this on I found no effect at all on the piece values when varying the thinking time by a factor of 10 (which would amount to a strength change of ~250 Elo), or when using a different engine that intrinsically was ~400 Elo stronger. Of course the total score for a given imbalance gets closer to 50% if play gets less accurate, but it does that for all imbalances in equal proportion. And in particular, imbalances that are 'neutral' stayed neutral at all levels of play.

Of course all this arguing still bypasses the most important point: if games played by Gods would have different statistics from those played by us mortals, we should use the values derived from the statistics of the mortals. Because we are mortals, and what Gods can afford has no relevance for us.

Piece values are just an aid for making statistical predictions on the outcome of a game (i.e. the probability for win, loss or draw) from the present material alone, without knowing anything about the board position. And even in that context they are an approximation, based on a model where the 'value' of a certain material balance can be obtained by simply adding the value of individual pieces. But that model is not very accurate; in reality presence of other pieces (in equal numbers) has an effect on the value of the imbalance. E.g. with an imbalance of Queen vs 3 minors, it is known that additional Rooks improve the chances for the minors. With B vs N it is known that the number of Pawns affects the result. And in Capablanca Chess Q is (much) better than R+B (and more like R+B+P, like in normal Chess) if the C and A are already traded, but on average it (slightly) loses against R+B if C and A are both still present, showing that this effect can grow rather large.

This makes it rather pointless to specify piece values to a precision better than ~0.1 Pawn. Who cares whether a Bishop = 3.5 or 3.49 in the grand average of things, if in any real position it will almost always count for 3.4 or 3.6 depending on what else is on the board, but you don't know which? Using 3.49 instead 3.50 will have no detectable effect on the typical error in the result prediction for a given total material on the board.

In addition, there are also positional factors that (by definition) cannot be taken into account for piece values (they can only be averaged over). But when we use the game values during game play we know of course the position we are in, or can get to, and we do take that into account. The difference between a centralized Knight with good mobility, and a Knight in a corner is much more than 0.1 Pawn. Strong players will also distinguish 'good' and 'bad' Bishop, and that difference can outweigh the advantage of the Bishop pair (i.e. 0.5 Pawn), leading tho a preference for exchanging the bad Bishop for a Knight. None of those decisions would be in the slightest affected by whether we set B=3.49 or 3.50. In physics we would say that extra decimal is just meaningless precision on a measurement with a much lower accuracy.

Also note what Ralph Betza called the 'levelling effect', which (in a somewhat course formulation) says the effective value of two pieces fighting on opposite sides cannot be very close if it is not exactly the same. The point is that a stronger piece loses value if it gets burdened by having to avoid 1:1 exchange for the opponent weaker piece. A Bishop that would be 'royal' w.r.t. a Knight (meaning that it cannot expose itself to Knight attack like a King cannot expose itself to any attack) would be less valuable than a normal Bishop, and the difference can easily be larger than 0.1 Pawn. So if the intrinsic difference in value would be less than 0.1 (or whatever the obligation to avoid trading costs you), you are better off treating them as if they are equal. Which again means that the imbalance will hardly be worth anything, as you will have little opportunity to cash on the intrinsic superiority before the imbalance gets traded away.

So it could very well be that the intrinsic value of a Bishop is larger than that of a Knight, when measured through the imbalances B+N vs R(+Pawns) and 2N vs R(+Pawns), where the B+N would then do better because the opponents has no Knights it has to fear. But that the leveling effect destroys this advantage for the imbalances B vs N, B+N vs 2N or B+2N vs R+N(+Pawns), where there is a Knight for the Bishop to fear.

Kevin Pacey wrote on 2018-12-12 UTC

I thought a little more about your second last post, and maybe I can add some insight regarding chess books and the uselessness of an advantage below a certain size for a given category of human player.

In the case of chess books, I haven't read much literature written recently for beginners, though you mentioned that level and it's a place to start. In old Reinfeld books meant for beginners (or low-level novices), he'd regularly give B=N=3. He probably knew even in his day that GMs considered a B just a shade better on average, though for Reinfeld to explain all the reasons that might be true would involve explanations over the head of his selected audience level (and the space limitations allowed by his editor, too, perhaps).

Another bit of half-truth that has been regularly used by chess authors, for similar reasons, is the explanation of the first few moves of the main line of the Scandinavian Defence, namely 1.e4 d5 2.exd5 Qxd5 3.Nc3. At this point authors often say something like Black will now have to lose time in the opening. But the truth is not so simple, as I have yet to see pointed out. First of all, 2.exd5 is not in any way a developing move. Second of all, after 3...Qa5 (or 3...Qd6) the level of development is equal (except White is on move, as usual): Both sides have a piece deployed, and each side has one diagonal open for a bishop.

However, Black has commited his queen rather early, and after good play by White the Black queen will have to move again, only now losing time, or Black will need to make some other sort of concession. Try writing all that, plus giving analysis, in a book with space limitations, though. So, a half-truth is conveniently told instead. There is even more to the story, as White's gain of time may not be won with a particularly potent extra move (e.g. Bc1-d2 played at some point after 3...Qa5), so that's why the Scandinavian Defence may not turn out to be so bad for Black - but to establish the evaluation of the opening takes in ever changing opening theory and evaluations (these days often contributed to by computers).

On top of all that, there is a lot of poor or obsolete chess writing out there. Fischer once 'wrote' a book that was to teach chess, but all it was was puzzle positions, if I recall correctly - i.e. not even piece values given in that book. There may also be overly complicated writing done for novices, too, at times (but not at all often) and it probably leaves them asking even more questions than e.g. simply if they were told B=N=3, which admittedly suffices to meet their needs at their level. However, they are also often told R=5, and I think (unlike you, perhaps) having a R for a N usually would be a significant advantage to have even at their level, even though a N is a tricky piece tactically speaking.

A further insight may be that Kaufman also once studied what material odds a lower-rated player might be given, and one of his conclusions was that a player rated 600 points lower than his opponent could be given the odds of a N, to evenly match the players. That study result I have little reason to doubt, for now, though I have not thought about it much.

The quest for piece values is clearly important to chess variant players, and a lot would prefer them to be as precise as possible, even if that matters not a bit at their level, and even if the quest for perfect accuracy is pointless (though there's also what value to state in a wiki or book, if one is forced to or wishes to). It's a bit like in chess where amatuers play complex openings like the Najdorf Sicialian where Black sometimes hangs on by his fingernails in certain sequences of moves played at elite level. At lower levels, the players wouldn't even find the moves over the board, or understand why they are played, just like having B=N=3 would suffice for them rather than N=3.49 and B=3.5, for example, if that happened to actually be the real values. Amatuers would be better off playing the safer, less trappy French Defence, for example, to end the rather imperfect analogy I first started. Sometimes a certain degree of precision for piece values may matter (at the least psychologically), though, say if one is debating whether to throw in a pawn, as part of a 3 for 2 trade, where the 3 includes said pawn.

Incidently, it's not clear to me you are a 'patzer' as at least once you corrected some faulty but very short analysis I hastily gave of a (fairy?) chess position (unless you somehow used an engine, which I doubted it was possible to in that particular case since the evaluation of whether an endgame was a draw was at stake, and you explained with words more than actual moves).

H. G. Muller wrote on 2018-12-11 UTC

Perfect play is usually not helpful at all in determining piece values. Because piece values are a heuristic aid to be used by inperfect players to begin with. In perfect play the only thing that matters is distance to mate, and you don't care at all about which pieces get lost or gained in the process of forcing or delaying the mate. And all moves that draw are equally good. There is no such thing as a 'nearly won' or 'nearly lost' position with perfect play; both are just draws. The difference is only determined by how large the chances are that an imperfect player will make the mistake that will push him over the edge.

We now do have perfect play (theough End-Game Tables) for all end-games of orthodox Chess with 7 men or less. Amongst those are B vs N with 1 vs 1 or 2 vs 1 Pawns. But you would be hard-pressed to deduce anything about the B vs N value from those. Counting the fraction of won and lost positions isn't very helpful, as these are completely dominated by the trivially won and lost positions, where either the N or the B is hanging and captured on the first move. And then there is the somewhat deeper tactics that loses a minor, through a fork or skewer. Even if you weed all those out the remaining positions are often of a type that you would never encounter in real games. To make any sense of it you would really have to know the probability that each 7-men position will be reached by simplifying from a more complex one in games. It tends to be that the 'silly' positions have the largest probability to be a win or loss.

I think this whole obsession with (near) perfect play is a fallacy. This is well known in technology. There exists for instance a device called a 'strip detector', used to measure the position where a particle (be it light or elctrons) hits a screen. It consists of a number of 'collector' strips next to each other. If you let the particles fall directly on the strips, you could never get a resolution better than the width of the strips: you would know which strip it its, but you would have no clue whether this was close to the edge or to which edge. So it is typically used behind a 'diffuser', which blurs the incoming particles (by disturbing their trajectory in a random way) to about the width of a stip. Then it always hits multiple strips, and the fraction that falls on one strip and that on its neighbor make it possible to see if it was just on the boundary between the two, or in the center (almost no overspill on either side). The imperfection of the imaging allows you to determine the point of arrival an order of magnitude more precise than with perfect imaging, to a fraction of the strip with.

Now this is exactly what we have in Chess: a detector with 3 stips, a narrow 'draw' strip in the center, and 'win' and 'loss' strips on either side of it. Precisely imaging a set of positions with a certain material imbalance onto this detector, by perfect play, will make it completely invisible whether you hit the draw strip in the center or near an edge. Blurring the imaging by imperfect play, however, increases your resolution, and makes it very easy to determine if the positions were on average 'nearly lost' or 'nearly won'. The thing of interest is how much perspective the material imbalance offers when you are up against an imperfect player, taking into account that you might make errors yourself as well.

Kevin Pacey wrote on 2018-12-11 UTC

I'm not sure how to argue with all that. I was hoping there just might be a clearly good way to establish objective, or even perfectly calculated (or estimated) piece values. Unfortunately, I suppose there's no way we can know what'd happen if God played God (or if He already knows the piece values we should all have, regardless of whether it helps our play enough at our particular level, if a given size of advantage is not significant enough there). Also, if chess is ever solved by man, there still might be some trouble isolating whether B=N on average.

Aside from that, I'd note I added an extra paragraph to my previous post just now.

H. G. Muller wrote on 2018-12-10 UTC

What you say is inconsistent. If top-200 players would get different results in B vs N imbalances as patzers like me, why should I (and other patzers) care the slightest what results they got? They might score 90% with the Bishop, but if at my level of play the Knight would win more often than not, I would do wise to consider the Knight more valuable.

Either piece values are a meaningless concept, because they are different at every level of play, or they are the same for everyone, in which case it wouldn't hurt the slightest to take the statistics from a pool of patzers.

Well, I have never seen a chess book for beginners that says: super-GMs consider a Rook worth 5 Pawns, but you are a beginner, so for you the Knight is more valuable...

Kevin Pacey wrote on 2018-12-10 UTC

I did recall the Kaufman study, although that I count (correctly or not) as a 'computer study' of sorts; it is relying on statistical analysis of games that may not have approached close to perfect play in a considerable number of the cases. Ideally, B vs. N studies that involve humans would include only, say, the top 200 in the world chess players playing each other. Maybe even just top 10, although the sample size would be too small I suppose for many years to come. Bobby Fischer's virtuoso exploitation of B over N in certain endgames of his is not something everyone could do, at least in his day. A statistical study of today's top engines playing each other a large number of B vs.N positions ought to be revealing, if it's ever been done. Otherwise, I had recalled that your own B vs. N study with engines playing themselves (Fairy-max?!) you wrote had a result that matched Kaufman's result, and that's from what is clearly a computer study.

Kaufman's study I admit I'm now unfamiliar with the lowest ratings of the human players in the games he included, but I was assuming the average rating of the players involved was not incredibly high (i.e. not even making it to 2500 FIDE, or minimum grandmaster, level), as I recall a very large number of games were involved in his statistical study, and any grandmasters involved apparently thus could not have played just with their own peers (or with super-grandmasters) in those games. I may have looked at a link to the study long ago, perhaps.

I'd heard or seen somewhere that Kaufman was in on giving his piece values for the programing of one engine (can't recall which), including the values that equate single B=N, so what you say about engines regularly slightly rating B's over Ns even nowadays is even more interesting to me.

H. G. Muller wrote on 2018-12-10 UTC

But the conclusion that lone B = N was originally not from a computer study at all: it was something Larry Kaufman noticed from a huge database of human GM games. In otherwise materially equal positions, the Knight won as often as the Bishop.

He detailed this result, though, by also classifying the B vs N positions by number of Pawns. There he found that the equality was only exact when each side had 5 Pawns; with fewer Pawns the Bishop advantage grows, with more Pawns the Knight advantage grows.

If you don't believe that (i.e. if you believe GMs in general don't know what they are doing when they are playing where the B vs N imbalance occurs), don't blame computer studies... Most engines set the value of a lone Bishop somewhat higher than that of a Knight, b.t.w.

Kevin Pacey wrote on 2018-12-10 UTC

Regarding my playing against Fairy-max at Sac Chess, I recall I played at a speed chess kind of time control for the first (4)  games, and I thought I was holding my own in most of them for a considerable time until I made shallow tactical blunders in each of the games that cost me a significant amount of material, and ultimately the game. I'm not that great at fast played chess (or fast played CVs by extention), especially against a computer, where psychology is also a factor for me, just being a tactically fallible human, and one who is getting easily agitated in his old age too. There is also that a 10x10 board is bigger to visualize and generally think about (than 8x8), and that Sac Chess must be more complex than chess in terms of tactics and strategy (much is waiting to be discovered, if the variant's played often enough).

I think I misremembered the (4) later games, which I played at a slower rate, and forced the engine to move after only about 2 minutes on a given turn. In fact I now think I just won 1 game and drew one, and lost 2. My misfortunes were always due to blunders, and the engine didn't see far enough ahead in the game it lost. I seem to recall that the drawn game was due to a draw by 3-fold repetition that might not have been clearly necessary. So, not much of a test of piece values.

I think I wrote in the Sac Chess thread long ago that I pitted the engine against itself for one game, and amusingly it came down to a pawn ending, but unfortunately where the winning side was two pawns ahead. A lot of tactics going on in that game, and towards the end the winning side swapped two or three pairs of pieces off just to force the pawn ending, even though it was already clearly ahead by more than 2 pawns worth. I doubted it was already calculating all the way to the point where it promoted a pawn to an amazon, as that was many moves ahead.

Also in the Sac Chess thread, Carlos Cetina posted a game where he decisively beat the engine; the game started off with the engine allowing the trade of one of its knights for two centre pawns. Although, Carlos didn't write what the machine's rate of play was like, and it may have been made to be just a bit too fast for it to play sufficently well. In any event, things got far worse for the machine from there onwards. Carlos is a regular player on Game Courier, and it seems to me he would have a FIDE rating of at the least 2000, if he were to play over-the-board chess seriously by itself.. Other than that, a lot (if not all) of the games of Sac Chess I'm mentioning had Fairy-max playing moves that seemed to me, as a chess player anyway, to be anti-positional, in that it would block its centre pawns with pieces, even very valuable ones, in the opening phase.

At the moment it's clear that my ways of estimating piece values are, as a general rule, less to be trusted than estimations derived from computer studies (accounting for Fairy-Max' choice of values, I assume), but that does not necessarily mean computer studies always get results that are absolutely correct, either. The conclusion of computer studies that a single B=N in chess is particularly hard for most experienced chess players to swallow, even though that feeling is based on books and top human player's philosophies. A computer equating a B to a N (on average) may regularly beat a world champion, but that particular valuation would seldom if ever prove to be why the human would lose any particular game to a machine, I suspect.

H. G. Muller wrote on 2018-12-10 UTC

I think the term you are looking for is 'color alternator'. I don't think there isn't much consequence of being a color alternator for the piece value, though. It merely means that there are some squares where you could not go in an even or odd number of moves, but for any piece that is not a 'Universal Leaper' there are always squares where you cannot go in a given number of moves, and it doesn't matter much which color they have. (There are some quirky positions, though, like { white: Ka8, Nh1, Pa7; black Kc8 } which are draw depending on who has the move because of this alternation, but that is just a coincidence, because the defending King here has to keep switching between c8 and c7 to keep up the defense, which happen to be of opposite color.) The significance of 'full' color binding that there are squares you can never go.

Note that 'color binding' is just a special case of 'area binding', more obvious than other forms because in western chess we happen to checker our boards. Pieces that move only in two opposite directions are confined to a single 'ray', and if the opposite steps are equal in size and not minimal, they can even only access part of that ray. If they have two pairs of such moves (like the Alfil) they just sample a subset of the squares in a subset of parallel rays. All point-symmetric pieces with 4 moves suffer from this, except the Wazir (which is the only piece with minimal step in both dimensions, so that it never skips any square). So also left- and right-handed Half-Knights, even though they alternate the 'checkering color' on each move, they still are restricted to 20% of the squares, some light, some dark in the usual checkering. You could paint the board with some 5-colored pattern to make this obvious ('meta-colors'), but the left- and right-handed Half-Knights would need different patterns.

As for the strength of computers: by my standards your rating is very high, and from what you say that you score somewhat below 50% against Fairy-Max in Sac Chess. So why would you consider Fairy-Max' experience considering the value of pieces any less reliable than your own? There is not only the level of play, but also the number of games that constitute this experience. Fairy-Max can play tens of thousands of games in a few months, by just letting a few copies of it run 24/7, you probably did not play more than a hundred in any particular variant. In addition, for determination of piece values I would force Fairy-Max to play with the relevant imbalances from the start, (like Q vs 2R or Q vs 3 minors) rather than just waiting for the small fraction of the games where they occur by coincidence, so that every game is relevant, rather than just some 10%, the other 90% being decided because one of the players gained a Pawn somewhere during the game and could convert that to an end-game (or the game ending in a draw without there ever having been a material imbalance).

Note that Fairy-Max plays variants just as strongy as it plays orthodox Chess. This must be so, because it doesn't have any knowledge specifically pertaining to orthodox Chess. This is why I considered it a good 'test bed' for evaluating values of unorthodox pieces. Also note that for some specific variants much stronger engines are available, and that for determining the value of the Capablanca pieces I used a dedicated Capablanca Chess engine that had a computer rating about 400 Elo above Fairy-Max. (And nowadays engines are available for free download that are 400 Elo stronger still.)

Kevin Pacey wrote on 2018-12-10 UTC

H.G. wrote:

"I don't understand what you mean by a 'fully once colorbound Knight', and thus whether your conjecture that it is worth half a normal Knight makes any sense. A Ferfil has 8 moves and is color bound. Would that make it worth half a Knight? In practice a pair of (unlike) Ferfils is worth as much as a pair of Knights, on 8x8.

Why do you think that computer programs are weak? Can you easily beat them? Why do you think that the level of play matters anyway? Isn't it true that in a game between purely random movers the side with a Queen versus a Rook would already have a significant advantage, in terms of win rate?"

In chess, as I think I may have read on this website, a N is in a way colourbound (but as I would put it, only 'once', i.e. by just one 'binding') for every second move that it makes (this is also illustrated, differently, in Alice Chess)., so it is not 'fully' (i.e. every time it moves) colourbound. I do not know if there's existing CV terminology that describes this differently, and in fewer words - I suppose if I simply say something is colourbound, that phrase is always understood to mean all that I wrote, but I unnecessarily tried to be more precise.

I have trouble imaging a CV where a N could be fully once colourbound (even if more than one board is involved in said variant, as in Alice Chess), barring that some weird board shape could make it physically possible, if that's even possible in itself, but otherwise I had meant to (yet again) illustrate in my previous post that my usual way of giving a 'binding' penalty is to divide by 2, at some stage of a calculation, although I don't bother with such a penalty for the case of a ferz, even on 8x8, or in the case of a B (as its total value is known on 8x8, or I simply increase it a bit in another [imperfect] way, for bigger board sizes).

Aside from all that, I'd note that a ferfil (on 8x8) for me equals (N-P)/4+ferz+P=3.125 (with my assuming N=3.5, as per Euwe, and ferz=1.5). Note if I valued N=3 and ferz still 1.5, I happen to get ferfil=3, like for a N, too, which would seem to have been rather nicer in this particular case. However, all this is using my Q=R+B+P analogy, plus my imperfect way of estimating an A (i.e. as (N-P)/4) which, to be kind to myself, I would say is not always fully appropriate (it's clear I have work to do if I ever aim to be a serious CV piece values authority, at least for all my given estimate cases and methods used to get them).

I had a couple of years ago noticed the chess rating of a program used for computer studies when it was entered into a computer chess tournament, and it's rating happened to be relatively low, i.e. around 2300, assuming I remembered the name of the program right. I'm currently about 2200 Canadian (2400 peak rating Cdn 8 years ago, maybe a bit of a feat for me since I was around 50 then), or almost 2300 peak FIDE rating when I was about 30 (under-rated juniors with low FIDE ratings killed that). However, I as a human would have trouble against a computer even if it had just my rating, as it would never blunder at a relatively shallow move level. Your old Sac Chess program, which may not have had relatively many heuristics, I played several times long ago, and I beat it just a couple of times, but only when I didn't let it look ahead for more than a couple of minutes for any given move. Sac Chess would happen to have a high number for the average number of legal moves available during a game, I'd note.

Even if some of the best chess engines and hardware become available for CV piece type studies (have they yet? I don't know), it would seem it might take some time to (extensively?) modify their algorithms to play CVs nearly as well as for chess, though calculating speed may compensate for that a lot in the beginning. Still, chess computers became much stronger than humans at first because of better and better heuristics for chess specifically programmed into them, I've heard - perhaps chess grandmasters' brains were picked for the heuristics. However currently there are no CV 'grandmasters' other than for a handful of CVs (i.e. chess, shogi, chinese chess, at the least). So, I have doubts about the strength of available CV engines for now, though you might be able to quickly inform me why I shouldn't.

A match based on games between very low level players where one side has Q for R at the start (e.g. 1000 FIDE rated adults) would be a significant edge, it would seem, for the side with the Q. Not sure it would win nearly as often as it should. In an extreme but rather imperfect analogy, a timed contest of monkeys with typewriters pitted against each other writing a million 'books' where half the monkeys get a twenty page head start might not get significantly better literary results (in the eyes of most humans) for the latter group. It would also be so usually for unrated 2 year old kids playing chess with one side having Q vs. R edge. For 2300 vs. 2300 computers, with tree searching allowed, but heuristics (other than assigned piece values) banned, with one side having Q for R edge, it should prove a decisive edge, and any typical errors made for that level would prove relatively insignificant in light of the now crushing material edge, and the games might even show examples of at least some sufficiently adequate methods to make use of the advantage. Otherwise, it's pretty tough for me to be categorical about such hypothetical situations as random move games (i.e. 1 ply search depth, no heuristics?!). I once had a BASIC program on a PDP/11 to do just that (but did not try Q vs. R handicap), and the results were not pretty. Possibly the 50 move drawn game rule might come into play almost every time in any game/match, almost no matter how great a material advantage one side might ever have at any stage.

H. G. Muller wrote on 2018-12-09 UTC

One quick side remark: Wikipedia pages have a 'history' tab, where you can see exactly who contributed what, over time. There also is a 'talk' tab, where you can see how several authors come to a concensus about issues, before they make modifications.

Indeed the empirical values of Man, Knight, or Ferfil found from computer games are roughly equal.

As you remark upon yourself, the formula you use for combining pieces cannot work both for combining two Quarter Knights to a Half Knight and for combining to Half Knights to a full Knight. So it must be obviously wrong in one of the cases, which means it can be wrong for some pieces, and thus could easily be wrong for all the pieces you applied it to. It is basically just a coincidence when it works. The problem is obviously that you add the same value of 1P all the time, irrespective of whether the combined moves actually cooperate well, or are sufficiently valuable to begin with.

I don't understand what you mean by a 'fully once colorbound Knight', and thus whether your conjecture that it is worth half a normal Knight makes any sense. A Ferfil has 8 moves and is color bound. Would that make it worth half a Knight? In practice a pair of (unlike) Ferfils is worth as much as a pair of Knights, on 8x8.

Why do you think that computer programs are weak? Can you easily beat them? Why do you think that the level of play matters anyway? Isn't it true that in a game between purely random movers the side with a Queen versus a Rook would already have a significant advantage, in terms of win rate?


Kevin Pacey wrote on 2018-12-08 UTC

I've edited my last post considerably, for any who missed it.

[edit: Did so yet again, on 8 Dec 2018.]

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