Comments/Ratings for a Single Item

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.

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)?

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.

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.

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).

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?

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.

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. 

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 :

The author, Kevin Pacey, has updated this page.