Comments/Ratings for a Single Item

My intuition tells me that you greatly underestimate the value of the FAD, Wizard and Champion. On 8x8 short-range leapers with 12 targets are as strong as a Rook.

Hi H.G.

I know I've estimated the value of a champion on 8x8 in the past, and I recall it worked out to be close to a rook for me. On 10x10 I rate the wazir and A, D components of it to be of considerably less worth (than on 8x8), which accounts for the lower estimate I got for the Champion on 10x10 compared to on 8x8 (it'd be a similar story for FADs and Wizards, I'd suppose). However, my methods of estimating are admittedly based on primative ways of calculating e.g. net values for compound pieces.

The point is that there seems no justification for devaluating the short-range Champion moves that would not also apply to the Knight. Yet you keep the Knight at 3. The other pieces have 1.5 times as many moves, and piece values are known to grow faster than linear with the number of moves because of cooperativity. So if Knight = 3, a 12-target leaper should be 4.75 to 5.

On 8x8 I value a N not at 3, but at Euwe's 3.5 (or I'd say unofficially 3.49, just to make it a hair less than a B, but 3.5 is a rounder number), and also I value R=5.5 (Euwe), A=D=(N-P)/4(approx.)=0.625 and W=(Man-P)/2(approx.)=1.5 (though I'd suppose it's actually a little less since a ferz is thought worth a little more than a W). Cooperativity (your own idea?) I'm not sure about the meaning of. One thing that somewhat influenced me to believe my value for a WAD (or Wizard) on 10x10 was about okay is that the strategy page for Omega Chess' commercial website advises that B=4, like for a Champion(WAD) or Wizard, but it was advised not to trade a B for a Champion, in that 10x10 game (plus its 4 extra cells), I seem to recall. Not sure if the advice was meant just for the opening phase of a game. Nor do I know who wrote the strategy page, either. I don't rate a B as worth 4 pawns, quite (prefering never to use more than e.g. 3.99 for any board size with at least some breathing room on it - otherwise the thought that in an endgame a B can sometimes restrain 3 passed pawns, but seldom 4 of them, affects my reasoning, correctly or not).

Regarding the above, an A has half as many targets as a N (which moves in a somewhat similar way, at least in terms of range and being a leaper), and is thrice 'binded' too (which I take as calling for a [further] halving, only, since as I see it there is a x2 leaper bonus offsetting a x0.5 penalty I would give for one of the bindings, and I also gave a x2 bonus for an A being able to move faster across a board to a certain square at times than a N, as a [possibly generous] way to offset one other binding x0.5 penalty). A D has a similar story, except it's only twice 'binded', but I also don't think of it as quite as often being speedier than a N to get to a given cell that both might want to reach eventually (aside from the square reached via a one move leap by the D - though that is not so bad for the N to get to as for it getting to a cell that an A moves to in just 1 turn). It may seem the above reckoning is fishy somewhere, but the value I get for A (and maybe also for a D - note it can be slower at times than an A to get to a given cell) seem about right to me. The wikis for As and Ds rate either as worth a bit more than a P on 8x8, which I find hard to believe in the case of an A especially. Anyway, a Man is a compound of a W and a ferz, the latter two being of roughly equal value IMHO (though your own results disagree with Man=4), and I value Man=W+ferz+P, similar to Q=R+B+P in chess (the latter is an equation I often depend on heavily as an analogy when I calculate/estimate values, quite possibly incorrectly at times, but it helps keep my life simpler).

Fwiw, for the WAD on 10x10, first I rate a (lone) Man there worth 2.5 approx. (using my rather unproven formula for Man value that doesn't apply to all possible board sizes - I have an even more complex such formula for N value), thus rating a W or ferz there worth 0.75, and an A or a D there as worth 0.5 each. Assuming these values aren't too far off, since Q=R+B+P in chess, I hazard to rate a WAD=(W+A+P)+D+P where P=1, to get WAD=3.75 (for 8x8 I get WAD=4.75 with such a calculation). Thus 2Ps worth of value is being added in by this calculation (I neglected to mention this detail earlier). I figure a camel is still worth 2 on 10x10 (as on 8x8) due to its considerable range, so a wizard=camel+ferz+pawn=3.75 I'd say similarly (for 8x8 I get wizard=4.5 with such a calculation). Unfortunately, for 10x10 at least, my results don't agree with the effects of the notion of cooperativity, as you described them. I have trouble understanding the effects described, too. For example, on 8x8 a camel is worth 2 and has 8 targets (max.). A wizard has 1.5 more targets, so I'm guessing by your example calculation that only camelx1.5=3 must be exceeded for the value of a wizard on 8x8, which isn't saying too much yet. Unless describing the effects of cooperativity depends on using a (normally more valuable) N rather than (e.g.) a camel for one's example calculated estimate of a 12-target leaper's approx. value (on any size board).

https://omegachess.com/strategy.htm

I've extensively edited my previous post in this thread, for any who missed it.

{edit: slightly edited again, and with link added for a commercial website's page 'Omega Chess Strategy' (featuring piece values, including for Champion and Wizard piece types).]

Well, it depends on what scale you use; I quote values on the Kaufman scale, where R = 5 and N = (lone) B = 3.25. Which is obviously different from the Euwe scale. The problem is that '1 Pawn' is a very poorly defined concept; because Pawns suffer their own pretty bad form of 'area binding' there are many different Pawns, spanning a factor ~3 in value. So depending on what you imagine to be the 'standard Pawn' you get different scales.

The point is that all symmetric 8-move leapers that are not 'sick' in some global way should have approximately the same value on a given board. In this case apparently 3. And that symmetric 12-move leapers should be more than 1.5 times as valuable. Within a class with a give number of moves the differences are only minor (due to over-all effects like speed, forwardness, mating potential). Even color binding appears to hardly affect the value, as long as you have the pair.

'Cooperativity' is the +P in Q = R + B + P; ignoring it you would have Q = R + B, which is obviously quite wrong. For pieces with sliding moves it is often hard to predict, e.g. why it is ~2P in A (Archbishop) = B + N + 2P and only 1P for Q. For short-range leapers on 8x8 the formula value = 33*N + 0.7*N*N (centi-Pawn) works pretty well for the average (symmetric) leaper with N move targets, and the quadratic term describes the cooperativity between moves that by themselves would be worth only 33 cP. Adding 1P just any time when you combine two disjunct pieces is completely arbitrary, violates known facts, and in fact makes no sense to begin with. Such cooperativity bonuses should be relative rather than absolute, or pieces that are worthless by themselves (e.g. pieces that have only a single sideway non-capture step) would combine to give at least a Pawn. (And you can be sure that a piece that cannot capture and only moves along its rank is worth much less than a Pawn, if it is worth anything at all.)

I am not sure why you dwell on the value of A or D. These are 'sick' pieces because of their heavy (meta-)color binding, which gives them a value far below that what you would expect from the average contribution of their individual moves. In Shatranj an Alfil is considered to be worth about a single Pawn (but since you have so many Pawns it is often better to hang on to the Alfil), and a Ferz about two. But Shatranj Pawns are worth significantly less than FIDE Pawns, because of their worthless promotion. But you cannot draw any conclusion from that as to how much these moves would be worth when added to a piece that is not sick to begin with. Detailed study has show that W, F, N, A or D moves are all roughly equally valuable, the more important effect being that forward moves are worth about twice as much as backward or purely sideway moves. A 'Half-Knight' (which has only the right-bending or left-bending moves of a Knight) would not be worth more than a Dababba; more-likely it woul be worth less (because it is confined to 1/5 of the board).

I would not have much confidence in what vendors of commercial variants claim. Or what I read in the internet in general.

I mentioned more about the A and D as I'd used a formula for them in the first paragraph of my previous post in this thread, and I thought I'd elaborate on my reasoning for the value I got for each of these pieces, which figured in my later estimate of the value for a WAD.

Sometimes it's easy to incorporate some of your observations (or other people's, such as Betza's) into the ways I use to estimate piece values. I seem to remember somewhere seeing a x0.5 penalty for non-capturing moves by piece types, for example.

Other times I'd have to go back and revise many estimates I've given for piece types here and there. I've sometimes gotten around to doing this. Meanwhile, I usually note that my estimates are tentative where I write them down. I figure people appreciate seeing something they can chew on, even if the values given sometimes turn out to be considerably off. Perhaps I can in time give your values (where known) in all places where I give mine, for the sake of comparison by the reader. The values of Archbishop, Amazon and BvsN are particular problem issues for me, on 8x8 at the least.

I suspect most players take values given anywhere (perhaps moreso when given by someone not into CVs seriously for that many years, like me) with a big grain of salt. Many/most CV inventors/commentators (e.g. Fergus, perhaps) never, or almost never, give numerical values (though, e.g. Fergus, may indicate a hierarchy for the piece types in the setup, and/or indicate mating potential of these), possibly to be safe, or to avoid disputes. Still, omiting such is naturally less interesting/useful than otherwise. At least you're regarded as an established authority on piece values, so people would most likely take yours (if you've given such) over mine when in doubt about piece types' values for on a given board. My main problem is I don't completely trust computer studies, by anyone, as I've written in the past. At the moment my primative methods at least do seem somewhat applicable to a range of board sizes, shapes and piece types, including those where computer studies have yet to go.

You overestimate my reputation; I am pretty sure most people playing chess variants haven't even heard of me.

Computer studies are not beyond doubt, like nothing in science ever is. But that does't mean that any voodoo method should be considered equally valid and reliable as serious scientific research. I don't know what your reservations are with respect to results from computer self-play. (After all, the level of play these can reach is often superior to that of human GMs.) But it seems to me that most other methods have very serious and obvious defects, and often are not better than educated guessing without any attempt at a ' reality check'.

Correct me if I am wrong, but it seemed to me that you concluded from the fact that the Alfil and the Dababba are practically worthless pieces that the A an D moves also hardly contribute to the value of a Champion. And that is totally flawed reasoning. Alfil and Dababba have low value because their moves fail to cooperate in a useful way, so that they can access only a small fraction of the board. Not because the individual moves were intrinsically less useful. But this can hold for any type of move, on a piece that only has a few of them. I already pointed out that a (say right-handed) Half-Knight is worse than a dababba in this respect. And replacing the sideway jumps of a Dababba by the vertical moves of a Wazir makes the piece even weaker (as it is then confined to a single file).

You can always think of a piece as a compound of a number of pieces with fewer moves, and the latter can be given so few moves that they are practically useless. Trying to correct that by arbitrarily adding a Pawn's value each time you combine two of them, because that seemed to work for a Queen, is no good and leads to totally wrong results: it would estimate a piece that has two diametrically opposite Knight moves as at least 1P,  the combiation of two of those moving along perpendicular hippogonals (the already mentioned Half-Knight) as >3P and a full Knight at >7P.

To have a method of piece-value prediction (as opposed to measurement) that makes any sense, one should take account of the fact that moves must cooperate, and that some specific combinations of moves fail to do so in a useful way, leading to some severe ineptness of the resulting piece (like color binding or lack of speed), which cannot be blamed on any of its moves in particular. E.g. a Camel is a very weak piece on 8x8 because it only has moves with long stride, forcing it to a useless and vulnerable location when it gets chased away from a good one. (And in addition it doesn't cooperate well with other orthodox pieces, which all have short-range moves, so that there is no possibility of mutual protection.) I am not sure at all that the Wizard would suffer from the same problems even on 8x8, as the Ferz moves it also has see to fully solve that problem. The Camel moves might be just as good as any short-range move. It might fall off board more often, but it can also attack deep into the opponent's camp. Even more so on 10x10.

I don't need to correct you regarding my calculation of e.g. a WAD. At least I added in 2Ps worth of cooperativity, as you put it in different words. The explanation you gave that included reference to N moves/parts is a bit over my head, as I'm still not that familiar with a lot of CV terminology (and being a bit lazy, I've tried to keep it that way for some time now). I would note that I estimate half a knight's value as (N-P)/2 since (as I would much prefer to end things with) as I see it a halfN plus a halfN plus pawn = full N, but only to appearances this is in line with my Q=R+B+P analogy. My estimate of the value a quarter of a N is, perhaps at first glance, based on a dissimilar calculation, however, and it's an illustration of a quandry of mine which I had much preferred not to write about, until your pointed query. I value quarterN as (N-P)/4, i.e. on 8x8 it is (3.5-1)/4=0.625 (i.e. also my value for an A there), so it's significantly less than 1 for that board size (which seems contrary to what your reasoning expected from the beginning, if I get the gist of it all the same).

I'd creatively noted that a full N equals N-P+P=((N-P)/4)x4)+(P/4)x4=3.5, i.e. I'm clearly not using the Q=R+B+P analogy in this case, since otherwise the value I'd get for a quarterN (or its equivalent, an A) would clearly (even to me) be way too small (i.e it would be (N-3P)/4, or 0.125 - note if we used eighths of a N, it would then be (N-7P)/8, which would produce a negative value!). Instead, by using a different way of thinking when estimating fractions of a piece type's total movements (including for that of a halfN), I get a value for a quarterN (or A) I can live with for now (i.e. 0.625), and an A is a piece of relatively low value regardless, as you observed. So, now having yet another tentative estimate method, perhaps an even more imperfect one (which I figure is better than not having it, unless I go questing for different way(s) to get values of CV piece type fractions, and with such apparently giving more accuracy, which may take ages hunting down or concocting a satisfactory number of them), for such a thing as a 'compound' of 4 quarterNs, which in effect is the same as having a N, I'd make the 'compound', of 4 quarterNs (no matter if some move differently), work out to equal a Ns value, by using 'compound' (or N)=4quarterNs+(P/4)x4=3.5.

One fly in the ointment seems to be that I value AD=A+D+P, where I've already put the value of D=A=quarterN. Fortunately for me, so far I can console myself with the thought that none of these piece types move (even almost) the same way as each other. Unfortunately, I discovered that a quarterN + quarterN compound's value would still seem rightly to work out greater than a halfN's value, unless I immediately treat that whole compound as simply known to be identical to a halfN (that having a value I'd work out instead, as I happen to have done already for 8x8); I am to say the least a little unhappy about using such an apparently fishy way to try to patch things up, unless in my toolbox there's another trick, or alternative method to choose from, in such a case, that I have forgotten - I don't always have such written down somewhere. Life is far from perfect for me, and all this looks horribly suspect (i.e. inconsistent, unsound), but so far the CV piece values I've come up with haven't looked too far off in my own eyes, and at least no one has put their finger on my quandry that I mention here, until now.

On the other hand, if a N (or a piece of about the same range and number of targets) were somehow (fully once) colourbound then such would be worth N/2 in my eyes (it's also less than what I got for a halfN above, i.e. (N-P)/2, which makes sense to me). In the case of a AD (which does have 8 targets, and is colourbound), it's also a compound of (known) pieces, so for that I choose to simplify my life and just use A+D+P to get it's value. Nevertheless, I'm very much still feeling my way, regarding such formulae, and guidelines on what to use them for (e.g. in the case of B vs. N I've tried, not quite conclusively, to finely weigh them against each other on my own, in terms of their having an equal number of roughly equally important assets and liabilities, as I see them, and my somehow applying a x0.5 numerical colourbound penalty against a B could complicate things, though I could try to do so one day if I could quantify all the other vital factors to my satisfaction, too). At times I look at piece types that seem similar, when estimating the value of a type that is new to me, to check if they have known values that are relatively close to what I get for the new piece type.

In the case of a WAD (or Champion), I still give some consideration to the page of Omega Chess Strategy's piece value of 4 (on 10x10+4 cells), as that game has been played for some time now, and I'm guessing people have established that estimate through many played games - though possibly largely between relatively low-rated people (if they were chess players). It's also convenient for me to accept (at least for a while) that that value is about what I get for a value, too. The alternative is to start re-working or trashing some or all of my primative methods, leading to a lot more work and perhaps the result that I have to refrain from giving even tentative estimates for things, at least for a long time, which is no fun (otherwise, I often can satisfy anyone who wants just any quick and dirty estimate, before a 'more scientific' valuation might become available). As an aside, I once saw someone opine somewhere on the internet that an A was likely worth less than a pawn, unlike the wiki for an A, so when in doubt I again chose to believe something that agreed more with my own methods' results to date, although that time I was even less sure of assuming that that person had done any sort of significant research than in the case of a Champion in Omega Chess.

I recall you once indirectly indicated to me you wrote at least one CV-related wiki entry. I wouldn't be surprised if you've written all or nearly all of them. One thing I don't understand about wikipedia entries in general is why the names of writer(s) of them are not attached (or at least not prominently displayed, if they are there to be found). I think that would be right regardless of the topic, and in your case it would certainly bring you at least some recognition, for any who noticed that you've written many CV wiki entries, especially if you were ever allowed to note in places results of your own research, and that it was yours, in regard to computer studies/formulae (though I seem to recall that extensively quoting independent research goes against established wiki regulations in itself).

Long ago I posted (perhaps via an edit) a summary of my some of biggest doubts re: computer studies, of which I think there were 3. One that was major was the strength of the engines involved (or of the humans, in the case of Kaufman) possibly being too low to establish the probable truth - best play with e.g. B vs. N only comes from consistently high level test games, otherwise it seemed to me it's a bit like having kids play kids and then taking down the results of very imperfect play with imperfect plans used. A second doubt (relatively minor, perhaps) was about the calculation of the margin of error used for such studies, which I thought might be bigger and bigger the larger the value of the piece type being tested (i.e. a B's margin might actually be considerably less than the margin for an amazon, and, e.g., a Ps margin [if such is/can be attempted to be measured] might be smaller than a Bs margin, in each case assuming the same number of test games were used when testing the value of each piece type). I think the third doubt (also relatively minor) was the idea that maybe the margin of error in general should be as much as double than assumed, since it's theoretically possible the materially inferior side might win more games in a random set of them (unlikely, but possible, especially if the engines are relatively weak). However, these last two doubts may just show that I don't get the statistical methods or math in general - but I still feel the whole field of computer studies (or even of concocting piece evaluation formulae based on its results) is immensely complex and relatively unproven, and somehow subtle wrong assumptions or reasoning might creep in, even for experienced mathemeticians. Aside from all that, the composition of the armies for a variant's setup (or possibly even the exact positioning of the pieces in that) can throw something of a wrench into the value estimations, and then there's opening, middlegame and endgame phase piece valuations, which aren't necessarily in agreement, though I know you're aware of all that - the possible issue is that there may be a theoretical need to examine all these cases with computer studies, too.

In my last post I more or less neglected to mention that a Man's (or K's fighting) value is another that is problematical for me, on 8x8 at the least. One way for people to test piece values for themselves is simply to play a variant involving a piece type with an assumed value that the player bases his calculations on. Doing so gives an insecure feeling though, and I'm often reluctant to have piece imbalances in regard to the position on the board or in my calculations, when possible. Moreso when the piece types are powerful, like in the case of swapping an amazon for a collection of pieces of clearly smaller value. In the Modern Shatranj variant, in my games I've been able to somewhat better appreciate the value of a Man on 8x8, relative to 'other' minor pieces, and it's not been often that it clearly seems to be by far the best piece type to own, compared to a N or Modern Elephant. Often in the middlegame or early endgame stages, the types all seem to 'feel' as if about of equal value, in my numerous games (on Game Courier) so far. However, in one endgame that I won, a Man I had proved superior when pitted against a N, as I was able to launch a mating attack at the end for one thing. The final position may give an impression, even if one does not wish to play over the entire game:

https://www.chessvariants.com/play/pbm/play.php?game=Modern+Shatranj&log=panther-cvgameroom-2018-112-744

Another thing I've gotten a bit of a feel for through actual play is the comparative value of a N to a Modern Elephant (e.g. in my 10x8 Hannibal Chess variant), and in the opening phase, at least, a ferfil (aka Modern Elephant) seems at least as influential as a N - contrary to my valuing a ferfil somewhat less than a N on 10x8 (or on several other board sizes).

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

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