Ratings & Comments

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).]

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.]

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?

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.)

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.

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.

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.

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...

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.

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.

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).

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.