Comments/Ratings for a Single Item

Sorry, I haven't played much around with modern engines. I'm not tech-savvy at all. I joined The Variants Page in order to give my ideas for chess variants at the least some more exposure on the net, though I haven't lots of free time these days to try to promote any of them vigourously, if I wished to.

I can try out The Variant Pages diagram generator at some point to update my submissions as requested, e.g. for Sac Chess, though I would note I'm virtually new to the chess variant world as far as conventions or terminology goes. Today I'm still recovering from giving blood for the first time last evening (partly for my own health benefit), so I'm going to try to take it easy for a while.

Take care, Kevin

Thanks for your kind words. I'll add just a bit more to what I wrote:

I forgot to note that I've read somewhere on the net that it is the Chancellor that is approximately worth a Queen, while the Archbishop is worth about what I gave its value for. Perhaps someone else may chime in if you still do not concur.

The Amazon, I also read, has been estimated as low as 11 or 12 points, but perhaps this does not take into account it being on a larger board. In any case, I found it hard to believe that a queen and knight acting seperately are as effective as an Amazon in general. Compare that by analogy to asserting that a queen would be worth only a rook plus bishop on a standard chess board. Still, as I wrote I am new to fairy chess.

I don't think one can be too dogmatic about the relative value of pieces across all contexts.

It is very useful in Chess to throw lesser pieces onto the front lines for profitable exchanges, while using the Big Guns for backup.  Whether a Big Gun is worth more to you than 2 lesser value pieces will probably depend on the balance between Big Guns and lesser pieces that you already have.

But I think that generally, a singe piece that combines the powers of two lesser pieces would be worth less than those two pieces.  As evidence of this, I offer the fact that a Queen controls twice as many squares as a Rook (and as many squares as 2 Rooks), but is worth less than two Rooks.

A Bishop has the same firepower as a Rook, in terms of the number of squares it controls.  The reason a Bishop is worth less than a Rook is that a Bishop is confined to half the board, and the Rook is not.  A Queen gains the Bishop's extra firepower without suffering from its limitations.  This is the reason why a Queen is generally worth more than a Rook and Bishop combined.

This logic does not apply to the Amazon.  Neither the Knight nor the Queen is limited in the way the Bishop is; so a piece that combines their powers should be worth less than the value of Knight+Queen, not more.

I agree that 10x10 board size will tend to increase the value of unlimited range pieces.  On the other hand, the SAC chess board is rather crowded, and this may tend to decrease the value of ranged pieces (while increasing the relative value of the knight's unblockable movement). SAC chess starts with 60% of the board occupied, as opposed to 50% for FIDE chess.

Test comment. Intended for deletion.

I can't delete this unless I do so as an editor.

> But I think that generally, a singe piece that combines the powers of two lesser pieces would be worth less than those two pieces. As evidence of this, I offer the fact that a Queen controls twice as many squares as a Rook (and as many squares as 2 Rooks), but is worth less than two Rooks.

You're doing your math wrong. A Queen at 9 points is worth more than a Rook (5 points) + a Bishop (3.25 points).

>  You're doing your math wrong. A Queen at 9 points is worth more than 
>  a Rook (5 points) + a Bishop (3.25 points).

There's nothing wrong with my math.  If firepower were the only issue, Rooks and Bishops would be worth the same (5).  A Bishop is worth less because of its limitations.  A Queen is worth more than Rook and Bishop because it gains the Bishop's firepower without suffering from its limitations.

Imagine a piece that had the power to move like a Rook and, in addition, the power to move diagonally but ONLY along black diagonals.  This piece would come closer to combining the powers of a Rook and (black-square) Bishop.  And its value would be less than the combined value of those two pieces. 

A Queen does not really combine the power of one Rook and one Bishop.  It might be closer to the truth to say it combines the powers of a Rook and both Bishops.

Let me examine what you said in more detail. You said,

> a single piece that combines the powers of two lesser pieces would be worth less than those two pieces.

Then as evidence for this you gave an example of something else,

> the fact that a Queen controls twice as many squares as a Rook (and as many squares as 2 Rooks), but is worth less than two Rooks.

If the Queen is your example, you must compare the Queen to the value of a Rook plus a Bishop, not to the value of two Rooks. Because you did not, there is no logical connection between your evidence and your conclusion.

>  If the Queen is your example, you must compare the Queen to the value of a Rook plus a Bishop, not to the value of two Rooks.

Okay.  But haven't I already done that?

A Queen does not really have just the power of a Rook and Bishop.  It has the power of a Rook and both Bishops.  It can move like a Rook, and in addition, can move diagonally on black diagonals, and diagonally on white diagonals.  

As to "firepower" (the number of squares it controls at one time), the firepower of the Queen is nearly equal to that of two Rooks.  Or to put it another way, it has the firepower of a Rook and a Bishop (both of which can control an almost equal number of squares), but does not suffer those limitations that restrict the Bishop to one half of the board (which limitation reduces the Bishop's value compared to the Rook, despite it's almost-equal "firepower").

It's a fair point to say that the Queen combines the powers of a Rook and both Bishops. But this might not make your case, because the lesser value of a Queen to a Rook + two Bishops might be due to the effect of diminishing returns when you start to combine more than two pieces or to the similarity of one Bishop's move to the other. To make your case, the Marshall is the most unambiguous example. It combines the powers of a Rook and Knight, neither of which were originally colorbound, and neither of which have moves similar to the other. In some tests I just ran on Zillions-of-Games, I had one side use a Marshall in place of its Rook and Bishop. I ran two tests so that each side could use the Marshall in place of its Rook and Bishop, and in each test, the side with the Marshall won.

Although you might say a Queen combines the powers of a Rook and two Bishops, this is not precisely true. At most, from any given position, a Queen may move as a Rook or as the Bishop on that color. If it's on a Black square, it cannot move diagonally along the White squares, or vice versa. So, for any given move, a Queen's possible moves are limited to those that can be made by a Rook or by one Bishop. This puts a Queen in a grey area between having the power of two pieces and the power of three pieces.

On consideration, it seems I must revise my position that the one-color only limitations of the Bishop seriously affect its value.

The Rook controls 14 squares on an empty 8x8 board.  The Bishop controls 13 at most (when in the center of the board) but more often only 7 (when at the sides).  The average for the whole board seems to be control of 8.75 squares, though in practice that number will be higher as players will tend to position their bishops to best advantage (i.e., they will tend to avoid sides and corners).

Divide both by a similar factor (2.8) and you get 5 for a Rook, and 3.125 for a Bishop (or perhaps higher in practice for the reasons stated), which is not too different from their relative values as found in practice.

To H.G.:

I'm not sure how you measured an Archbishop's relative point value for every case that you mentioned. Did you always base your measurements on, say, the outcomes of a large number of games that were play-tested which involved a variant using that piece? If so, the results might at least somewhat depend on the average strength of the players involved, even if all/many were computer playing engines, though I would assume you took that into account if that was always your method. Hopefully the method can be described briefly, if you are happy to do that.

To John:

It looks like you have a valid point about saying an Amazon = Q + N is fundamentally different than objecting that a Q is greater than a R + B, in comparing by analogy. I hadn't thought about a B's limitations when a piece by itself.

[edit: though a vital follow up question could be: are 2 Queens only = 2 Rooks + 2 (different coloured) B's, or are 2 Q's in fact = 2 Rs + 2 (different coloured) Bs + 2 Pawns? Looking at your last post before this one of mine, I assume you now would put the 2Qs value as closer to the latter.]

[edit: Another follow up question could be: are 2 Amazons = 2 Qs + 2 Knights? The reason I thought of asking is that either Amazon might (using just its bishop and/or rook type powers) be able to double attack the 2 enemy knights, or else attack one enemy knight and another enemy piece. In fact, a single Amazon could do the latter in case of having a Queen and Knight for it, so again the real question becomes: is an Amazon really just = Q + N (or even less)? To strengthen my doubts a little more, I would note a single Amazon can also use its knight type power to double attack an enemy queen and another enemy piece, say even another queen, though such a queen might often have a good chance to move and guard the other piece that is being double attacked by the Amazon. As an aside, fwiw long ago I read in some chess book that it is the great mobility and double attacking capability of a queen that makes it such a powerful chess piece, and a knight of course is awesome at forking.
]


In any case, I hope it's not too objectionable to anyone if I leave my own estimates for Sac Chess pieces (which I admitted were tentative) the way they are for at least a little while longer. I can edit my submission to change my estimates after I think about it more if necessary, and perhaps have even play-tested variants using Amazons or Archbishops myself at some point. 

[edit: I had forgotten that I've already played a quite small number of games of Seirawan Chess (8x8 chessboard variant, with first rank drops of Hawks [aka Archbishops] and Elephants [aka Chancellors] in the opening phase) and it seemed to me based just on these games (two of them with a fellow chess master) that the Chancellor seemed at least as dangerous a piece as the Archbishop, especially after the opening phase, if nothing else. Also, I wonder if there really has been enough human experience playing with such fairy chess pieces yet for even a very strong human chess player to really know how to defend (when necessary) against their unusual movement capabilities.]

Muller, thank you for your analysis.  You gave me some things to think about.  However, I'm still not sure we're entirely on the same page.

You mention the Wazir, and how it's barely worth more than a pawn.  So I asked myself, why would that be?  I now hear that the analogous-but-diagonal Ferz is considered more valuable, despite being colorbound.  Why would that be?  They have equal "firepower".  A little thought produces the answer.  The Wazir is devalued by its lack of mobility, especially on a board crowded with pawns (and others).   A Ferz can easily slip through pawn formations (which depend on diagonals), but a Wazir cannot.

These same considerations will impact a Rook when compared to a Bishop.  With that in mind, it seems likely that the colorbound nature of the Bishop does affect its value, but this probably does little more than balance the pawn-bound limitations of the Rook.

This also explains the phenomena you discuss.  A rook/knight combo breaks the rook's pawn-bound status, and is probably more valuable than a knight+rook as separate pieces.  A bishop/knight combo breaks the bishops color-bound status, and is probably more valuable than a knight+bishop as separate pieces.

But a queen already breaks both these limitations, and a Knight's powers probably don't break them much further than they are already broken.  I therefore still doubt very much that an Amazon is worth more than the combined value of Queen + Knight.  

The pawn-synergy factor brings me back to my original point.  The value of a piece will depend on what else is on the board and the synergy or lack thereof between them.  "Sac Chess" throws us all for a loop by radically altering the other pieces on the board.  It's guesswork, and the value of a piece as measured in one context (such as a close approximation of FIDE Chess), will not necessarily apply here.

H.G. posted some time ago:

"Some of your piece values are off, especially Archbishop, which is about C - 0.25 P = Q - 0.75 P (so 9.25 on your scale). The Amazon seems to be worth only Q+N, so 13 on your scale.
..."


I'm still wondering about the value of an Amazon asserted to be only Q + N, in your opinion (besides that of other people). Maybe it is since I am a fairy chess newbie, but I'm not clear on subsequent remarks you have made regarding synergy, in regard to them being fully in line with saying that an Amazon just = Q + N. Furthermore, assuming that value is the measurement your method yielded, that result is still a red flag for me presently, as far how infallible the method or its playtesting conditions might be. 

As I alluded to earlier, in particular the supreme double attacking powers of an Amazon make me value it more than just a Q + N. Currently I am more willing to accept that I gave a tentative value for an Archbishop that was too low, by contrast (though it may be that the value of an Archbishop in the context of Sac Chess, rather than when it is used just in an 8x8 variant approximating standard chess, still ought to be measured from scratch by testplaying using Sac Chess games, if anyone is willing). There is another red flag for me concerning the method, regarding comparing a bishop to a knight by such measurement, resulting in asserting they are of equal value. All about that further below.


H.G. posted more recently:

"The values were indeed measured by play-testing through self-play of computer programs. To measure the value of, say, an Archbishop, I set it up opening positions where one side has the Archbishop instead of a combination of other material expected to be similar in value (like Q, R+B, R+N+P, 2B+N, R+R). For any particular material imbalance the back-rank pieces are shuffled to promote game diversity. I then play several hundred games for each imbalance, to record the score. This is rarely exactly 50%, and then I handicap the winning side by deleting one of its Pawns, and run the test again. This calibrated which fraction of a Pawn the excess score corresponds to. E.g. Q vs A might end in a 62% victory for the Q, and if Q vs A+P then ends in a 54% victory for the A+P, I know the P apparently was worth 16%, so that the 62% Q vs A advantage corresponds to 0.75 Pawn. 

I tried this with two different computer programs, the virtually knowledgeless Fairy-Max, and the 400 Elo stronger Joker80. The results are in general the same (after conversion to Pawn units), and also independent of the time control. (I tried from 40moves/min to 40 moves/10min.) Typically they also are quite consistent: if two material combinations X and Y exactly balance each other (i.e. score 50%), then a combination Z usually scores the same against X and Y. 

The results furthermore reproduce the common lore about the value of orthodox Chess pieces. E.g. if I delete one side's Knights, and the other side's Bishops, the side that still has the Bishop pair wins (say) by 68%, and after receiving additional Pawn odds, loses by 68%. Showing that the B-pair is worth half a Pawn. Deleting only one N and one B gives a balanced 50% score, showing that lone Bishop and Knight are on the average equivalent. This is exactly what Larry Kaufman has found by statistical analysis of millions of GM games.
..."


First off, I think I see how the Archbishop's value was measured, in terms of being 0.75 Pawns less than a Queen, based on the percentages given. Whether just hundreds of games is a statistically satisfactory playtest sample size, I am not sure (note that in chess White is thought to have a standard statistical edge, by about 54% or 55% over Black, so I assume half the time the Archbishop was with White thoughout the playtests). It also could be important how highly rated the computer programs were. By way of illustration, in chess it takes a good degree of skill for human players to know how to defend against a queen using, say, R + B + P, in situations where they are worth at least the queen objectively, based on the current position on the board.

Larry Kaufmam is an International Master as far as chess goes, which puts him below Grandmaster or certainly world champion level, and such players have in the past and present certainly believed that though a bishop is close in value to a knight in terms of relative value, in general unless there is a special reason to prefer having a knight, situations favouring a bishop tend to happen more often - whether in actual game play, or in the many calculated variations that could have arisen from them (these alas do not appear in the playtesting process). So, if a grandmaster willingly gives up a bishop for a knight, he has reasons to do so based on other factors in the position. 

In any event, I have not heard of any reasonably strong human chess players changing their strategies in regard to trading bishops for knights in over the board play, based on Kaufman's result from his method. In regard to the millions of games Kaufman looked at, I am not sure all the chess Grandmasters in history have played close to a million games yet, especially against just each other. I have chess game databases with over a million games in them, but they include vast numbers of games played by players who were below Grandmaster level at the time.